\ Hi àç ) ^ d^\ H^ dw) " '''
(') Lamé, Sur l'équilibre des températures dans un ellipsoïde à trois axes inégaux {Journal de Liouville, t. TV, p. 126; 1889 ).
CIIAP. VII. - TRANSFORMATION OF THE EQUATION AV = G. 'iOI
Let's put
The equalities (i5) will then give us
H, H, I s/M7i)
- {v-w),
{w - m),
and the equality (lo), transformed from AV = o into elliptical coordinates, becomes
-(..-.)v/=:iH:(^^[v/=Â(7)f]
H ~
^- v/- .ïi.t.v^'^^l.v'*'/'
Ha H
I '/- ^(p)
Hi
* V^ù>i.(tvyè)l(^a)
HHi H,
2 ./ - 'R ( I, K'rt / ,. \
(3o)
Let
V a function of the sevde variable u : t? a function of the sevde variable ç; ^ a function of the single variable iv,
these three functions being determined respectively by the second order differential equations
(3i)
W
dw
where A and B are two arbitrary constants. Let us then pose
(32) Y = V)'<>^
and substitute this expression in equality (3o); this one of
a02 MVRK II - DISTRIDUTION AND PROBMiMK OF DIRICIILET.
will come
t)\!P'C^[(A + BiA)(p-"0-4-CA + Bp)((P - f/)-f-(.V + Bw)rfi - t')] = o.
She will be equally satisfied.
Now, Lamé has shown that the equations (3i) can be integrated by certain series developments whose coefficients can also be determined in such a way that the function V, obtained by the combination (32), reduces to Q for u =^ c^. The Dirichlet problem is thus solved for the space outside the ellipsoid.
The fruitful path that had led Lamé to this beautiful result, namely the transformation of the equation AY = o into orthogonal coordinates, allowed Lamé and other analysts to solve Dirichlet's problem for a large number of cases, and in particular for any space, bounded or unbounded, that confines only to a surface of the second order (' ).
(' ) See Heine, Handbuch der Kugelfunktionen, -i" edition; Berlin, 1878
niAP. vni.
rA METHODE DK L IXVEKSION.
-20'i
CHAPTER VIII.
THE METHOD OF INVERSION.
v5 1 - The method of inversion or electrical images.
We know the immense role played, in modern geometry, by 1(!S methods of transformation which make it possible, when a problem is solved for a certain figure, to deduce other figures for which one can immediately solve the same or similar problems.
Sir \V . Thomson (' ) has shown, and this is certainly one of the beautiful discoveries of this great geometer, that when one knows how to solve the Dirichlet problem for a certain space, one can form an infinite number of other spaces for which one can also immediately solve the Dirichlet problem. The method of transformation which he used under the name of the method of electric images is also known as the transformation by inversion or the transformation by reciprocal vector rays.
Let .r, j', z be the coordinates of a point M in a certain space
Fis. 37.
and p its distance to Torigine O ifig' 37). Let's match it
(') Extract from a letter from Mr. William Thomson to Mr. Liouville {^Journal de Liouville, t. X, p. 364; i9,!\b) - Extract from two letters addressed to Mr. Liou\?ille by Mr. W. Thomson (Journal de Liouville, t. XII, p. 256; 1847) - Liou
2o4 LIVIŒ II. - DISTRIBUTION AND DIRICHLET PROBLEM.
a point M'{x',y\z) of a second space by the following formulas (,) a^^psf:., y^p^r^, --p-y^'
A" being a positive or negative coefficient.
Let p' be the distance OM'. It is easy to deduce from the formulas (i) that the product pp' is equal in absolute value to k, and that the point M' lies on the line OM, on the same side of the point O as the point M if A" is positive, and on the opposite side of the point O if k is negative. The transformation in question is reciprocal, in that we can write, instead of the formulas (i), the formulas
(i bis) a7'=p'2|, y=p'2^;, ^'=P''|'
Some of the simplest properties of this transformation are studied in the elementary courses.
It is known that a plane transforms into a sphere in general, and into a plane in the particular case where the pole of transformation is on the plane; a sphere transforms in general into another sphere, which is concentric to the given sphere when the pole of transformation is the center of the given sphere, and which, when the pole is taken on the given sphere, reduces to the plane tangent to the given sphere at that point.
This transformation has, from a geometrical point of view, a fundamental property that we will first highlight.
Let MMi be a linear element of the first space, joining the point M(a;,y, z) to the point Mi (x 4- dx, y -+- dy^ z -^ dz).
It will be transformed into a linear element M'M'j of the second space,
joining the point M' {x',y, z') to the point M', (x'-h dx' ,y' 4- dy' , z'-\- dz') .
According to the equalities (i), we have
-, , / dx 9. X due \ dx = A - ^ ,
Vp2 p3 ;
p2 p^
, , , l' dz "xz da
dz ^ k { - - - - T
p2 p3
VILLE, Note on the previous article {ibid., p. 265). - All these articles are reproduced in W. Thomson, Reprint of papers on Electrostatics and Magnetism, Art. XIV
cil VI". VIII. L.V METHOD OF iNVERSIOX. 2o5
which gives
A'2 r
dx"^-^dy'-^-\-dz"^= ~ dx'^-h dy'^-t- dz'^ P L
- ^{x dx -i-y dy -*■- z dz) -
52)(^p)2J
r
But we have
X dx -^- y dy -^ zdz - Ç)d^, so that the previous equality reduces to
^ ■!. ) dx">- -f- dy"^ -I- dz!^ - -- ( f/j^s _i_ dyi- -f- c?^^ )
P or
2 lï 2
M'M'i = _, MMi . OR
Let's see the consequences of this equality.
From point M, let us radiate an infinity of linear elements MMi, MMo, MM;),.... They will have as correspondents an infinity of linear elements M'M', , M'MJ,, M'M!, all coming from the
point M', and we have
M' m; _ M' M', _ M' m; _ _ \k\
"MM7 ~ "MmJ ~ "MM^ - - - - - =^2 -
The infinitely small figure formed by the points M,, M2, M3, . . . will transform into a similar infinitely small figure formed by the points M',, M.,, M'^, ....
We express this property by saying that the space (^', 7'', z') (which the inversion transformation makes correspond to the space [X, y, z) is a conformal representation of it.
11 It is easy to see that this result leads to the following ones:
Two curves or two surfaces that intersect at point M at an angle a are transformed by inversion into two curves or two surfaces that intersect at the same angle a at point M' transformed from point M.
In particular, three families of orthogonal surfaces transform into three other families of orthogonal surfaces.
Uo6 I.IVRK 1 . - DISTRIBUTION AND DIRICIILET PROBLEM.
Let be a point M' of the transformed space, and let u^ v, w be its cartesian coordinates. The position of the point M which corresponds to it in the first space is fixed by the knowledge of these three quantities u, v^ w. We can therefore consider these three quantities as three curvilinear coordinates fixing the position of a point in the first space.
The point M' is at the intersection of the three orthogonal planes
The point M must therefore be at the meeting of three orthogonal spheres. This is easily verified, because the three previous planes give, by transforming themselves, according to the equalities (i his)^ the three surfaces
'^ V , , ^^
x^- -+- Y- -^ [ z^ - = O,
\ 2(1'/ 4 H' 2
which are three spheres intersecting at point O and tangent respectively at this point to the planes ZOY, XOZ, YOX.
The coordinates m, c, w constitute, for the point M, a system of orthogonal curvilinear coordinates. The preceding equations provide the transformation formulas that allow us to go from Cartesian coordinates {x^y^ z) to curvilinear coordinates (m, p, w) or vice versa.
These formulas can be written again
(3)
i X'- ^ y^- -\- z'- -
k
- a: = 0,
II
' a:---HjK-+2' -
x-^y'^+ z"' - \ ^
k
- z -0,
w
Equality (2) gives us But, if we put
du^ -+- di>- -\- dw^ = j- ( dx^ -h dy''- -\- dz-).
p'2="2-1- {>^--\- w'i
CHAP. MU. - I.A .MKTHODE DK L INVERSION. 'loy
we know that
p2p'2= A:2,
nd the previous equalizer should be
(2a) dx'^ H- dy'>- -\- dz^= -^ ( du^ -+- dv^ -\- dw^- ).
If we refer to equality (8) of the previous chapter, we see (|that we have, for the change of coordinates that we consider,
(4) H2= H?-=r H|= 4J Now, let us consider a function \ (.r,jKj ^) which is harmonic inside a certain surface S belonging to the first space; for all values of x^y^ z, which correspond to the points inside the surface S, we have
W(x, r, z ) - o.
^the points {x,y, z-) inside the surface S correspond to a set of values of the quantities u, v, w. Let ^{ii, p, a) be what the function V becomes when we replace a:, y, z by their expressions in terms of u, v, w. We will have, according to equations (i bis),
For the set of values of u, v, (v, that we have defined, this function will verify the transformed equation of AV = o, that is, according to equations (4),
) / I of I> >
du \p'2 du /
to / I d<\>
l ^ '^ ,
/ i ')1>
1 -t- y- 1
\p'- div
This can still be written
p' Idu^ Vp' / "^ dv^ \'f') '"' à^^\ô'
2Ï El 1 ^ ^ 1 - ^^ 1
p' [^àu^ p' ' dv^ p' ~^ àiv^ p'
If we put
ai d' d^
208 BOOK II. - DISTHIBUTION AND PROBLEM OF DIRICIILET.
and if we notice that
A' "
A - = o,
P
this equation will simply become
(6) ^'(?)="
The result we have just obtained is susceptible to an interpretation that makes it important.
The coordinates u^ v^ w can be considered as the Cartesian coordinates of a point M' in a space transformed by reciprocal vector rays of the space to which the point M(^,j^, ^) belongs. This transformation makes the surface S correspond to a surface S'; any point inside the surface S to a point outside the surface S'^ to the points infinitely close to the origin in the first space the points infinitely far from the origin in the second space.
If the function Y(.r,jK> ^) is harmonic inside the surface S, the function
iy) Wiu, V, w) = -, *(", P, w) = -, V i-^^, ^, -,-^j
will be, according to equalities (6) and (5), harmonic outside the surface S'.
If the function \(x,y,z) takes, at point O, the value Vo, the
function W(", ç, w) will tend to zero as ~ when p' grows beyond any limit.
If the function Y[x,y,z) takes the value A at the point P of the surface S, the function W (m, t^, w) will take at the point P', transformed from P, on the surface S', the value
OP' OP,
These results easily lead to the following proposition:
If we know how to solve the Dirichlet problem for V space inside a closed surface S, we know how to solve it for the space outside the surf ace S' transformed from S by reciprocal vector rays.
C.llVl'. Mil. - l,.V METHOD OF INVERSION. 209
Let Q be the function, defined at each point of S', to which the sought harmonic function must become equal at any point of S'.
Let us form, which we know how to do by hypothesis, a function V(x,y,z), harmonic inside the surface S, and taking at any point P of the surface S the value
OP
Then let's form the function
Wiu,i^,w)=^,\('^^, ^.^
I -. /Ali kv kw
P Vp'' p
The function W(", v^ w) will be harmonic at any point outside the surface S'; it will become equal to O(P') at any point P' of the surface S'; it will behave at infinity as a potential function; it will thus solve the Dirichlet problem for the space outside the surface S'.
It would be shown in the same way that, if Von knows how to solve the Dlrichlet problem for V space outside a surface S, we know how to solve it for the space inside the surface S', transformed from the surface S by reciprocal vector rays.
Some consequences of this important proposal:
We have seen that the Dirichlet problem can be solved for the space inside a spherical surface and also for the space outside this same surface; by what we have just demonstrated, each of these two solutions can be deduced from the other.
We know how to find the Green's function for the inner space of a dihedron commensurable with u; we know how to solve the Dirichlet problem for the outer space of two spheres which intersect at an angle commensurable with tt.
We know how to find Green's function for the space between two parallel planes; we can therefore determine the electrical distribution on two equal or unequal spheres that touch each other.
The determination of the electrical distribution on two touching spheres is indeed one of the beautiful results to which is D. - I. i4
DISTIUBUTION AND PROBLEM OF DiniCllLET
reached Poisson (' ). A great number of geometers, among whom one can quote especially Plana, Sir W. Thomson, G. Kirchlioff, have subsequently dealt with this problem. A remarkable exposition of the solution can be found in the work devoted by É. Mathieu to the exposition of the Theory of Potential (-).
§ 2 - Application. - Electrical distribution on a spherical cap.
One of the beautiful applications that Sir W. Thomson (^) has made of the method of inversion is the determination of the electrical distribution on an infinitely thin conductor having the shape of a spherical cap. We shall state the solution of this problem in the form given to it by Lipschitz ('■), confining ourselves to the case where the spherical cap is removed from all influence.
The solution we are going to develop is based on a few remarks that we will indicate first.
Let S be a spherical surface having its center at the origin of the coordinates and a radius equal to R. Let W be a function which behaves at infinity as a potential function, which is harmonic inside the sphere, harmonic outside the sphere, but discontinuous on the sphere, so that in the vicinity of a point M on the surface of the sphere it takes a value close to W/(M) inside the sphere and a value close to We(M) outside the sphere.
Such a function is not the potential function of a certain
(' ) Poisson, Mémoire sur la distribution de l'électricité à la sur/ace des corps conducteurs {Mémoires des Savants étrangers, p. 1811, p. i).
(^) E. Mathieu, Théorie du potentiel et ses applications à l'Électrostatique et au Magnétisme, t. II, p. 65; Paris, 188G.
(') W. Thomson, Extracts from two letters to M. Liouville {Liouville's Diary, vol. XII, p. 243; 18^7. - W. Thomson, Reprint of papers on Electrostatics and Magnetism, 1" edition, p. 162 ). W. Thomson, Determination of the distribution of Electricity on a circulçir segment of plane or spherical conducting surface, under any given influence ( W. Thomson, Reprint of papers on Electrostatics and Magnetism, 2" edition, p. 178).
(*) Lipschitz, Uebkr die Vertheilung der stâtischen Electricitàt in einem kreisformig begrenzten Segment einer Kugelflàche {Rorchardt's Journal, JBd. LVIIl, p. i52; 1861).
CII.VP. VIII. - L.V METIIODK OF INVERSION. 211
tle of electricity distributed on the spherical cap. Let's transform the space by inversion by taking the pole of inversion at the center of the sphere and the square R^ of the radius for the coefficient of inversion; any point M of the sphere will be transformed into itself; any point inside the sphere will be transformed into an external point and vice versa.
Consider the function
w'(^',y,.') = ^w(î^
'7T
R2
It will be harmonic inside the S sphere, harmonic outside the S sphere, but discontinuous on the S sphere. In the vicinity of the point M of the sphere S, it will take, inside
of the sphere a value close to ^ We(M), and outside the
sphere a value close to ^ W,(M). Consider then the function
= W{x,y,z)
It will be harmonic inside the S-sphere, harmonic outside the S-sphere, and continuous on the S-sphere, so that this function will be the potential function of a certain amount of electricity distributed on the S-sphere.
An analogous method would apply to the more general case below:
A closed surface is composed in part by a portion S
Fig. 38.
RW'ix,y,z)
R",/R2- R2.r
R2g
P'
(the spherical surface, partly by another surface S of shape* ([uelconcic (/?^. 38). The function ^/(x, y, z) is harmonic both inside and outside this closed surface; it is
ai-i
DISTRIBUTION AND PROBLEM OF UIRIOIILET.
behaves at infinity as a potential function; it varies in a continuous way when passing the surface S and in a discontinuous way when passing the surface S. Let O be the center of the sphere S taken as the origin of the coordinates; let R be its radius. The function
(8) W(^,7,.') + -W( - ,^,-
V(^, j, -s)
will be the potential function of a certain amount of electricity distributed on the closed surface composed by the surface S, the surface S and the surface S' transformed from the surface S by reciprocal vector rays.
It is a process of this kind that will allow us to determine the potential function of electricity in equilibrium on a spherical cap.
Let S be this spherical cap {^/îg- Sg); let R be its radius; soil
H is the distance of the base circle S from the center O of the sphere. The radius of the base circle is the quantity
(9)" = (R2-H2)i
Consider the quantity u defined by F equation
(lO)
(z-BY^
"2-1- u
and consider the function
(it) V(a:-,7, s) = K( ^ - ai-ctang^
where K is any constant
CHAP. VIII. - THE METHOD OF INVERSION. "ilS
According to what we have seen in Chapter VII, § 3, this function is harmonic in all space, except on the base circle (the cap; it behaves at infinity as a potential function; it varies in a continuous way when we cross this circle, at a finite distance from its edges 5 on this circle, it takes
the value -;- - When crossing the circle, the partial derivatives
of the first order of this function change sign without changing absolute value.
Let T be the closed surface formed by the cap S and the plane S. Let us take a function 'W[x,y,z) equal, inside the surface T, to
and, outside the surface T, at
bow tane
a /
K \^ ^ - arc tang ^
This function will be harmonic inside the surface ï ; it will be harmonic outside the surface T ; it will vary in a continuous way when crossing the surface S and in a discontinuous way when crossing the surface S. Its first order derivatives will not undergo any discontinuity when crossing the surface S.
Let's consider the function
If we define the quantity u' by the relation
(10 bis)
l.
u z
H =
P*("'^-f-i<') \ I p'^ / R It is easy to see that we will have, inside the surface T,
K
"2
KR 1
' Tt
u
arc tang - "a y
/H
-
- arc tang -
P
K-^
" a
(--2) \{x,yz)
inside the volume between the surface S and the cap 2'.
-il 4 BOOK II. - DISTRIBUTION AND DIRICHLET PROBLEM.
into which it is transformed by inversion,
/ - \ ' - \
, . ",, ,A T. u- \ KR / TT a'- 1
[vihis^ V(a:,jK,z) = K l - - arc tang - / -; - \- arc tang - ^
and, outside the surface T', formed by the caps S and S',
s Tr, ■ ^ri"^ u-\ KR Tz * u'"' ( 12 ter) y (x. )-, ^ ) = K \ arc tang - / -r- - \ - f- arc tang -
^ ■ \ 2 a; p \ 2 o,
This function will be, according to the above, the potential function of a certain amount of electricity distributed on the surface S, on the surface S and on the cap S', into which the plane S is transformed by inversion.
But the first-order partial derivatives of this function vary in a continuous way when one crosses either the surface S, or the surface S'; the electric density is thus null on each of the two surfaces S, S', and the preceding function is the potential function of a certain quantity of electricity distributed only on the cap S.
Moreover, on this cap, \ ix^y^ z) takes the constant value Ktt. The distribution in question is therefore an equilibrium distribution.
To complete the determination of the potential function of a quantity Q of electricity distributed in equilibrium on the cap S, it is sufficient to establish the relation which connects the constant K to the total load Q.
To obtain this relation, we will write the value of the potential function at the point O in two ways.
This value can be written, on the one hand.
On the other hand, it can be deduced from the formula (12). Equality (10) shows that at point O, we have
u = H2.
Equality (10a) shows that at the same point O we have
, R2
CIIAP. VIII. - THE METHOD OF INVERSION. '215
In fact, we have
i
it "'" a
arc tang - = arc tang - - >
2 a ,\
and, in the current calculation,
I / TT u '■ \ a
- \ arctang - / "^ tt'
p \ 2 a J H
So we have
Vo = K ( - -h arc tang _ ) -h K cr . \'i. a j K
Let y be the angle under which the radius of the base circle is seen from the center of the cap; the equalities
a . a
-^^sinY, jj = tangY
allow to write
Vo = K(7: - Y -+-sinY).
Let's compare the two values of Vo, and we finally find
Q
(i3) K
R(Tr - Y "T~ sinY)
This equality completes the determination of the potential function of a quantity Q of electricity spread in equilibrium on the spherical cap S.
This amount of electricity raises the potential level of the spherical cap to a value Kt:, or, according to equality (i3), to a value
Q 7t
R T. - Y"^ sinY The capacity of the spherical cap is therefore
(i4) C = R î ".
§ 3 - Liouville's and M. P. Painlevé's theorems.
One can, in an infinite number of ways, obtain a conformal representation of a plane (^, y) on another plane (X, Y), that is
2l6 BOOK II. - DISTRIBUTION AND DIRICHLET PROBLEM.
find transformation formulas
^=?(X,Y),
such that if the two curves represented with respect to the axes of ce and j" by the equations
intersect at angle a, the two transformed curves, represented, with respect to the X and Y axes, by the equations
F(X,Y) = /[c?(X,Y), .KX,Y)l = o, G(X,Y) = ^[(X,Y,ZK x(X,Y,Z),j
is regular and satisfies the equation
of ^W a^W (92 W _
at any point in the domain D, transformed from d into V space
rfe,v(X, Y,Z).
In answer to this question, we can say that the transformation formulas we are looking for necessarily represent a transformation by reciprocal vectors followed by a change of axes of rectangular coordinates.
This important theorem, by clearly marking the exclusive role of the method of inversion, narrowly limits the field of Analysis in the way opened by Sir W. Thomson. It is due to M. Paul Painlevé (' ).
(') Paul Painlevé, Sur la transformation des fonctions {x, y, z) qui satisfont à l'équation AV = o et sur les coordonnées curvilignes orthogonales ( Travaux et Mémoires des Facultés de Lille, l. I, n° I ; 1889).
DISTRIBUTION AND DIRICHLET'S PRONLEM.
CHAPTER IX.
THE METHOD OF MR. CARL NEUMANN.
§ 1 - The theorem of Mr. Vito Volterra.
Lejeune-Dirichlet's demonstration does not allow us to affirm with complete certainty that the problem to which this geometer's name has been given admits, in all possible cases, a solution. This demonstration only has the effect of replacing the hypothesis that this problem admits a solution by other hypotheses which are perhaps more probable, although also not very well demonstrated.
Green posed the same problem in a different form, but without advancing the solution.
Thus, far from having a general method to solve the Dirichlet problem, we do not even know how to demonstrate the general existence of a solution of this problem.
It is therefore necessary to try to solve it in particular cases.
There are a very small number of special cases for which the Dirichlet problem can be solved immediately (Chap. VI).
The use of appropriate orthogonal curvilinear coordinates has enabled Lamé and other geometers to extend considerably the number of cases in which it is possible to solve Dirichlet's problem. These methods, however, have so far proved fruitful only for surfaces that are part of an orthogonal and isothermal curvilinear coordinate system (Chap. VII).
The method of inversion, created by Sir W. Thomson, has greatly increased the number of cases in which the Dirichlet problem can be solved; but the power of this method cannot be shared by any other method of transformation, as M. P. Painlevé has shown (Chap. VIII).
CHAP. IX. - THE METHOD OF MR. CAHL NEUMAXN. '121
The efforts we have just traced only allow us to solve the Dirichlet problem for particular classes of surfaces, very large classes, it is true.
The methods we are now going to explain are profoundly different from the previous ones, in that the bodies to which they apply are not required to belong to certain classes, but only to verify certain restrictions. They are therefore infinitely more general in scope than those we have explained so far.
They have, it is true, a disadvantage over the previous ones.
To solve the Lejeunc-Dirichlet problem for the sphere, for the ellipsoid, etc., the methods we have indicated lead either to integrate over the whole surface an analytical combination in which the given function appears on this surface; or to determine, by means of the known values, that the function sought takes on the surface the coefficients of a certain development in series. The actual use of these methods gives rise to arduous, but nevertheless practicable, calculations. Thus, the development in series of spherical functions which are used to solve the Dirichlet problem for the space inside or outside a sphere give rise to numerical calculations in celestial mechanics and in the study of terrestrial magnetism.
It is different for the methods that we are going to expose.
These methods lead to the representation of the sought function by a series of integrals extended to the given surface. Only the first of these integrals depends directly on the values given on the surface. Each of the other integrals requires, in order to be performed, that we know the one that precedes it.
Such a procedure undoubtedly escapes any numerical calculation. Therefore, it loses all practical interest; but it is nevertheless of great theoretical interest, because it demonstrates, without any possible dispute, the existence of a solution of the Dirichlet problem for the extended classes of surfaces to which it applies.
A method, invented in i833 by Muiphy (') to treat the problems of electrical influence, gave the first idea of
( ' ) MuRPHY, Elementary principles of the theories of clectricity, heai, and inolecular actions^ Pari, l, Cliap. V, p. gS; i833.
■i-Vl I.IVIŒ 11. - DISTUlHliTION lîT PnOBLbME Olî UlRICIILET.
this class of processes. This idea was further developed by Béer (') in i856. But it is to M. CarlNeumann (-) that we owe the legitimacy of such processes.
The convergence of the series employed can always be demonstrated by means of a fundamental theorem which, implicitly contained in the work of M. Schwarz and M. Cari Neumann, seems to have been stated for the first time by M. Vito Volterra (-'). Mr. Axel Harnack ('') and Mr. Paul Painlevé (^) have insisted on the role that this theorem must play in the study of harmonic functions of two or three variables.
Here is the statement of this theorem:
Let Vt, Uo, ---, U", ... be an unlimited sequence of harmonic functions inside a certain space E bounded by a surface S. Let Ut, u^-, --., Un, ... be the values towards which these functions tend when the point at which they are brought tends towards the surface S. If the series
(l) Ui-\- ll^-T-. . .^ U,i-\- ■ - ■
converges uniformly at any point on the surface S, the series
converges uniformly at any point in V space E and represents^ at any point in this space, a harmonic function.
Let us first prove that the series (2) converges uniformly in the space E.
(') BTÊ.Y.B., Allgemeine Méthode zur Bestinimun g cler elektrischen und inagnetischen Induction {Poggendoff's Annalen, B. XCVIII, p. 187; 1806).
(^) C. Neumann, Untersuchungen ûber das logarithmische und Newton'sche Potential (Leipzig, 1876).
(') Vito Volterra, Sopra alcune condizioni caratteristiche délie funzioni di una variabile complessa {Annali di Matematica, series II, t. XI, p. Sa: 1882).
(' ) Axel Harnack, Die Grundlagen der Theorie des logarithmischen Potentiales un der eindeutigen Potentialfunktion in der Ebene, § 20; Leipzig, 1887,
{') Paul Painlevé, Sur les lignes singulières des fonctions analytiques (Annales de la Faculté des Sciences de Toulouse, l. II, p. B. i5; 1887).
ClIAP. IX. - THE METHOD OF MR. C.VRL NEUMANN. 223
Let's consider the function
*"/. = U" -T- U,i+i -h . . . + U "+p.
It is a matter of proving that we can always take n large enough so that we have, whatever p is,
l*"pl<^,
î being any positive number given in advance.
Now, the series (ij being uniformly convergent, if we pose
we can always take n large enough to have, whatever/?
On the other hand, the quantity ^np, the sum of a finite number of harmonic functions in E space, is a harmonic function in E space. The values it takes in E space are, according to the general theorem of M. Cari Neumann (Book II, Chap. III), between two of the values it takes on the surface S. On the surface S it becomes identical to cp . If therefore, at any point on the surface S, we have
I ? "p I < ', whatever/?, we will have, at any point of the spaceE,
the ^npKh
whatever/?. So let it be
V = Ui-+-U2H-...+ U"^...
the continuous function whose existence is now proven. We have to prove that this function is harmonic at any point inside the space E.
Let M be a point inside the space E. Let us take it inside a whole sphere S contained in E. Let G(M) be the Green's function for the point M and the sphere S, a function which is known (see Chap. VI).
Let's put
w" = u, + u, + ...+ u",
V = W"-f-R".
224 BOOK II. - DISTRIBUTION AND DIRICLLET PROBLEM.
The function W" being obviously harmonic inside the sphere S, its value at any point M inside the sphere is given by
This equality can be written
The first member does not depend on n; on the other hand, we can take n large enough so that R", and consequently the second member, are as close to O as we want. The previous equality can therefore only exist if we have
^"-<-)=S.^-'-^'''^
The function V is, therefore, a harmonic function at any point inside the sphere S.
The function V is harmonic inside any sphere contained in the space E; that is to say that it is harmonic at any point of the space E.
Our theorem is thus proved.
Let us add to this theorem the following proposition:
When the point M of V space E tends 'towards the point m of the surface S, the value V(M) that the series
at point M tends to the value v[ni) that the series
Ml -+- Mo -I- . -1- M" -t- .
at point m, and it reaches this value when point M comes to point m.
To demonstrate this, let's say
W',, - "1 -i-"2 -^ - - .+"rt,
We can take 'e n large enough to have, for
THE METHOD OF MR. CARL NEUMANN.
Any position of the point M in the space E, and for any position (the point m on the surface S,
any neighbor of o that is e.
Moreover, when the point M tends to the point m, U(, Uo,
\},i tending respectively towards w, w., ---, w", W" tends towards iv" . We can therefore, around the point m, draw in the space E a domain small enough so that, for any point M of this domain, we have
|W"(M)-"'"(m)| -^ - - > - ^ - , - j-^ - > - - -
are obtained by multiplying V(P) by factors which are finite for any point M in the space E.
Having proved this lemma, let's consider functions in numbers
unlimited
U,(M), U2(M), UaCM), ....
Suppose that each of these functions is harmonic within V space E and that each of them has, in all this space, a constant sign which is the same for all. If the series
Ui(P) + U2(P)-i-U3(P) + ...
is convergent at any point P of V space E, the
series
Ui(M)-4-U2(M)-+-U3(M)-i-...
converges uniformly at any point in this space and represents, in this space, a harmonic function.
According to the previous proposition, we have
U,(M) = X,U,(P), U2(M) = X2U2(P),
dUi(M)
= [^iU,(P), )
= viUi(P),
t)U2(M) d.T
= M.2U2(P), * - ?
= V,U2(P),
\^, lo, ..., jx,, [Xo, ..., V,, V. point M of the space E. Then we have
. being finite factors for all
= X "U"(P)+X "+iU "+i(P)-.-.
dx ôx
dx
.a "U"(P)+[Ji "+iU "+,(P)
X "+/>U;i+p(P),
^"+pU "+;,(P),
d2U"(M) a2U"^,(M)
()2U "+,(M)
dx^ dx* " dx^
= v "U"(P)4- v"^,U"^-,(P)
^hi+p^ ii+i>{^ )i
228 BOOK II. - DISTRinUTION AND PROBLEM OF DIRICIILin.
But the functions U,j(P), Un+i{^)j - - -, U/,^^(P) all have the same sign. If therefore 4^ is a finite factor between the largest and smallest value of X^, \i^i, - - -, "^n+p '■, if likewise OÏL is a finite factor between the largest and smallest values of [x", p."^,, . . . , [J-^+y?; if finally SIL is a finite factor between the largest and smallest values of v",
^n+\ !
n+p
we will have
U"(M)-hU "+i(M)
dx
dx
■U"^/,(M)
^U "+;,(M) ÔT
-C, [U.(P)= 01L[U"(P)
-U "4-i(P) + . U "+, (?)+..
.+ U"^"(P)1.
d2U "m) . a2U "+i(M)
dx'^
a^2
.+ ^^^^^^f^^=<î>C,[U"(P)-+-U "+,(P) + ...^U "+p(P)|
Once these results are obtained, the proof of the stated theorem is completed without any difficulty.
§ 3 - Some definitions.
If we cut a closed surface by any straight line, we can obtain as intersection points isolated from each other and also whole line segments. For example, if we consider the surface generated by the rotation of the line FABCD {fig. \o) whose side AF is perpendicular to DF, a line
Any straight line will generally cut it into two or four isolated points; to the straight line AI will correspond as an intersection a segment; to the straight line AB, a segment AB and a point K, etc. Whenever the surface is cut by a line and one or more segments are obtained, it is advisable to count only the points of intersection as the points of intersection.
CHAP. IX. - THE METHOD OF MR. CABL NEUMANN. 229
According to this convention, the line AI will cut the surface at only two points A and I; the line AB at three points A, B, K. .... In what follows, we shall constantly assume that a closed surface is intersected by any line in space at a finite number of points, the maximum of which is necessarily even, and we shall call this maximum number the rank of the surface.
The surfaces that we will specially study are the second rank surfaces.
We will admit that, except at certain isolated points or along certain isolated lines, the surface under study admits a tangent plane that varies from one point to another in a continuous manner. It is easy to see that, if the surface is of second rank, the tangent plane at each point will leave the whole surface on one side.
At the isolated points where there is no tangent plane, we will admit that there is a conical tangent sheet; if along a line there is no tangent plane, we will admit that at each point of this line there is a tangent dihedral.
We will say that a surface, open or closed, is multi-layered of order n when it will be possible to imagine in space n fixed points such that any plane lying on the surface passes through one of them, but when at the same time it will not be possible to assign less than n fixed points enjoying this property.
A uniwoven surface is necessarily composed of a portion of a cone and, therefore, is necessarily open. Examples of closed bi-structured surfaces are the parallelepiped, the octahedron, the tetrahedron, the surface generated by the revolution of a rhombus around one of its diagonals, etc. In this last surface, the two stars, i.e. the two fixed points of the definition, are the ends of the diagonal considered; in the parallelepiped and the octahedron, the ends of any diagonal; in the tetrahedron, any vertex and any point of the opposite face.
Non-biased second rank surfaces have a property that we will demonstrate and that will be useful later.
23o BOOK II. - DISTRIBUTION AND DIRICHLET PROBLEM.
Let us take a point [jt. on a similar surface; let m be another variable point of this surface; let dS be an element surrounding the point m; let N^ be the normal to the element û?S toward the interior of the surface S; let p be the line m [ji. The quantity
n COs(p, Ne) ^g
has a value given by Gauss's third lemma; this value is air if the surface S admits at the point [x a continuously variable tangent plane; if the tangents at this point form a conical sheet of spherical opening a, it has the value a; if, at the point u, it admits a tangent dihedral whose opening is
-j it still has the value a. We can therefore write, in all the
possible cases.
S
If we observe that, for the considered surfaces, cos(p,]N, is never negative, we see that we will also have
S
'^'^^^p'^^^^.s.^,
integration extending only to a portion o- of the surface S. If p. and [jl' are any two points, distinct or coincident, on the surface S, and if the surface is divided into two portions '
inequalities that certainly give
Scos (p,Ne) , n cos(p', N,)^^,^ Ar p2 "^ O p'2 - L2
This inequality is established by assuming that the two points [x and a' have a given position on the surface S. But the quantity L has a value independent of the position of the two points jx and [x'. It is obvious that we can always make sure that the two quantities A and S" do not fall below a certain limit. It will thus be possible, whatever the positions of the two points [x and jx' and the two segments o-, ([i)cos(r, N^)
S.
'^ '^ 1^ 3 '
On the other hand, the quantity
S.
[v(m) - v(ix)]cos(r, N^)
d^
obviously varies continuously, even when the point M crosses the surface "t. We can therefore, around the point [jl, draw a domain d, contained in D, small enough so that, for any point Mj of this domain, we have
.)] =- ap([ji;.
If the two points M/, M^ tend towards the same point of the surface S, one remaining inside this surface, the other remaining outside it, we have, in all cases,
(5) lim[U(M,)-U(Me)]-4Trp([x).
§ 5 Definition and properties of subordinate functions.
Let '^{m) be a quantity defined as follows: If at the point m the surface s admits a unique tangent plane that is continuously variable, we have '^{m) ^^ o. If, on the other hand, the point m is a singular point, we have àÇm) = 2tz - a, a having the same meaning as in formulas (3a) and (4a). Let's put
(6) ZTzv'(m) ^ u{jn) -^'b{m) -h v{m)
and
i>'(m) cos( r, N^)
(7) U'(M}=g
rî
dS.
This equality defines a function which exists at any point in space, and which we will call the subordinate function of the function U(M).
It is easy to see from the inequalities (3) and (3 bis) that we have
2Trf'(/n) - lim[U(M,-)] - 7.Tzv(m).
The subordinate function is thus a continuously variable quantity on the surface S. Also the function U'(JVl), defined by equality (7), is a function of any analogous point
236 BOOK II. - DISTRIBUTION AND DIRICHLET PROBLEM.
to the function U(M) defined by equality (2). It is harmonic in all Fespace, except on the surface S. It is discontinuous on this surface; if w'(a) denotes the value it takes at the point [a of this surface, we have
(8)
Just as we formed the subordinate function U'(M) from the function U(M), we can form the inner subordinate function U"(M) from the function U'(M), We will then have a regular way to form an unlimited sequence of functions as shown in the following Tableavi:
U (M)=§
v(tn)cos(r\ Ng) r2
^S,
iT.v" (m) - u' (m)-'\i(77i)v' (m), 2Tzv"'(m) - u"{m)-h'\i)o" (m),
It is easy to see, by the equalities (3 bis), (8) and the analogous equalities, that we can write
1' lim[U (M,-)] =2t:[p ([jl)-^p' ([x)J,
Iim[U'(M,)]-27i[p'(»-4-(."(fx)].
Mim[U"(MO] - ■27r[p"(!x) + p"'([Ji)],
On the other hand, according to equalities (4a), (8) and similar equalities, we can write
( liin[U (Me)] = 27r[p'([a)-P (fji)], lim[U'(M,)]==27r[p"(jx)-(>'(|x)], lim[U"(M,)]-27r[p"'([Ji)-(^"([x)],
(IIJ
Let us now turn to the fundamental proposition of this whole theory; it is stated as follows:
If G is the largest and H the smallest of the values that
CUAP. IX. - THE METHOD OF MR. CARL NEUMANN. 237
takes v(ni) on the surface S; if u. and jjt,' are any two points on the surface S, we always have
i.'(}x')-i.'((x)^(G-H),
which is the announced inequality.
Let be, in particular. G' the largest and H' the smallest of the values that ç'(m) will take on the surface S; we will have
iG'-U')<'k(G - U).
If G, G', G", G"', . . . are the largest values and H, H', H", H'", . . . are the smallest values of the functions
v{m), v'{m)^ t'"(m), v"'(m), ... on the surface S, we have
(IV)
(G'- H')f.X(G - H;, (G" - H")1X(G'-H'), (G'"- H"')"{m), . ..., U, U', L", ..., according to the rules indicated in table (I); we will look for the constant C, limit of the values of the quantities i^{m), v'{m), ç"{m), ...and the function we are looking for will be given by the series
(VIII) j V(M,) = C+^|[U(M,)-U'(M,)] + [U" (M,,-IJ"'(M,)]
The Dirichlet problem is thus solved for the space inside any non-biased second rank surface.
§ 7 - Solution of the Dirichlet problem for the space outside a non-biased second rank surface.
The Dirichlet problem can also be solved for the space outside a non-biased second rank surface; but the solution presents some complications that were not available in the previous problem.
Let w{m,) be the set of values given on the surface S. Let us place inside this surface, at an arbitrarily chosen point, an electric charge k. At the point m of the surface S, the potential function of this charge takes the value q(^m). Let's put
\>{}n) =■ iv{ni) -\- q{}n),
and form the function U and the subordinate functions as shown in table (I).
The function v{m) is the sum of a function w(^ni), independent of A, and a function q{m'), proportional to k; it is therefore a linear function of k. It is easy to see that it is
CH.VP. IX. - I..\ METHOD OF M. CARI. NEUMANN. 24 1
is the same for the functions v'{/n), v"{m), ..., so that the limit of these functions is L + PA", L and P being two constants. Let us now consider the series
- [U(Me) ■+■ U'(M,) -h U"(M,) 4- U"'(M,) - . . . J.
27t
Each of its terms is a harmonic function outside the surface S, behaving at infinity as a potential function, and, according to the equalities (III), when the point M^ tends to the point m of the surface S, each of these terms tends in a continuous way to the corresponding term of the series (lo).
Consequently, the series considered represents a harmonic function in all the space outside the surface S, behaving at infinity as a potential function, taking on the surface S hi value
v{m) - G = iv(m) -^ q{/n) - L - P k
and continuously tending towards this value when the point M^ tends towards the point m.
This result leads us without any difficulty to the following rule:
If we want to form/' a harmonic function at V outside the surface S, behaving at V infinity as a potential function, continuous up to the surface S and taking at any point m of this surface a given value w(m), we will operate as follows:
At a point O arbitrarily chosen inside the surface S, a charge K will be placed whose potential function will take the value q{ni) at point m. Posing
v{m) - w{m) -r- g{>n),
we will form the table (I).
The functions v'{m), v"(m), ... will tend to a limit of the form
L + PK,
h and V being two constants.
D. - I. i6
242 BOOK II. - DISTKIBUTION AND PROBLEM OF DIRICIILET.
In the functions U, U', U", ..., we replace by the value - ^- If Von denotes by R the distance from the point M,. to the point O, and if we form the function
(IX) V(M,)=^ - _L[U(M,)-f-U'(M,) + U"('Me)H-...J, r ri air
it will solve the problem.
We can thus see that the method of M. Cari Neuinann allows us, given a non-biased surface of second rank, to effectively solve the Dirichlet problem both for the space inside this surface and for the space outside it. It is not true that we have the solution of the Dirichlet problem for all possible cases; but, at least, we obtain the solution for a case infinitely more extensive than all those where it had been possible to give it until now.
NATURAL DISTRIBUTION. 9.f'i
CHAPTER X.
NATURAL DISTRIBUTION.
§ 1 - How the solution of the Dirichlet problem for all bodies can be reduced to the search for the natural distribution over all bodies.
If we know how to solve the Dirichlet problem for the space inside a certain surface, we also know how to solve it for the space outside the surface into which the first one is transformed by inversion. If, therefore, we know how to solve the interior Lejeune-Dirichlet problem for all bodies, we also know how to solve the exterior Lejeune-Dirichlet problem for all bodies. We can only propose to solve the first problem.
We can solve the Lejeune-Dirichlet problem for the space inside a certain surface S, if we can find the Green's function relative to this surface S and to any pole P inside this surface S.
Finding the Green's function for the surface S and the pole P is the same as finding a function TÇa^y, z), harmonic inside
of the surface S and becoming equal to ( - -^=- j at any point M of
the surface S.
Let's transform the surface S by inversion by placing the transformation pole at the point P and taking the unit for the transformation coefficient. Let S' be the transform of the surface S. Suppose we know how to find the function -^{0:',^', z'), which is harmonic in the space outside the surface S', equal to unity on the surface S' and, at infinity, equal to zero like a potential function. If we transform the surface S' into the surface S again by inversion, the function f{x',y, z')is transformed into a function g{^,y, :■)■ If we denote by /the distance of the point
a44 BOOK II. - DISTRIBUTION AND THE DHUCIILET PROBLEM.
{■r,y, z) at point P, the function
will be harmonic at any point inside the surface S and will take
the value ( - -=- ) at any point of this surface. This will be the
\ MP /
function sought Y{x,y, z).
One will therefore know how to find the function Tl^x^y^z) for the space inside any surface and for any position of the pole in this space, if one knows how to find, for any surface, the function Y(^, jK? z) which is harmonic in all the space outside this surface, which takes the value i on the surface, and becomes, at infinity, ('■gale to zero as a potential function.
This function ^(^x,y,z) is the potential function of a certain quantity of electricity distributed in equilibrium on the conductor subtracted from any influence. The quantity of electincity which admits for potential function y(c,jKj ^) is the quantity
C
- j C being the capacitance of the capacitor and £ the constant of the laws of
Coulomb.
Let us suppose that we know how to determine the equilibrium distribution that alTecle, on the conductor subtracted from any influence, an electric charge equal to the unit. We can then calculate the potential function u{x, y, z) of this distribution. This function takes, at any point of the conductor surface, the same value A. We will have
and
^=é
G j((t, V. z)
A
We will name natural distribution on a conductor the (distribution that affects a charge equal to the unit to the surface of this conductor subtracted from any influence, and we can state this theorem:
The Lej eune-Dirichlet problem can be solved for V space both inside and outside any surface, if one can determine the natural distribution on any conductor.
CHAP. X. - THE NATURAL DISTUIBLTIOX. 1^5
§ 2 - M. G. Robin's method for determining the natural distribution on a conductor.
Any method suitable for solving the Lejeune-Dirichlet problem in general is obviously suitable for determining the natural distribution on a conductor in particular; but the search for the natural distribution gives rise to certain methods which cannot be extended to the solution of the Lejeune-Dirichlet problem in general.
13e this number is the elegant method indicated by M. G. Robin to determine the natural distribution on the surface of any body bounded by a surface of second rank.
The starting point of M. G. Robin's research is the consideration of a remarkable functional equation which is satisfied by the electric density at any point of a conductor in equilibrium.
Let
t(M) the electric density at a point M on the surface S of an electrified conductor;
N,. the normal to the surface S, at point M, to the outside of the conductor ;
j, the component along N^ of the electrostatic action at point M.
We know that we have (p. 88)
"1>K,. = 27r£n/r 2 L7_ Plu"
s PM 'T PM
CHVP. X. - NATURAL DISTRIBUTION. '^49
But, according to equality (2). we must have
/AIN C e(P)cos(PiVl, N^) ,ç 2re(M) = V - - ^ ab.
PM
If we put
(6)
e(P)cos(PiVl,Ne) , . ,..,
^ ^ -p- ^ - - di - 2Tree(M),
Pi\r
e(P)cos(PM,N")
-) It is not assumed that we have given values to the function v{m) at the various points on the line L. We only assume that these values must be finite (').
It is easy to see that, under these conditions, the demonstration
(' ) This restriction is essential. If the function v{in) is not subject to being finite along the lines of discontinuity, the proposed problem can admit an infinite number of solutions, as M. Painlevé has shown for the analogous problem relating to functions of two variables [Paul Painlevé, ^wr les lignes singulières des fonctions analytiques [Annales de la Faculté des Sciences de Toulouse, t. IL p. B.22; 1888)]
CH.VP. XI. - THE ALTERNATING PROCESS. 253
by which we proved, in Chapter V, that the Dirichlet problem could not admit more than one solution remains entirely valid.
But a circumstance arises to which it is necessary to call attention.
Let us first take a point m in the neighborhood of which the values of v{m) do not experience any discontinuity. Let M/ be a point inside the surface S. Let V(M,) be a function which solves the Dirichlet problem for this space. We can always draw a domain D around the point m, inside the surface S, and small enough so that, for any point M/ of this domain, we have
|V(M,-)-i^(/n)i<£,
î being a ([uuntité positive given in advance.
Let, in efi'et, s be the part of the surface S which forms the limit of the domain D (Jîg'. i'i) with a surface a- included inside do
Fi
the considered space. The function V(M) is continuous on the surface t and on the surface s. It is the same for the function
U(M) = V(M)-t^("i),
since v(/n) does not vary with the point M. We can therefore take this surface (-î + '^) small enough so that, for any point M on this surface, we have
|U(M)|<£.
Then, as the function U(M/), harmonic inside the domain D, takes its absolute maximum value at a point of the surface {s -t- a-) which limits the domain D, we see that we will have, as we had announced,
|U(M,)|{m) and on the surface S4 the values m" This function takes, on the surface S3, values that we will denote by w™.
(' ) Schwarz, Programm der Polyteknikum-Schule in Zurich; 1869-70.
(") C. Neumann, Berichte iiber die Ver/iandlungen der K. Sàchsischen Gesellschaft der Wissenschaften zu Leipzig; 21 April and 3i October 1870.
(') C. Neumann, Untersuchungen iiber das logarithmische und Newton'sche Potential. Leipzig, 1877.
(*) Axel Haknack, Die Grundlagen der Theorie des logarithmischen Potentiales, .... Leipzig, 1887.
D. - l. m
258 BOOK II. - DISTRIBUTION AND DIRICIILET PROBLEM.
Let us form again a function l]3{x, y, z), harmonic in space (En- e), taking on the surface S, the given values p(m) and on the surface S3 the values u"^.
This function takes, on the surface S4, values which we will designate by u\J.
Continuing in this way, we will successively form the functions contained in the two tables below:
I. - Harmonic functions in (E h- e).
Functions. S,.
Ui(^, jr, -s) i'(m)
1^3(^,7,-5) i'irn)
Us(a?, jK, ^) i>(m)
II. - Harmonic functions in (E'-+-e).
S^ functions.
Viix, j, z) ç(m)
\],,(a!;,y, z) i>(m)
l}6{oe,y,z) v(m)
Values on
S,
S3.
..V
It"
"y
u'I
"y
u "l
We see that we must alternately form a function of the first Table, then a function of the second, which justifies the name of alternating process given by M. Schwarz to this method.
We will see that the solution we are looking for can be obtained by taking, at any point M, the space E,
(2) V(M/) = Ui(M,-) + [U3(M,)-Ui(M,)] + [U5(M,)-U3(M,- )]+...; at any point M/ in space E',
(3) V(M,) = U2(M,0+ [U4(M,-)- U2(M,-)] + [UeCM/) - UiCM,-)] + - - - ;
finally by taking arbitrarily, at any point of the space e, one or the other of these two expressions.
Cn\P. XI. - THE ALTERNATING PROCESS. 9.Sçf
§ i. - Demonstration of the above theorem.
To prove that the previous method provides the solution of the Lejeune-Dirichlet problem for the space (E + E'-t-e), it is obviously sufficient to prove the following propositions:
1° The series given by equality (2) is convergent and represents a harmonic function in the whole space (E + e).
2" The series given by equality (3) is convergent and represents a harmonic function in the whole space (E'h- e).
3" These two harmonic functions are identical in all of e-space.
4" When the point M/ tends to a point m of the surface S|, the function V(M/), defined by equality (2), tends uniformly to r(m).
5" When the point M,- tends to a point m of the surface Sj, the function V(M/), defined by equality (3), tends uniformly to v{m).
The fact that the series given by equality (2) is convergent in the space (E -h e) and j represents a harmonic function can be deduced from the theorem of M. Axel Harnack, proved in § 2 of Chapter IX.
The function U( (.r, y, z) takes on the surface S) the values p(/?i), which are all between G and K, and on the surface S3 the value K.
The value of [^{x^y, z) at any point in the space (E + e) is therefore between G and K. This is especially true for the values denoted by u^^.
The function \]2{x,y, z) has, in the whole space (E'+e), values between G and the smallest value of ('(m) or of m','. It is in particular the case of the values u!^ that it takes on the surface S3.
It follows that the difference \\]Muiphy's problem kXdiTecheTchc of the combinatorial methods suitable for performing this reduction, because the first method of this kind, a method which later gave rise to the researches of Béer, of M. Cari Neumann, of M. Schwarz, on the Lejeune-Dirichlet problem, was given by Murphy (') in i833.
§ 2 - Fundamental laws of electrical condensation.
Murphy's method was intended to study a special problem, the problem of electrical condensation, which can be stated as follows:
Given a conductor C) {fig- 47 )> held at
At the potential level Pi.', a conductor C2 connected to the ground and, consequently, maintained at the potential level O, find the distribution of V electricity on these two conductors.
Before studying the method by which Murphj reduces this problem to the study of the distribution generated by given charges on each of the conductors put in communication with the ground, we will expose the fundamental properties of the distribution produced on the two conductors G| andCa- Several of
(' ) Murphy, Elementary principles of the theories of electricity, heat and moleculars actions, Part. I, Chap. V, p. 98.
266 BOOK II. - DISTRIBUTION AND PROBLEM OF DIRICHLET.
these properties will be invoked in the justification of Murphy's method.
We already know (Chapter IV, theorem IX) that the distribution must be monogenous on each of the two conductors Cl and C2. On C, it has the sign of A and an opposite sign on C2.
For the sake of clarity, we will assume that A is positive; the G| conductor will then be charged with positive electricity, and the C2 conductor with negative electricity.
Lemma I. - If the conductor C2, put in communication with the ground, is subjected to the influence of a positive electric mass Y- concentrated in an external point M, it is covered with a monogenous and negative distribution.
Let S2 {fig- 48) be the surface of the conductor Co. Let us surround the
Fis. 48.
point M of a small surface S. The potential function, equal to o on the surface Sa and at infinity, is positive and very large on the surface S. It is therefore positive in all the space outside the
surfaces So and S. The quantity -7^ is then positive at any point
of the surface S2, which proves the stated theorem:
Lemma IL - If the conductor Co, placed in communication with the ground, is subjected to the action of the electric charges jji, [x', p.", . . . concentrated at the external points M, M', M", . . ., it is covered with a distribution which is the superposition of the distributions with which it would be covered, if it were subjected successively to the action of the only charge [x concentrated at the point M, of the only charge \k' concentrated at the point M',
The proof of this lemma is immediate.
CHAP. XII. - MURPHY'S PROBLEM. 267
Corollary. - If Vonjixe at the various points M', M", M'", . . . of the surface of the conductor Ci charges pi.', fx", ^" ^ . . . all positive, the sum of which is equal to Qi, these charges will induce on the conductor Co, put in communication with the ground, charges all negative, the sum of which will be equal to ( - Qo)? and we will have
(1) Q2;=KQ"
K is a positive constant, less than unity, and depends only on the mutual position of the two conductors Gj and Ga. It is first obvious from Lemmas ï and II that all charges induced on the conductor C2 are negative. The last part of our proposition therefore needs only a demonstration.
A unit of positive electricity, in equilibrium with itself on the isolated conductor G2, brings it to the positive potential level A. Its potential function has, at the points M', M", M'", . . ., values A', A", A"^ . . . all positive, and lower than A. We have therefore
A'
Conductor Lo { . , ,/
I potential level ... H
" . ,,, 1 charge O
Point M' I .
potential level ... B
Point M"
load O
potential level ... B"
This Gaussian identity will give us
BQ, = BV-i- BV'-f- B'V'H-. ...
All quantities in this equality are positive. Moreover
;jl' -h ;j." -I- [jl'" -4- . - 1 .
The inequalities (2) give us
Q2SKQ,.
This is what we wanted to demonstrate.
It would be shown in the same way that, if we fix at the various points of the surface of the conductor C2 charges all negative and whose sum is equal to ( - Q_-i), these charges will induce on the conductor Cj, put in communication with the ground, charges all positive whose sum will be equal to Q3, and Von will have
(3) QsSK'Q^,
K.' being a positive constant, less than V unit and depending on the mutual position of the two conductors C| and C:>.
These propositions will serve to demonstrate the legitimacy of Murphy's method. They also provide us with information on the Gaugain coefficients (*) whose definition we will give.
These coefficients are defined by means of three experiments.
First experience. - The C| driver being maintained at the
(' ) This theory is due to Mr. J. Moutier [.J. Moutier, Sur la condensation électrique {Bulletin de la Société Philomatique, 1878)].
CIIAP. XII. - THE PROBLEM OF MURPHV. 269
Potential level A, the C2 conductor is connected to the
with the ground. The first takes a positive charge a, the second
a negative charge - b.
Let's put
b - ma.
The coefficient m depends only on the situation of the two conductors Cl and C2 and not on the potential level A. It is the first Gaugain coefficient. If the two conductors C|, C2 are outside each other, this coefficient is at most equal to K, and, consequently, less than unity.
Second experiment. - The conductor Ci, maintained at the potential level A, is put in presence of the conductor C2, isolated and carrying a total charge equal to O. This one is put at the potential level B and the conductor Ci takes a total charge "q Gaugain calls the condensing force of the device the ratio
"0
Third experiment. - We isolate the conductor C2, carrier
of the load - b ei we put the conductor C| on the ground. It takes a
positive charge"(.
Let's put
"1 = ni'b.
The coefficient m' depends only on the situation of the two conductors C, and C2 and not on the load b. G isvXthe second Gaugain coefficient. If the two conductors C,, Co are outside each other, this coefficient is at most equal to K', and, consequently, less than unity.
Gaugain found, through experience, that we had
<4) a = "0 -+-"!-
This result can be found by the theory. The Gaussian identity, applied to the first two experiments, gives
ao A - a A - èB.
This same identity, applied to the second experiment and to the
9-70 BOOK H. - DISTRIBUTION AND DIRICHLET PROBLEM.
third, gives
O = "lA - 6iB.
These two relations give the equality found by Gaugain. As we obviously have
a, = mm'a, the relation (4) gives
1
F
- 1
relation that expresses the condensing force as a function of the two Gaugain coefficients.
We will see later, in the study of the discharge of the capacitor, the importance of these two coefficients.
§ 3 - Murphy's method.
Murphy's method has Lut to find the electrical distribution on two conductors C,, C2 ifig- 49)? one of which, d.
is maintained at the potential level A, while the other, C2, is maintained at the potential level o, when one knows how to determine the distribution that given electric charges generate on each of these conductors, considered in isolation, and put in communication with the ground, or maintained at a given potential level.
The conductor C, being at the potential level A and subtracted from the action of any external electric charge, is covered by an electric distribution that we can determine by hypothesis. Let Q be the total quantity of electricity it contains, a quantity which is positive if A is positive; let A| be the density of this distribution at the point M( of the surface Si of the conductor Ci; we know that this density A) is positive.
Let us imagine this distribution yZ^ee on the conductor C); let us place in its presence the conductor C2, put in communication
CHAP. X". - THE MURPHY PROBLEM. 27I
with the ground. This conductor will be covered with an electric distribution that we know, by hypothesis, to determine; this distribution is monogenic and negative; ( - Q2) is its total mass; at a point Ma of the surface S2 of the conductor Co, it has a density
(-A.) Let's fix this distribution on the C2 conductor and place it in the presence of the Ci conductor in communication with the ground. This one is covered with a distribution that we know how to determine by hypothesis. This distribution is monogenic and positive. Let Q3 be its total mass and A3 its density at point M, .
Continuing in this way, we will determine, on the conductor C(, a series of monogenic and positive distributions having respectively for total mass
Qi, Q3, Q", ..., and for density at point M,
^1, A3, As,
On the Co conductor, we will determine a series of monogenic and negative distributions having respectively for total mass
-Qî, -Qv, -Qs, ...,
and for density at point M2
- A2, - A4, - Ag,
The proposition stated by Murphy is as follows:
If conductor C|, maintained at potential level A, is brought into contact with conductor C2 maintained at potential level o, the electrical density at point M, of conductor C) will have the value
(5) 8, = A,-+- A3+ Aj-f-...,
and the electrical density at the point Ma of the conductor C2 aut^a has the value
(6) - §2 = - A2-Ai- Ae - ....
This proposal, to which Murphy was obviously led
272 BOOK II. - DISTRIBUTION AND DIRICIILET PROBLEM.
by the old theory of concealed electricity, is demonstrated by M. Cari Neumann (*) in the following way: Equalities (i) and (3) give
Q2^KQ" Qv^KQ3, Qe^KQs, QaSK'Qa, Qs^K'Qi, ,
and, therefore,
Qa^KK'Q,, QslKK'Qa, ...,
The two series
Qi + Q3-i-Q5-...,
are therefore convergent.
Let us now observe that we have
Q,=:§Ai^S" Q3=§A3t/Si, Q5-§A,6?S,,, .. ,
Q^^^A^^S,, Qv-§A4^S,, Q6=gAcû?S2,
and that the quantities A, , A3, A^, . . ., Ao, A,, Ae, . . . are all positive, and we will see immediately that the series (5) and (6) are convergent.
These series thus determine a true electrical distribution on conductors C) and C2. H remains to prove that the potential function of this distribution is equal to A at any point of conductor C) and to o at any point of conductor C^.
Let
V" V3, \" ...
the potential functions of the distributions
A,, A,,, A" ...,
and
--V2, -V4, -Vfi,, ...
the potential functions of the distributions
- A2, -Ai, - Afi,
(' ) Carl Neumann, Untersuchungen ùber das logarithmische und Newton'sche Potential, p. 3io. Leipzig, 1877.
OHAP. XII. - THE MURPHY PROBLEM. 278
t is easy to see that the potential function of the distribution given by (5) and (6) will be given by the formula
(7) "" --= V,- Vo-- V3_ V,-f- V,_ Vc^. . . .
But, referring to the determination of the successive distributions, it is easy to see that we ?i, at any point of the conduc~ their C| ,
and, at any point of the Co,
V, - V2=:0, V:,- Vi-:(), Vi- Vo=0,
Formula (7) shows that we have, at any point of the conductor C|,
V-= A
and, at any point of the C2 conductor,
r = o,
which further legitimizes Murphy's method.
§ 4 - Combinatorial method of M. Cari Neumann.
Murph's method solves, in a particular case, the general problem to which we have given the name Murphy's problem: knowing how to solve Dirichlet's problem for the space outside the conductor Ci and for V space outside the conductor Go, solve it for the space outside the set of conductors Ci and C2. Lipschitz (') and M. Cari Neumann (2) have shown how one can extend Murph's method}', in order to obtain the general solution of the previous problem. But, in spite of this generalization, Murphv's method leaves something to be desired in one respect: it has no analogue in the theory of harmonic functions of two variables (^) and, consequently, interrupts the parallelism between the theory of the
(' ) Lipschitz, Crelle's Journal fiir reine und angevandte Mathematik, Bd LXI, p. 12.
(' ) Carl Neumann, Untersuchungen iiber das logarithmische und Newton'sche Potential, p. 3i2. Leipzig, 1887.
(' ) The corollary on which the justification of Murphy's procedure is based has no analogue in the p!an.
D. - L 18
3.74 UVRE M. - DISTRIBUTION AND MANAGEMENT PROBLEM.
logarithmic potential and the tliéoric of the Newtonian potential function.
Mr. Cari Neumann (*) has re-established this parallelism by creating, to solve Murpliy's problem, a combinatorial method which also applies to harmonic functions of two and three variables.
Let E be the space outside the two conductors C,, G2. H is to form a function V(M), hai^monic in the space E, and ])renant at the various points m, of the surface S| given values r(mi), and at the various points m-i of the surface S2 given values also (^(mo). The values ^^(m,) vary continuously on the surface S, and the values vi^m-^) vary continuously on the surface S^.
Suppose we have determined a function W)(M), harmonic in E space, taking on the surface S| the given values ('(w,) and on the surface S2 the value o; then a second function Wo (M), harmonic in E space, taking on the surface S^ the given values (^(ma) and on the surface S( the value o.
The function
\Viai)^-\V2(M)
will be harmonic in the whole space E, and will take the values given on the surfaces S|, So.
The problem we seek to solve can therefore be reduced to two problems of the following type:
Find a harmonic function in V space E, taking on the surface S, given values^ continuously variable, (^(/Wi), and on the surface S^ the value o.
Let G be the largest absolute value of the quantities (^(/?î,).
Let us form a first function, W,(M), harmonic in E space, taking on the surface S^ the value o and, on the surface Sj, the values, all positive, r(m))-|- G; then a second function, W2(M), harmonic in E space, taking on the surface S2 the value o and, on the surface S(, the constant and positive value G The function
Wi(M) - WîCM)
will obviously solve the previous problem.
(' ) Carl Neumann, Untersuchungen, ..., p. 3i3.
CHAP. XII. - THE PROBLEM OF MURPIIV. 7.yS
We are thus brought back, in the final analysis, to the following problem:
Find a function V(M), harmonic in the space E, taking at any point of the surface Sj the value o and, at the various points of the surface S, given positive values, <'(/??,), continuously variable.
Let us form, which we know how to do by hypothesis, a function Vf (M), harmonic in the space (E + Co) and taking on the surface S| the given values p (/;?,).
At the various points of the surface S2, this function takes certain values u\ .
Let us then form, which we also know how to do by hypothesis, a function U2(M), harmonic in the space (EH-Ci) and taking on the surface So the values u].
At various points on the surface S, this function takes on certain values u'.^.
Let us form a function U3 ( M ) , harmonic in the space (E -h G2) and taking on the surface Sj the values u'.^.
At various points on the surface Sa, this function takes on certain values u"^.
Continuing in this way, we will alternately form the functions contained in the two tables below:
I. - Harmonic functions in space (£-+-€2).
Values on
Functions. S,. S^.
U,(M) v(ini) u'\^
UsCM) u', u"^
U5(M) lù "5
II.- Harmonic functions in the space (E-f-Ci).
Values on
Functions. S^. S,.
U2(M) u] u'.2
Ut(M) u":i U;
Uc(M) u',
'6
UyQ BOOK II. - DISTRIBUTION AND DIRICHLET PROBLEM.
We will demonstrate that the equalized
(8) V(M) = U,(M) - UîCM) + U3(M) - U^M) -^. . .
defines a harmonic function in the space E, taking on the surface S, the values v{mt) and on the surface S2 the value o.
According to the theorem of Mr. Vito Volterra, it is sufficient to show that, on the surfaces S^, So, this series converges uniformly to the values we have just indicated.
Suppose that the function W(M), harmonic in the whole space (E + G2), takes the value i at any point of the surface S, ; it will take, at any point of the space (E + Go), values less than I . In particular, at any point of the surface So, we have the inequality
WSK,
R being a positive quantity certainly lower than the unit.
Let us denote by G the largest of the positive values ç(m') given on the surface S). 11 It is easy to see that at any point of the surface So we will have
u\ 1 KG.
Let us consider the function
GW(M) - Ui(IVl).
It is harmonic in the whole space (E + Co); it is equal to o at infinity and cannot have any negative value on the surface Si; it cannot therefore take any negative value in the space (E -+- Co), and in particular on the surface So, from which the demonstration of the previous inequality follows immediately.
We have therefore, on the surface S| ,
t^(m)Ç G, and, on the surface Sa,
u\ ^KG.
If we denote by K' a positive quantity, less than i and analogous to K, we can easily deduce from the previous equality that at any point of the surface S) we have
u'^iKK'G.
Continuing in this way, we will arrive at the results contained in the following tables:
ICVP. XII.
THE MURPHY PROBLEM.
-^77
i" On' the surface Sj,
takes the values
positive function
U,(M) v{m)
UîCM) a'.^
U3(M) "3 = l<'.,
Uv(M) u
U5(M) W'5 = î*4
U6(M)
who have
for liniile
superior
G
KK'G
KK'G
K2K'2G
K^K'îG
K3K'3G
1° On the S^ area,
takes who have
The values for limit
upper positive function
Ui(M^ u\ KG
UiCM) u".= u\ KG
U3(M) ul K^K'G
Uv(M) ul=ul K^K'G
U5(M) ul ' K3K'2G
U6(M) ul^u's K3K'2G
Inspection of these two Tables immediately shows that the series which appears in the second member of equality (8) converges uniformly both on the surface S) and on the surface So; that it takes the value v(m) at any point m on the surface S, and the value o at any point on the surface So; this series therefore represents the harmonic function sought.
This justifies the combinatorial method of Mr. Cari Neumann. It is hardly necessary to notice that this method lends itself indefinitely to extension. For example, let us consider three conductors C| , Co, G3. If we know how to solve the Dirichlet problem for the space outside each of them, we can, by a first application of Mr. Cari Neumann's method, solve it for the space outside the set of two conductors C|, C2; then, knowing how to solve it for this space on the one hand, and on the other hand for the space outside the conductor C3, we will solve it for the space outside the set of three conductors Ci,
-278 BOOK I'. - DISTRIBUTION AND PROBLEM OF DIRICHLET.
C2, C3, by a new application of the procedure of Mr. Cari Neuniann. We thus arrive at this beautiful theorem:
Given n conductors outside each other in space, if Con knows how to determine the distribution that electricity takes on, on each of them alone, under the action of arbitrarily given fixed charges, we know how to determine the distribution of electricity on V all of these n conductors subjected to V action of arbitrarily given fixed charges.
We know, for example, how to determine, in all possible cases, the distribution of electricity on a sphere subjected to a given influence. We will therefore be able to determine the distribution of electricity on two spheres outside of each other and mutually influencing each other. This beautiful problem was, in fact, solved directly by Poisson (' ), and has since occupied a great number of geometers, such as Plana, W. Thomson and G. Kirchhoff (-).
(') Poisson, Second Mémoire sur la distribution de l'électricité à la sur/ace des corps conducteurs (read at the Académie des Sciences on September 6, i8i3). (") Foi> Mathieu, Théorie du potentiel. 2° Part, Applications, p. 65. Paris,
BOOK m.
EXPERIMENTAL STUDY OF ELECTRICAL DISTRIBUTION.
CHAPTER ONE.
THE BODY PROVES.
§ 1 - Theory of the body of test.
After having seen how Poisson had, by means of questionable hypotheses which we will have to return to later, reduced the study of the distribution of electricity on conducting bodies to a problem of analysis, we have outlined the efforts by which geometers have attempted to solve this problem. We have now arrived at another part of our task; it is to take advantage of the analytical methods that we have studied, to deduce from them consequences that can be used for the experimental control of the theory of electrical distribution. The study of some of these consequences and their comparison with the facts will be the subject of the present book.
We will begin by examining the method by which Coulomb experimentally studied the electrical distribution on a conducting body, the test body method.
If a very small conducting body, carried by an insulating shellac needle, is electrified, the torsion balance in which this small body is taken as the fixed ball, while the moving ball is electrified by its contact with this small body, will easily allow to determine numbers proportional to the electric charges taken by this small body in various circumstances.
28o BOOK MI. - EXPERIMENTAL STUDY OF DISTRIBUTION.
Therefore, it is easy to see that Ion will have a method to determine the distribution that affects the electricity on a conductor, if we admit the accuracy of the following proposition:
A conductive body, of extremely small dimensions, being put in contact with another conductor of finite dimensions, takes an electric charge which is the product of two factors: i° the electric density which was at the point of the conductor which the test body touches before V approach of this body; i° a coefficient which depends on the shape of the test body and its situation with regard to the tangent plane to the conductor at its point of contact with the test body.
This proposition was stated by Coulomb who sketched out a demonstration (' ). As we shall see, this demonstration can be given a more precise form, although some criticisms can still be made of its absolute rigor.
This demohstration will be based on the following lemma ("):
We consider two systems of homothetic conductors V one of V other, whose ratio of homothety is \. We suppose that the electric charges that these conductors carry in their homologous points are also in the ratio \. The potential function will have the same value at the homologous points of these two systems.
If, instead of placing at the homologous points of the two systems charges which are between them in the ratio of homothety, we consider two systems containing only surface electricity, and if Von supposes that at the homologous points of these two systems the surface densities are the same, it is easy to see that homologous elements will carry charges which will be between them as the square of the ratio of homothety; at the homologous points of the two systems, the values of the potential function will be between them as the ratio of ho
(' ) Coulomb, Sixth Memoir on Electricity. Continuation of the research on the distribution of the electric fluid between several conducting bodies; determination of the electric density in the different points of the surface of these bodies {Mémoire de l'Académie des Sciences for 1788, p. 676.)
{') J. MouTiER, Cours de Physique, t. I, p. 4o6.
ClIAP. I. - THE TEST MOVE.
281
moletia; the first order partial derivatives of the potential function will have the same values at the homologous points of both systems.
This last consequence shows us that, if the electrical equilibrium is established on the first system, it is also established on the second.
Gela posed, that is to say an electrified and isolated body G (/?^. 5o). Tl carries
Fig. 5o.
At each point, the surface density D is given. An isolated body E, in a neutral state, is brought close to it and touches it at the point P, occupying a given position with respect to the tangent plane TT'. The electricity is distributed in a certain way on the body E. Let A be its density at the point m of the surface of the body E.
If we keep the body G invariant, but if, by homothety, we reduce the dimensions of the body E in a certain ratio "k, the center of homothety being in P, the density A tends to a certain limit d which we propose to determine.
For this, let us take the state which corresponds to a certain value A of the homothety ratio. The body E has been transformed into a certain body e whose dimensions are those of the body E in the ratio À. Let be the value that the density A takes in this state.
Let us form a system homothetic to this one, the homothety ratio being y The body e will become again the body E, but the conductor G will be replaced by a similar and larger conductor. If, on the latter, the density before the contact of the body E is the same as at the homologous point of the body G, we can easily see that after the contact the density will be the same at the homologous points of the bodies E and e.
We can see that, instead of dealing with the original problem, we can deal with the following problem: without changing the
282 BOOK III. - lirCDE EXPERIMENTAL OF DISTUIBUTION.
dimensions of the body E, we transform by homothety the body C, the center of homothety being in P and the ratio of homothety being a certain number [x which increases beyond any limit. It is assumed that the electric density before the contact of the body E is the same at two homologous points.
We ask to what limit d the density A tends at a point of the body E after the contact.
Now the limit state of the problem is as follows:
An indefinite conducting mass is bounded by the unlimited plane TT'; before the contact of body E, the plane TT' is covered with an electric layer having at any point the density D(P) of the electricity which was at the point P of body C before the contact; body E is approached isolated and in the neutral state; what is, after the contact, the electric density d[m) at the point m of body E?
We can admit that we have
d{m) = Â-(m)D(P),
k[ni) being a function of the position of the point /;?, a function whose form is fixed by the shape of the body E and by its position with respect to the plane TT'.
From this expression of the electric charge in each point of our infinitely small test body, we see that the total charge Q taken by our test body will have the value
Q=:KD(P), .
K being a coefficient which depends only on the shape of the test body and its position with respect to the tangent plane at the affected point. This is the result we announced.
Although, to determine the value of the coefficient K, one can arbitrarily dispose of the shape of the body C and the position of the point P on this body, the calculation of this coefficient can be carried out only for a small number of test bodies.
Poisson ('), aj'ant solved the problem of electrical distribution on two spheres in contact, was able to determine the coefficient K for a test body formed by a small sphere of radius R.
(' ) Poisson, Mémoire sur la distribution de l'Électricité à la surface des corps conducteurs {Savants étrangers, p. 62; 181 1).
THE BODY OF KPRELVK.
283
It has Iroiivé
K= -^4TrR2
4Tr3R5
Mr. Beltrami (' ) examined the case where the test body is formed by a hemisphere whose base BB' lies on the plane
Fi". 5i.
langent TT' {^fig- 5i). He found, by designating the hemisphere's rajon by R,
K = 37iR2.
Mr. G. Robin (-) treated a third case. The test body is the body of greatest attraction, whose polar equation is
COS Ci I
The constant a is the diameter; the body, which is of revolution around its diameter, presents at the pole an infinite flattening. We have then
K = 27ra-.
§ 2 - Use of the test body.
The calculation of the K coefficient, which is often very difficult, is fortunately unnecessary when one simply proposes to compare, at
(' ) Beltrami, On the experimental determination of the electric density at the surface of conducting bodies (Il nuovo Cimento, 3" series, t. I, p. 2i5; 1877).
(' ) G. Robin, Sur la distribution de l'électricité à la surface des conducteurs fermés et des conducteurs ouverts {Annales de l'École Normale supérieure, 2" série, t. III. Supplement, p. 9; 1886).
■28 1 LIVHE m. - KTIDE EXPERIMENTALALK OF THE DISTRIBUTION.
by means of the same test body, the electric densities at the various points of the surface of the same conductor or of different conductors, without trying to know the total value of this density at each point.
The preceding considerations are therefore sufficient to justify the use that Coulomb made of the test body in his research on electrical distribution (' ).
This research has the particular interest of having provided the first experimental verifications of the theory, imagined by Poisson, of the distribution of electric fluid on conducting bodies.
We have seen, for example (p. 191), that, according to this theory, at each point of an electrified ellipsoid, the electric density was inversely proportional to the distance from the center to the tangent plane at that point; this proposition is easy to submit to the control of experiment. Although Coulomb did not leave anything to this control, we know from the writings of Poisson (2) that he had tried it and found it satisfactory: "... This law, he says, was, in fact, verified by Coulomb on a wooden ellipsoid, covered with a metallic blade. This ellipsoid of revolution had been made on the lathe by President de Saron, an honorary member of our former Academy. "
We have also seen (p. 209) that Poisson (^) had completely determined the electrical distribution on two conducting spheres, equal or unequal, placed in contact. On the other hand, Coulomb ('') made an experimental study of this distribution
(' ) Coulomb, Cinquième Mémoire sur l'Électricité. On the manner in which the electric fluid is carried between two conducting bodies placed in contact and on the distribution of this fluid on the different parts of these bodies {Mémoires de l'Académie for 1787, p. 421)- - Sixth Memoir on Electricity. Continuation of the researches on the distribution of the electric fluid between several conducting bodies: determination of the electric density in the various points of the surface of these bodies {Mémoires de l'Académie for 1788, p. 617).
(') Poisson, Mémoire sur l'attraction d'un ellipsoïde homogène, read at the Académie on October 7, 1833 (Mémoires de l'Académie, t. XIII, p. 5oi).
(') Poisson, Mémoire sur la distribution de l'Électricité à la surface des corps conducteurs, read at the Académie on May 9 and August 3, t8i2 {Savants étrangers, p. 1811, p. I.)
(' ) Coulomb, Cinquième Mémoire sur l'Électricité, ..., 1" Section, p. 437
CH.vp. I. - THE PROOF BODY. 285
whose results were compared by Poisson with the numbers deduced from his calculations.
Coulomb put in contact two spheres whose radii were in a certain ratio b. After the contact, they were separated and moved away from each other. The electricity was distributed uniformly on each of them, taking on the smaller one a density [3 times greater than on the larger sphere. Here are the values of [S given by calculation and observation:
Density ratio
Electrical
DifTerence
Report
of the shelves.
on both
spl
icres following
between the calculation
and observation.
the calculation.
observation.
b^\....
. p ^ r , I Go 1
?-.,o8
- 0,07
b^l...
. p^ i,3i68
?. [,3o
-*- 0,01
b=^l...
.. 3-1,4443
P^.i,G5
- , l >
"The first two differences fall within the limits of the errors of which observations of this kind are susceptible; the third, which is in the opposite direction to the other two, can still be attributed to these errors and proves nothing against the theory. However, it is good to observe that, in the case where Fun of the rays is the eighth of the other. Coulomb did not immediately determine the ratio according to which electricity is divided between the two spheres: to make the effect more sensitive, he made the large sphere touch the small one twenty-four times in succession; then he concluded from this complicated experiment the division of electricity in each contact. It is probably for this reason that the difference between calculation and observation is greater in the case of 6 = | than for other values of 6 (*). "
Let's place two unequal spheres in contact. At the pole opposite the point of contact, on the "smaller sphere, will be the maximum electric density. Let y be the ratio of this density to the average density of electricity on the larger sphere when the smaller one has been moved away. The values of y observed by Coulomb and calculated by Poisson are contained in the following table:
C) Poisson, loc. cit. p. 61.
286 BOOK m. - EXPERIMENTAL STUDY OF DISTRIBUTION.
Difference y-values between the calculation
Values - ^ and
of b. calculation. experience. experience.
b - \ ï = 1 ) 3'^^- ï = T , 27 -T- o , O4
h = \ Y = 1.834 Y =^ 1.55 -f-o,i5
h = \ Y = 2,477 Y = 2'35 -f- o,o">
Z> = I Y = 3.087 Y = 3.18 - o.o3
Two equal spheres being in contact, if we designate by h the electric density on one of them at 90° from the point of contact, the density will have for values at 180°, Co" and 3o°, a/<, a'/i, a"/i.
A sphere S touches another one S' of double rajon. If we denote by ]i the value of the density on the sphere at 90° from the point of contact, the density on this same sphere at 180" and 60" from the point of contact will have the value [i/i, ^'A. At 90° from the point of contact, on the sphere S', this density will have the value ^' h.
A sphere S touches another one S' of quadruple radius. If, on the sphere S, the electric density at the point of contact has a value h at 90° from the point of contact, on the same sphere, at 180" from the point of contact, it will have the value yA.
Coulomb had determined by experiment the quantities a, a', a", [3, P', P", y. Poisson calculated them in his formulas. The following table summarizes the comparison between calculation and experiment:
Ratio of electrical densities at different points of two spheres Differences
which are affected between the calculation
^- - - ^ -^ - and
calculation. observation. observation.
a = 0.877 o' = 0.95 - 0.08
a' =1.342 a' =1.25 -+-0.07
a" =5.837 "" = 4.80 -h 1.06 1
P = 0.739 P = o.75 - 0.01
P'= 1.797 P'= 1.70 + 0.09
P"= 1.238 P"= 1.25 - 0.01
Y = 1.673 Y= 1:43 + 0.24
(:ini>. II. - OPEN CONDUCTORS. 287
CHAPTER I
OPEN CONDUCTORS.
§ 1 - Electrical distribution on an open conductor subject to any influence.
Coulomb (' ) was the first to state, as a consequence of experience, that electricity in equilibrium on a conducting body resides entirely on the surface of this body. As this proposition is one of the most immediate and general consequences of Poisson's theory, one would be in possession of an excellent verification of this theory if one could demonstrate that, in the equilibrium condition, there is no electricity inside the conducting substance. But it is easy to see that this truth cannot be demonstrated either by Coulomb's experiments or by any experiment. It is true that one can dig into a conductor cavities communicating with the exterior through small openings and prove that the walls of these cavities are not electrified; but the points inside the conductive mass itself are inaccessible and one cannot directly prove that they are in a neutral state.
The experiments described in the Physics Treatises as suitable for verifying Coulomb's proposition are, in reality, experiments suitable for studying the distribution of electricity on open conductors^ that is, on conductors limited by two surfaces infinitely close to each other, bounded by the same contour that we call the edge of the conductor. Such is a spherical cap formed by a sheet of foil.
It is therefore interesting to search in the study of the distribution
(' ) Coulomb, Quatrième Mémoire sur l'électricité, in which two principal properties of electric fluid are demonstrated: the first, ...; the second, that in conducting bodies the fluid, once it has reached a stable state, is spread over the surface of the body and does not penetrate the interior (Memoirs of the Academy for 1786, p. 67).
288 BOOK III. - EXPERIMENTAL STUDY OF L/V DISTRIBUTION.
electric on open conductors the explanation of the experiments in question; this explanation results from some beautiful theorems due to Mr. G. Robin (' ).
Through the contour KK' {fig. 02), we pass an area limited to two sides S. At any point M of this area, we raise a normal on which we take, on both sides of the surface S, infinitely small lengths MM,, MMo. The points M,, Mo describe two surfaces S,, So, infinitely close to the surface S and passing through the contour RK'. J^the interval between the two
surfaces S), So is filled by a conductive material. This conductor being charged with electricity and subjected to the action of given electric charges, the electricity distributes itself in such a way that its density is a, at the point M, and o-o at the point Mo. The problem of distribution on open conductors consists in determining towards which limits S,, So tend to o-,, 0-2, when the two surfaces S(, So tend towards the surface S.
LetW(m)be the potential function at the point m of the surface S of all the acting charges. The potential function at the point /?i will be
^s, Ml m >^Sj M7 "i
The surface S being entirely inside the conductor, this quantity must have the same value at any point m of the surface S.
(' ) G. Robin, Sur la distribution de V électricité à la surface des conducteurs fermés et des conducteurs ouverts {Annales de l'École Normale supérieure, 2" série, t. III. Supplement, p. 9; 1886).
CIIAP. II. - LliS CONDLCTELRS OPEN. "^89
The limit towards which it tends when the two surfaces S,, So tend to merge with the surface S will also have a value independent of the position of the point m on the surface S. This limit being obviously
s
\(/n) -+- \ =z=-
we obtain the following proposition
If, SU7' the surface S, one distributes an electric layer whose potential function, added to the potential function of the acting niasses, forms a sum which has the same value at any point of the surface S, and whose total mass is equal to the electric charge communicated to the open conductor, the density of this layer has the value (S, + So).
This principle will be used to determine (S, + Ilo); this is how, for example, in Chapter VllI of the previous book we were able to determine this quantity for a spherical cap.
(S, + S2) being assumed known, it remains to determine S, and S2.
Let iN be the MM direction, and Ng the MMj direction.
If an electric charge equal to the unit is placed at the point M,, it undergoes, from the layer of density o-, distributed on S,, an action whose component following N) is/, (M,); from the layer of density 0-2 distributed on So, an action whose component following N, is^2(M,); from the external charges, an action whose component following N, is 'i>(M,). We have, according to a known theorem,
27rscr.(M,)=/,(M.)-i-""'2(M,)-r-
CllAI'. II. - THE OPEN CONDCOTEURS. 29I
The second member of equality (i) has therefore the same limit as
/i(Mi)+/2(M2)H-tî;(M2), and we have
(2) 27:£[S,(M) - i:2(M)] = F,(M)-hF2(M) + ^f(M).
Let q be one of the acting charges and Pie the point where it is concentrated. We will have
S,(M'Uos(M 'M,N,)
s rtm'
S.2(M')cosrM 'M, Ni) 's W^\
7cos(PM, N,)
F.(M) = sC ^i(M-)cos^M'M,lN,)^^g^ '^s MM'
F,(M) = .CME>^2liM2UV><,s,
^(M) = £"y
^*^ PM
and the equality (li) will become
[2.[x,(>i)_s,(M)]=y5L<^-^^i^^
I ^^ FM
^^ ' '^ [S, (M')-4-S2(M ')] cos(M'M,N,)
I -S.
s M'M
We thus arrive at this theorem of Mr. G. Robin (' ):
When Von has determined the sum (Si-f-Sj) of the electric densities at the corresponding points on the two sides of an open conductor, a quadrature suffices to determine separately the densities S| ei S2 at these two points.
In the previous demonstration, we have confused, for brevity, the normals in M|, Mo to surfaces S|, So with
(') If the surface S, is closed and the surface S^ is inside it, we have, at any point of the latter, a^= o; hence S^= o, and the equation (3) becomes
,... V 'ZCosCPM.N.) C\ S(M')cosfM'M.\ ) ,^
2^-,(M)=: > ^ -2 - + \ =2 ^^'^-
^^ PM ^s MM
This functional equation determines the electrical distribution on a closed confliicteur. Mr. G. Robin has made great use of this equation. He used it to solve the beautiful and difficult question of the electric distribution on a sjjheroid appreciably different from the sphere. Poisson had stopped at the case of the spheroid little different from the sphere (G. Robin, loc. cit.; see also, above, Book II, Chap. X).
292 BOOK III. - KTLDIi KVPÉRIMlîXT.VI.t: DE L.K UISTIUBI TIO.N.
the normal in M to the surface S. This confusion is obviously made legitimate by the quasi-identity of direction of these normals, except at points very close to the edge RK' of the conductor. As, moreover, in the vicinity of this edge, the densities S,, ^o can be infinite, it is permissible to wonder whether the previous demonsti*ation is not thereby put in default. However, it is easy to see that the previous demonstration remains valid if we admit, as we shall, that at an infinitely small distance 8 from the edge KR' the densities 2l)i, S^, while being infinitely large,
are infinitely small compared to < - In all known examples, they are of the oi'drc of - --
§ 2 - Conductor o ivert removed from any influence.
Let us suppose, in particular, that vm open and insulated conductor, to which a known charge of electricity has been communicated, is removed from any influence, and let us look for the electric distribution on a similar conductor.
We know, first of all, that this distribution will be monogenous, and that the electric density will have in each point the sign of the total charge.
Equation (3) will become
(4) .,[i,(>l)-^.("i)] = Q f^.(^^^)H-I.(M^)^o.(M^^■.N,,^3^
^s M'M
Let us assume the surface S is everywhere convex in the direction of the normal N,. The second member of equality (4) will have a constant sign which will be that of [S, (M') + 112 (M')], i.e. the sign of the total charge distributed on the conductor. Hence this theorem:
If an open conductor, whose surface is convex, is removed from any influence, the electric density has a greater absolute value at a point of the outer surface than at the corresponding point of the inner surface.
If the open conductor carries a total load Q divided into a load Qj distributed on the face Sj and a load Qo distributed
CIIVP. n. - THE OPEN CONDLCTEl'RS. 2Ç)3
on the Sa side, it is interesting to know in a simple way
the limits between which the ratio ^- This result
is obtained by the theorem that we will prove.
Let us assume, for the sake of brevity, that the conductor opening is flat {/ig. 53), although the theories that will be expounded in
Fie. 53.
the study of curvilinear integrals (t. III, Book XIII) allow us to generalize the following proof. On the plane P of the opening, the edge. KK' cuts a bounded area A. Suppose that the surface S does not cut the plane P inside the area A. In the vicinity of the edge KK', the surface S is all on one side of the plane P; suppose that it is below this plane.
The action exerted at the point M of the surface S by the electricity distributed on the surfaces S), Sj is equal to o; it is the same of the component following N) of this action and of the sum of these components for all the surface S; let us write the equality at o of this sum
S.--.)[S
cos(M',M, N,)
^S f/Si
s Al", iVl J
M'jM
Let }JL be a point of area A and v the normal at this point to the upper face of plane P. The area A and the area S together form a closed surface to which the point M'j is external and the point M'^
-2'j4 LIVRK m. - EXPERIMENTAL STUDY OF DISTRIIÎLTION.
integer. So, according to Gauss's lemmas, we have,
n cos (M')M, Ni) ^^g _^ n cos(M'i;jL, y) ^^ ^ _ ^ -^s M', m' '^a M^'
Scos(M',M,Ni) -^ n cos(M; [ji. V) c Ali' IVl ^-'i \l' .
The previous equality then becomes
1 cos(M', u, v)
fM
l^S,
^à, L^A m;[jl' J
Let us make the two surfaces Si and So tend towards the surface S; the two quantities
^A mY|I' ' ^A M^'
tend towards the common limit
ç\ ros(lM'[j., v)
^A,
A M'fx'
i.e. towards the angle to under which from the point M' we see the area A, this angle being counted positively or negatively according to whether the point M' is below or above the plane P. We thus have
Let Q be the maximum and Q' the minimum of the values of w. The previous equality gives us
JT.-iï ^ Qi ^ 4^-û ^^) - ?- >q;>-I> -
These inequalities can still be written
a' Os a (6) . >^< -.
4tc Q 4''^
CIIAP. II. - OPEN DRIVERS. 29"
As an application of the previous inequalities, we propose
to determine two upper and lower limits of the ratio-^
for a conductive cap resulting from the section of an ellipsoid of revolution elongated by the plane of a parallel {fig- 54)
Let a be the angle of the base plane with the tangent plane all along the contour; 2 ^ the angle at the apex of the cone of revolution circumscribed by the contour from the pole of the cap. It is easy to see that we have
ii = 2a, i2'=27r(i - cos^)
and, therefore,
cos3 Qî
2 ^ Q
We see that when the dimensions of the opening are infinitely small of the first order, the total charge distributed inside the cap becomes infinitely small of the second order. A test body, put in contact with the interior walls of the cap, will not bring back any electricity.
To the preceding theorems, M. G. Robin has added a large number of others; we will refer to his Memoir for the demonstration of the following ones, which we are only stating:
i" A closed surface S {fg. 55) is formed by two caps C and C; if one knows how to determine, on the one hand, the influence exerted by any electric charges on the surface S; on the other hand, the influence exerted on the cap G by a charge placed in a
296 BOOK III. - KXPERIEN'TALE STUDY OF DISTRIBUTION.
current point M' of the cap C, a triple quadrature allows to
Fi g. 55.
determine the influence exerted on the cap C by a load placed at any point in space.
'>-" A closed surface S {^fig- 56) is formed by a zone Z and two caps C and C. Let us assume that we know: i" the
Fig. 56.
electrical distribution on the surface S subtracted from any influence; 2" the influence on the cap (Z + G) of a variable point M' of the cap C; 3° the influence on the cap (Z + C) of a variable point M of the cap C. From these data, we can conclude the distribution of electricity in equilibrium of itself on the zone Z.
This beautiful theorem allowed M. Robin to deduce from the results obtained by Sir W. Thomson for the spherical callus the distribution of the electricity in equilibrium of itself on the spherical zone.
CHAP. III. - LEVEL SURFACES AND ORTHOGONAL TRAJECTORIES. 297
CIIAPTER m.
-VEAU SURFACES AND THEIR ORTHOGONAL TRAJECTORIES.
§ 1 - General theorems.
The potential function of certain electric charges is, . a,
which correspond to values of the potential function varying in arithmetic progression of infinitely small reason ( - a ) Let M be a point of the surface V; let ô be the normal distance of the
l'-is;. 37.
V+2aV+a> V V_a,V_
point M to the surface (V - "); let N be the direction of the normal to the point M to the first surface led to the second; this is the direction of the electric force at the point M, a force whose magnitude is
F = - E
£)N
But we can, if we want, take c/N--o; then we have dS =^- - rt, and, [)ar consequently.
The magnitude of the electrostatic force is, at each point, inversely proportional to the distance between the level surface that passes through this point and the infinitely close level surface.
The surface of an electrified conductor is a level surface. Let us suppose that it corresponds to a certain value Vof the potential function. In the space surrounding this body, let us draw a level surface infinitely close to the surface of the body;
3oo Livniî m. - experimental study: of l.v distribution.
it corresponds to a value (V - a) of the potential function, a being an infinitely small quantity, positive or negative. Let S be the normal distance from a point M of the conductor surface to this level surface. We will have at this point
ÔN _ a
But the surface density of electricity has the value at point M
__ _ J_ dY_
47t dNe'
So we have
a
~~47r8*
The electric density is, in each point of an electrified conductor, inversely proportional to the normal distance between this point and the surface of level infinitely close to this conductor.
Let's give, right now, an application of these general theorems.
Let's consider an electrified ellipsoid removed from any influence.
The potential function of this ellipsoid [Book II, Chapter VIÏ,
equality (17)] depends only, in elliptical coordinates, on the
parameter u. The level surfaces of this ellipsoid are therefore
surfaces
u = const.
Hence these two proposals:
The level surfaces of an electrified and uninfluenced ellipsoid are ellipsoids homofocal to it.
The orthogonal trajectories are lines of intersection of the two families of homofocal hyperholoids at a given V ellipsoid.
The general theorem proved last gives this proposition:
At each point of an electrified and isolated ellipsoid, the electric density is inversely proportional to the distance between this ellipsoid and V infinitely close homofocal ellipsoid.
CH.VP. ni. - DK LEVEL SURFACES AND ORTHOGONAL TRAJKCTORIES. 3oi
Let's go back to the general properties of level surfaces.
Consider an element on a level surface {fig. 58). Through all points on the contour of this element, let us conduct orthogonal paths. These lines generate a surface that bounds an infinitely unbound channel. Chasles, who pointed out the remarkable properties of such a channel, gave it the name of orthogonal channel; it is sometimes also called a force tube.
Let's cut a similar channel by two normal sections c/o>
and c/d)', sections which are obviously located on two level surfaces. The section thus formed is bounded by a closed surface to which we can apply Gauss' lemmas. If we suppose that the orthogonal channel has not met any electrified body between the two sections diù and d\ù we will have
^ F>" - dm -H V's diù' - G.
From this equality it follows, in the first place, that F^ and FJ, are of opposite sign. If, therefore, the force enters the channel at one end, it leaves it at the other. The direction of the force in the channel therefore marks a constant direction of travel.
3uZ BOOK m. - EXPÉKIMENT.VLE STUDY OF L.\ DISTRIBLTIOX.
Second, if we denote by F and F' the absolute values of the force at a point of the elements cIm and c/co', equality (i) can be written
F doi - F' /- being the distance from point (^, J", z-) to pointM.
Apply equality (i) to these two functions and we find
S,
^s'
Let us consider the third integral. Let dS be the angle under which, from the point M, we see the element û?S". We will have
di." ^ R2 of,
r \
[\ being the radius of the sphere. If this radius tends to o, we can easily see that the third integral tends to - 4'^V(M), and the previous equality becomes
"' I ù\\... n i ,. "7, àV
In the second integral, we have replaced the symbol N/ designating the normal to the surface S' towards the interior of the space considered by the symbol N^ designating, which amounts to the same thing, the normal towards the exterior of the surface S'.
In the particular case where the surface S contains no acting charge, equality (2) becomes
(3)
'- OY
^^^'w=SA"^-^^/^^
CIl.VP. IV. - THE LEVEL COUCIIKS. 3o7
From equality (a), we deduce another analogous equality. Let us apply it, in fact, to the space between a closed surface S, outside of which certain acting charges may be found, and a sphere of very large radius. The term relating to the surface of this sphere will vanish when its radius increases beyond any limit, and we will have the equality, true for any point outside the surface S and the acting charges,
'"^-<^')-SA^5^-^i;;''^-S.
If all the acting charges are inside the surface S, this equality becomes simply
(5)
Let us examine the role of equation (3).
When we know the existence, inside the space bounded by the surface i], of a harmonic function V, and when we know
dY in addition the values of V and - - on the surface S, it allows to
calculate the value of V at any point inside S.
The functions of three real variables which verify the equation AV = o have properties which are very similar, in many cases, to those of functions of imaginary variables ('). The previous theorem is one of the main elements of these analogies.
It is known that if a function y(5) of the imaginary variable z is finite, uniform and continuous within a certain area bounded by a closed contour 5 and if x denotes the affix of a point inside this contour, we have
^r.)
^l^fU-)r-..l f /-[^Idz,
(' ) Among the important works for the study of these analogies, let us quote: !'. Appell, 5m/- the /onctions of three real variables satisfying the equation AF = o (Acta matliematica, t. IV, p. 3i3; i884). - P- Painlevé, 5m/- the singular lines of analytic junctions {Annales de la Faculté des Sciences de Toulouse, t. II. B.; i888).
3o8 BOOK III. - EXPERIMENTAL STUDY OF THE DISTRIBUTION.
the contour s being traversed so as to leave the limited area on the left.
The proof of this fundamental Cauchy theorem can be traced (') to the proof of equality (3) given by Green and reproduced above. This theorem shows that one can compute the values of the function f(z) at any point inside an area, if one knows the values of the function at the different points of the contour of this area.
But there is an important difference between equalities (3) and (6). Equality (6) determines the function /(.r) inside the area when one knows only the values of this function at the various points of the contour. On the contrary, in order to calculate the value of V at a point in a certain space, equality (3) requires that we know in Ions the points of the surface that bounds this space, not
not only the value of V, but also the value of its derivative -r^rr
oN/
following the normal to the surface.
Now, it is easy to see that this introduces superfluous elements (-) into the determination of V(M). As we have seen, the demonstrations given in Book 1, Chapter V, § 3, lead to the following result:
For the harmonic function V to be determined unambiguously in all the space inside the surface S, it is sufficient that we know the values of V at all points of the surface S; or the values of V for certain regions of this surface and the
values of ^^ for the other regions. If only the values of -rrr were known at any point on the surface S, the function V
would be, inside this surface, determined to a constant.
However, this apparent difference between the equalities (3) (;t (6) is transformed again into an analogy by a closer examination, because Cauchy's theorem requires, also, for the
(') V. Hkrmitk, Cours d'Analyse de la Faculté des Sciences de Paris, written by H. Andoyer.
{") G. KiRCHHOFF, Vorlesungen iiber matliematische physilc. Median il, , p. i85.
ICVP. IV. - LAYERS OF xrvE.vi". 309
calculation of /(ar), superfluous elements. To determine /(x), it is not necessary to know the values of f{z) at all points of the contour, but only either the real or the imaginary part of f(z).
In any case, the previous remark leads to this problem: Make disappear from the second member of the equalities (3)
,., . d\ . ,.
and ( o) soil -t:-> or V.
It is to make the second member of these eaa liles disappear that Green created the function we studied in Chapter VI of Book II, and, consequently, posed for the first time the problem which received the name of Lejeune-Dirichlet.
F disappearing area V leaving only -r^ is the object of a
similar problem, to which we shall return in Volume II of this work when we study the magnetic distribution, and which we shall call the Lejeune-Dirichlet derivative problem. Depending on whether we want to remove V from the second member of equation (3) or from the second member of equation (5), we find ourselves in the presence of the interior derivative problem, or the exterior derivative problem of Lejeune-Dirichlet.
§ 2 - Fundamental properties of the level layers.
The exterior derivative problem of Dirichlet usually presents much greater difficulties than those presented by the Dirichlet problem. We shall see, however, that there is a special case where it can be solved immediately.
Let's imagine that the surface S is an over/under level enclosing all the acting charges inside. Let A be the constant value of the potential function at its surface. Equality (5) will give, for any point M^, outside the surface S,
But we know, by one of Gauss' lemmas, that the first of the two integrals in the second member has the value o. We have
3lO BOOK III. - EXPERIMENTAL STUDY OF DISTRIBUTION.
so
equality which solves immediately, in this case, the external derivative problem of Lejeune-Diriclilet.
By which equality should we replace equality (7), when, instead of considering a point Me outside the surface S, we consider a point Mj inside the surface S, but outside the acting masses?
We can always lead the surface S' in such a way that the point M,- is external to it; the equality (5) will then give
Equality (2) will then become
S,
i4,
[-
\
)d.^
0,
or else,
à
cause
of the
cheerfulness
to'r
dz =
47:A,
which respects
Ite
imm€
îdially d(
he Gaussian lemmas,
(8)
Uv r
dz^
.iTlA.
Let us look for an interpretation, of equalities (7) and (8).
We will name a level layer an electric layer distributed on the level 2 surface and having in each point a surface density determined by the formula
I ^V _ _i_ ^ 4ti d\e "^411 àNe
The potential function of this electricity will have as value at any point M in space the quantity
U(M) = C ?<^X.
CIUP. IV. - LEVEL LAYERS. 3ll
Then, according to equality (8), at any point inside the surface S,
(9) U = A,
and at any point outside the surface S, according to equality (n),
(10) U = V.
It is concluded that a layer of level exerts, outside C the surface on which it is spread, the same action as the electric charges located inside V this surface and that it rC exerts no action in a point interior to this surface.
Equality (9) shows that this layer, distributed on a conductor limited by the considered level surface, j would form an electric layer in equilibrium. Hence this new theorem:
If Von knows how to find the level surfaces of a system of electrified masses, which contain these masses in their interior, we know how to find the distribution that would affect V electricity on a conductor limited by any of these surfaces. It is enough to give to the density in each point of this surface a value in inverse reason of the distance of this point to the surface of level infinitely close. This distribution admits the same external level surfaces as the charges originally considered.
The total amount of electricity that forms the level layer has the value
The Gaussian lemmas immediately lead to the following theorem:
The total mass of a level layer has the same magnitude and sign as the masses whose action it can replace for the outer points.
3l2
BOOK III.
ETCDE EXPERIMENTALE DE LA DISTHIBITION.
These beautiful theorems were discovered by Grcen ('). Green's memoir was unknown when these propositions were found, almost simultaneously, by Cliasles (2), Ganss (-') and Sir W. Thomson (^).
Complete study of an electrical influence case.
The previous theorems lead us immediately to the solution of an interesting influence problem.
]^ bodies G, G',- G" are charged with electricity {fîg. 61); i] el ^'
are two charge level surfaces distributed over G, G', G"; ^ envelops the electrified bodies and S' envelops '^. These two surfaces limit a conductive layer. N/, N^ are the dii'ections of
( ' ) G. Grken, An Essay on the application of matheniatical Analysis to the Tlieories of Electricity and Magnetisni. Art. l'i. Noltingham, 1828. - Grcen's Matheniatical Paper's, p. 63.
(^) Chasles, Mémoire sur l'attraction d'une couche ellipsoïdale infiniment mince et les rapports qui ont lieu entre ces attractions el les lois de la chaleur dans un corps en équilibre de température {Journal de l' École Poly technique , t. XV, 25' Cahier, p. 3o4-3i6). - Énoncé de deux théorèmes généraux sur l'attraction des corps et la théorie de la chaleur {Comptes rendus des séances de l'Académie des Sciences, t. VIII, p. 209; iSSg). - Théorèmes généraux sur l'attraction des corps {Addition à la Connaissance des Temps for i845: published in 1842).
(') Gauss, Alegemeine Lehrsàtze ûber die im verkehrten Verhàltnisse des Quadrats der Entfernung wirkenden Kràfte. Art 37 {Magnetische Verein; 1889. - Gauss Werke, Bd. V, p. 241).
(*) W. Thomson, On the uniform motion of heat {Cambridge Matheniatical Journal; 1842. - Reprint of Papers on Elecirostatics and Magnetism, p. i el p. i3i).
CIIAP. IV. - LEVEL LAYERS. 3l3
the normal to the surface -; N) , N^. are the directions of the normal to the surface S'.
On the surface H', let us distribute a level layer; its density at each pointJ has the value
_ j ^)r _ I d\'
On the surface S, let us distribute a layer of level changed of sign; its density in each point has the value
_ I 0\ _ I d\
Let be:
U', U the potential functions of the two layers we have just defined; A, A' the values of V on the surfaces S, S'; P a point inside the surface i]; P' a point between the surfaces S, }i]';- P' a point inside the surface S'.
According to the theorems of the previous paragraph, we will have :
1" At point P,
(II) U(P) = - A. U'(P) = A':
-^° Au |)oint P', {VI) U(P') = -- V(P'), U'(P')=:A';
3" At point P",
(iS) U(P')--- VC-P"), U'(P'') = V(P").
Let W be the total potential function, defined at each point by the equationC
W = V -^' U + U'.
i" According to the equalities (i 1), at point P, we have
(14 ) W(P)-= V(P)-i-A'-A.
2** According to the equalities (12), at point P', we have
(i5) W(P')-.A'.
3l4 LIVIIE m. - EXPERIMENTAL.VLK STUDY OF LV DISTRIBUTION.
3" According to the equalities (i3), at point P", we will have
(i6) \V(P") = V(P").
Equality (i5) leads to the following theorem:
i" The electrical equilibrium is established on the conductive layer limited by the surfaces S, S'.
The properties of the level layers give iminediatomenl these propositions:
2" The layer o-' is equal in sign and in quantity to the charge OÏL spread on the conductors C, C, C"; the layer a is equal in size and of opposite sign to the charge SW. The equilibrium standard obtained on the conductor limited by these two surfaces is therefore the equilibrium state of this conductor supposed to be isolated and in the neutral state before having been influenced by the OÏL charges.
3° The layer i' would be in equilibrium of itself on the conductor which carries it.
4" The layer a- would be in equilibrium by itself on a solid conductor limited externally by the surface S.
Equalities (i4) and (i6) still give this proposition:
5° The electrostatic action exerted either in a point of the enclosed space of the hollow conductor, or in a point of the space outside it, is reduced to the action of the inductive charges.
§ 4 - A particular class of capacitors.
The properties of the level layers will also allow us to study in a complete way a particular problem of electrical condensation (' ). ,
If electricity were in equilibrium of itself on the conducting mass G {fig. 62), it would admit certain level surfaces; let S, S' be two of these level surfaces, the surface S' surrounding the surface S; between these two surfaces S, S', let us flow a conducting material C. We will have a system of two conductors (') J. MouTiKU, Cours de Physique, t. I; 1881.
CHAI". IV. - THE COLCIES OF .MVKM:. 3i5
their of which one, C, bears the name ([''internal armature, the other, C, of^ external armature. The spherical Lejde bottle realizes a similar system; a cylindrical and very long Lejde bottle realizes it approximately.
By placing one of the armatures in communication with a
Fis. 62.
If the source is at a constant potential level (') and the other armature is in communication with the ground, we will have a capacitor; this capacitor will take two different forms depending on whether the external or the internal armature is in communication with the source.
i" The internal armature is in communication with the source at potential level A. The external armature is in communication with the ground.
To determine the electrical distribution on the system, let us designate by V the potential level at which the conductor C would be carried by a charge equal to the unit in equilibrium of itself at its surface. The surfaces S, 2' would be level surfaces of this charge, and they would correspond to values u and u' of the potential function.
Gela posed, let us distribute on the conductor C a charge a in
C) The internal armature can only be connected to the source or to the ground if the external armature has a small hole. We will neglect the disturbing influence of this small hole.
3l6 BOOK III. - EXPERIMENTAL STUDY OF DISTRIIILTION.
equilibrium of itself; on the surface S vme level curve changed sign of mass - 6 = - a; on the surface ^' let us place no electricity.
Let Via be the potential function of the charge "5 let U be the potential function of the charge b. If we notice that the function V takes the value av at a point of the surface C and a value au at uu point of the surface S, we will easily see, from the properties of the level layers, that the total potential function has the value o at any point between the surfaces 2 and S', and a(v - u) at any point of the conductor C. If therefore we have been careful to determine the charge a by the equality
, .%
(17) a = >
i^ - u
we will have obtained the electrical distribution on the capacitor.
The first coefficient of Gaugain [Liv. II, Ghap. XI] is defined by the equality ;;? := -- We have therefore, in the present case,
(18) m - i.
To obtain the coefficient m' , we must isolate the external armature with a load - b, equal here to - a, and put the internal armature in communication with the ground. This one will take a
ciiarge "i and m' will be the ratio -j-' We can easily determine a,.
On the surface of the body G, let us distribute a charge a, in equilibrium with itself; on the surface S, a layer of level changed of sign of total mass - a, ; on the surface S', a layer in equilibrium with itself, of total mass - 6 + "). It is easy to see that the equilibrium will be established on the system; that the conductor G' will carry a total charge - b\ that inside the conductor C, the total potential function will have a value
"i c - ttiU - {b - ai)u .
We will determine ", by equating this value to o. We will thus have
(19) m - -y = ; j
CH.VP. IV. - THE LEVEL LAYERS. 'ilj
The condensing force of the device has the value
I - mm or, according to equalities (18) and (19),
(20) F = n ^-
V - u
■.i° u external armature is in communication with the source at potential level =1. ; V internal armature is in communication with the ground.
We will then have
( 17 OIS ) a =^ - , ,
II
(18 /day) m = r >
v -^ u - u
(196/.V) m'.^i,
{10 fjis) r - I H .
V - u
The condensing force of the device is the same in both cases. But the load taken by the collector is not the same.
Let us apply the previous results to the case where the internal frame is a sphere of rajon R and the surfaces 2, S' are spheres, concentric to the first one, of radii p, p'. We have then
The condensing force of the device has the value
I I
ÏÏ~p
When we take the internal armature as a collector, it is charged with a quantity of electricity
a = -
I I
ÏÏ-p
I
1
u = -,
u
=:
p
P
3i8
LIVRB m.
EXPERIMENTAL IDEA OF DISTRIBUTION.
If, on the other hand, we take the external armature as a collector, it is charged with a quantity of electricity
a - pM,.
§5.
Absolute electrometer by Sir "W. Thouison.
The electrometer of Sir W. Thomson's electrometer may be compared to the capacitors we have just mentioned. However, we shall expose directly the theory of this device as the last application of the properties of level surfaces.
Two conductors C, F are present (//^. 63). These two
Fig. 63.
conductors are partly located in a region of space bounded by a surface S, but they extend to distances from the surface S that are very large compared to the dimensions of this surface.
Not only inside the surface S, but also at a great distance from this surface, the surface which limits the conductor C has the shape of two parallel planes P and P', the surface which limits the conductor F has the shape of two planes II, II', parallel to these two.
Conductor F is connected to the ground and conductor C to a source at potential level a. It is easy to find the electrical distribution on these two conductors - we are only interested in the distribution on conductor F.
Within the surface S, the level surfaces can obviously differ only slightly from parallel planes. Let x be the
CUAP. IV. - THE LAYERS OF MVE.VU. 3 KJ
dislance of a point from the plane P. The Laplace equation, which the potential function must verify outside of the acting masses, is then reduced, for those points which are inside the surface S, to the form
dyy _
If one designates by D the distance OA of the two planes P, II; by E the thickness 1111' of the plate F, one sees without difficulty that, in the space included between the planes P and D, one has
(21) \- -x^o.
In the space beyond plane II', it is easy to see that
(29.) V = 0.
Let N be the exterior normal to the conductor F at a point on the plane n. Let N' be the exterior normal to the same conductor at a point of the plane n'. Equality (21) gives
5N ~ 1)' and equality (22),
dy_ _ dN' ~ ^'
The plate F does not carry electricity on its face II'. On its face n, it carries an electric layer whose density at each point has the value
(7
Let's suppose that a cylinder MM'NN', normal to planes II, II', divides the conductor F in such a way as to isolate a moving disk. In order that this disk and the remaining part of the conductor F (guard ring) continue to form a single conductor, electrified as we have just indicated, a wire F joins the disk to the guard ring. Let 2 be the base surface of the mobile disk. This disk, carrying electricity only on its lower base, is subjected to a force, directed along N, and having for magnitude
320 BOOK in. - EXPERIMENTAL STUDY OF DISTRIBUTION.
In other words, the moving disk is attracted by the fixed disk, and the attraction has the value
^^8^ï^"'
It will suffice to balance this attraction by a known force, to obtain the determination of the absolute value of the potential level.
CHAP. V. - THE GREEN PROBLEM AND FARADAY'S THEOREMS. 32t
CHAPTER V.
THE GREEN PROBLEM AND FARADAY THEOREMS.
§ 1 . - Green's inner problem. - Green's solution.
The properties of level layers give the immediate solution, in an extended case, of the following problem:
Given electrified masses and a surface surrounding these masses, distribute on this surface an electric layer exerting, at any point outside the surface, the same action as the given masses.
This problem was first tackled by Green ('); Green showed that this problem could be solved when one knew how to find, for the space inside the given surface, the function to which Riemann gave the name to Green's function.
Let's prove, first of all, that the problem we have just stated cannot admit several solutions.
Let us suppose, in fact, that on the same surface, two distinct layers have been distributed, having for density at point M one a-, the other t'. Let V be the potential function of the first distribution and V the potential function of the second.
The two distributions exert the same action at a point outside the surface; therefore, outside the surface, the two functions V and V admit the same partial derivatives. If we add that they are both equal to o at infinity, we see that at any point outside the surface, we have
V-V'=o.
(' ) G. Green , An Essay on the application of mathematical Analysis to the theories of Electricity and Magnetism. Arl. 5. Nottingham; 1828 {Green's Mathematical Papers, p. 3i).
D. - I. 21
322 BOOK III. - EXPERIMENTAL STUDY OF DISTRIBUTION.
The two functions V and V having the same value on the surface in question and being both harmonic inside this surface are identical between them inside this surface. We have therefore, in all space, the identity
V - V = o.
Let N/, Ne be the interior and exterior directions of the normal in M to the given surface. We will have
OR, by virtue of the previous equality,
This is what we announced.
Assured thus of the equivalence of all the solutions of Green's problem that can be obtained by various procedures, let us first expose Green's solution.
First, we propose to solve Green's problem for the case where the surface S contains only one electrified point P {/ig'- 64 ) carrying a charge /?z.
Fig. 64.
Let G(M/) be the value at the point M/, inside the surface S, of the Green's function of pole P. We know that, if /* designates the distance M/P, we can write (Liv. II, Chap. VI, §2)
G(M,-) = r(M,) + ;J:,
r(M/) being a harmonic function inside the surface S and equal to ( ) on the surface S.
CHAP. V. - THE GREEN PROBLEM AND FARADAY'S THEOREMS. 3^3
This being so, let us consider a function V(M) thus defined: 1° At any point M/, inside the surface S, we have
V(M,)=-/nr(M,);
2° at any point M^, inside the surface S, we have
The function V(M) is continuous in all space, equal to o at infinity, harmonic outside the surface S, harmonic inside the surface S; it is thus the potential function of an electric layer distributed on the surface S with density
This layer having, at the points outside the surface S, the same potential function as the mass m, solves Green's problem for the case where the surface S contains a single acting mass.
If the surface S contains several acting masses, we will solve Green's problem by superimposing the distributions which would solve this problem for each of the masses taken in particular.
We can see that we know how to solve Green's problem if we know how to find Green's function for the space inside the surface S. It is easy to recognize the exactness of the reciprocal.
§ 2 - Gaussian solution.
Without knowing Green's Memoir, Gauss was led, for his part, to ask himself the same problem (' ); he was the first to propose a general demonstration of the existence of one and only one solution to this problem; according to what we have just said, it is there, by
(') Gauss, Allgemeine Lehrsâtze ùber die im verkehrten Verhàltnisse des Quadrats der Entfernung wirkenden Kràfte, Art. 29-34 [Magnetische Verein, 1889 {Gauss Werke, Bd. V, p. 281)].
324 BOOK III. - EXPERIMENTAL STUDY OF DISTRIBUTION.
as a counterpart, a demonstration of Texistence of Green's function and of the correctness of the so-called Lejeune-Dirichlet principle. This demonstration by Gauss is not free of criticism; nevertheless, it presents fewer contentious points (' ) than the demonstration of Lejeune-Dirichlet, reported in Book II, Chap. V, § 2. We shall therefore explain this demonstration here.
Theorem I. - A surface S {fi g- 65) contains in its interior electric charges whose potential function is U. One can always, on the surface S, distribute a mo
nogen of given total mass 011, so that the difference between the potential function V of this layer and the function U has the same value at all points of the surface S.
Let us consider, in fact, a monogenous distribution, of given mass 31i, on the surface S; let us suppose, to fix the ideas, that this distribution is positive. Let u be the density of this distribution at the point M of the surface S.
The quantity V will certainly be positive at any point of the surface S. The same will be true for the quantity
This quantity is therefore limited to a smaller amount. The same applies to the quantity
- CaUcrcfô,
(' ) See, on this subject, Carl Neumann, Untersuchungen iiber das logarithmische und Newton'sclie Potential, p. XI and p. 107; Leipzig, 1877.
CHAP. V. - THE GREEN PROBLEM AND FARADAY'S THEOREMS. 325
because the latter can never become less than
X being the largest of the values of U on the surface S. The quantity
Q= C(V - 2U)c7o.
If the external forces which solicit the system admit a potential, this proposition can be transformed into the following one:
For V equilibrium of the system, it is necessary and sufficient that the value of the total thermodynamic potential
is a minimum among all the values that this quantity can take at the same temperature.
The internal thermodynamic potential has a large number of properties, some of which will be useful to us later.
i" Assume the state of the system defined by the temperature T and by a number of independent parameters a, , ao, . . . , a^^.
CIIAP. r. - NOTIONS OF THERMODYNAMICS. 34 1
Let's put
To maintain the system in equilibrium in the state defined by the value system
ai, (Xo, ..., %n, T,
it is, by hypothesis, necessary and sufficient to place it in an enclosure at the same temperature as it and to apply certain external forces to it. These forces must be such that the work done by them in any isothermal modification of the system has the value
Pi 5ai -f- P2 072 -H . . -^- P" 02t" In particular, to maintain in equilibrium a homogeneous body whose specific volume is r, it is necessary to apply a normal and uniform pressure whose value is
2° Let us suppose that the parameters a,, ao, ..., a^ are chosen in such a way that, if T varies, ai, ao, ..., a" remaining constant, no external work is done, and no live force taken by the system. This is what will happen if all the parameters on which the shape of the system depends are among ai, ao, ..., a". We will then have
(9) ^=-^^'
(10) ,:?-T^ = EU.
3° In these conditions, let g?Q be the quantity of heat released by the system during any isothermal modification. We will have
(") ErfQ = -3(.f-T^f)+^5.-S2=^ In the particular case where the modification is reversible, we have
8> = o, oj + c?fce=o.
34-;i BOOK IV. - THE THERMODYNAMIC POTENTIAL.
and the previous equality becomes simply
(,.) E "rQ = a(T^).
4*^ We will often give the name of compensated work done in a modification to the quantity
(i3) t^& = - ET8S.
If we put (i4) H=ETS,
in any isothermal modification, we will have (i5)
3" Of the second group of added forces; in the considered modification, these forces perform, by hypothesis, the work of^].
CIIAP. I. - CONCEPTS OF THERMODYN.VMIQUK. 345
So we have
f/8 = d<£e - divg -+- dxB'i = d(E'i,
and equality (19) becomes
dz = - E 0(1] - TS) -h dS'i.
According to the inequality (18), this quantity is zero or negative; therefore, according to the propositions of Thermodynamics that we recalled at the beginning of this paragraph, the corresponding modification, leading to zero or negative uncompensated work, cannot occur; the system, subjected to the mechanical actions that really act on it and to those that we have added, cannot undergo any displacement that does not alter the state of its various parts; the connections that forbid any modification other than displacements of this kind ensure the equilibrium of the system.
But, thanks to these connections by which each of the bodies that constitute the system is supposed to be maintained in an invariable state, the propositions of Rational Mechanics relating to systems formed of solid bodies are applicable to the preceding system. In particular, the principle of virtual velocities can be applied to it.
From the above, the system is in equilibrium under the action of four given force systems:
1° The external mechanical forces, which acted primitively on him;
2° The internal mechanical actions, that the various bodies which constitute it exert on each other;
3" Equal and directly opposite forces;
4° Forces performing, in any virtual displacement, a work cKb'^ equal or lower than Eô(U - TS).
The first and third group of forces destroy each other. We can therefore say that the system is in equilibrium under the action of the second and fourth group of forces.
But, in any modification of the system where its various parts change position without changing state, the forces of the second group perform a virtual work d(Ei; the forces of the fourth group eff'ect a virtual work d(B'^; by virtue of the principle of
3^6 BOOK IV. - THE THERMODYNAMIC POTENTIAL,
virtual velocities, the system cannot be in equilibrium under the action of these two groups of forces if we do not have
d(s>i -+- d^'i le.
This must take place whenever d^\ verifies the inequality (i8). We must therefore have
6/S,-^- E8(U - TS).
If we now suppose that to any virtual displacement of the various parts of the system we can make the inverse displacement correspond, we will easily see that the preceding inequality reduces to the equality (17),
i - say,
which can be stated as follows:
When a system undergoes a displacement without change of state of its di^'crses parts, the uncompensated work produced in the system is equal to the work produced by the given mechanical actions, both external and internal, acting on the various parts of the system.
CorollaryW. - We have seen that the entropy of the system was related to its internal tliermodynamic potential by the equality (9),
ES - - .
If therefore d^ designates the compensated work that accompanies a
CHAP. I.- NOTIONS OF THERMODYNAMICS. 847
modification without change of state, we have
But ( - ùj) can, in this case, be replaced by the variation that the potential of the internal mechanical actions of the system undergoes. If these mechanical actions are independent of the temperature, it will be the same of their potential and we will have, consequently,
/y, assume that this charge is uniformly distributed in volume vi. Gela posed, it is easy to demonstrate that the quantity /y depends neither on the shape of the surface which bounds the volume Vi, nor on the orientation of this surface with respect to that which bounds the volume Vj.
Whatever the shape of the surface which limits the volume p/, q,uatever the orientation of this surface with respect to that which limits the volume Vj^ we can always subdivide the volume vi into an unlimited number k of infinitely small cubes ", b, . . . , l, all equal to each other and to an infinitely small cube taken as type, having all, with respect to the surface which limits Vj, the same orientation independent of the orientation of the surface which limits Vj. l, all equal to each other and to an infinitely small cube taken as a type, all having, with respect to the surface which limits Vj, the same orientation independent of the orientation of the surface which limits vj.
Let (^aj be the quantity analogous to ^ij relative to the cube a and the volume çj. It is obvious that we must have
*/y = ^aj H- Ç/v + -..-+- O/j. The amount of electricity contained in each of the small cubes a, b, ... ^ /has the value ^- It is the same for all. The parameters which define respectively the systems
a, /; b, j; ...; l, /
have therefore the same value or approximately the same value for each of them; therefore, we have approximately
and
11 It is easy to see that the second member does not depend on the
CHAP. II. - THERMODYNAMIC POTENTIAL OF AN ELECTRIC SYSTEM. 35 1
the shape of the surface that limits the volume P/, nor the orientation of this surface with respect to the surface that limits çj; the stated proposition is therefore demonstrated.
We would also see that $/y does not depend on the shape of the surface that bounds t'y, nor on the orientation of this surface with respect to the surface that bounds c/.
Let /;//, nij be the masses of the two parcels ç^/, vj; let T be their temperature; let a, [3, ... be the parameters, such as the densities, which fix their physical and chemical state. From the above, we will have
"ï>/y= *(m,, /n;, Ci, qj, nj, T, a, p, . . .).
We will show that the quantity ^ij is a linear and homogeneous function of the niasse nii of the parcel vi and the electric charge qi it carries.
Let us divide the volume Vi in any way into a number of equal or unequal volumes ",6, . . . , /. Let m^, m^, . . . , m/ the masses of these elements; let qa-, qt-, - - - , <7^ be the electric charges they carry. For all these elements, the parameters ï, a, p, ... are approximately the same; we have also approximately
raj = r/,j - ...= nj = rtj.
We must therefore have
*(m/, qi) =i^(/na, qa)-i-^{mb, q/,) H-... -H ^{ni/, q/),
equality in which we have highlighted only the parameters that do not have the same value for all elements. But :
1° The masses ma-, mi,, ..., mi are subject to this single
provided that
nia -{- ntb-r. . .-+■ mi= mi;
2° Since the distribution of the charge qi on the element Vi could be chosen arbitrarily, we see that /y is a linear function and homo
35jt BOOK IV. - THE THERMODYNAMIC POTENTIAL.
£>ene of the mass mj of the element py, and of the electric charge qj that it carries.
The results we have just obtained are expressed by the following formula
(3) *,y= mimjOijirij) -\- mjgi'i^ij{rij) -+- miqj^ji{rij) + q/gjyjjirij),
in which it should be noted:
i" That the functions cp, y, "]; can change their form when the parameters T, a, ^, - . . change their value ;
2" That one has well, by reason of symmetry,
but not
It is easy to see that, to form the quantity ^^', given by equality (2), we can, by virtue of equality (3), operate as follows.
First, for the volume Cj, we will form the quantity
[§\= - /ni["ï2 9i2(''i2)-i- "Î3 ?i3(^i3) + - .-4- /n "çi"(/-,")]
(4) I + qiV'n^'^iiirii)-^ nii<]^],z{rii)-^. ..-^m,i<]/ui{rin)]
f ■-+- - 9'i[5'2Xi2(''i2) + S^s/isC^'ia) -H...+ ^"Xi/i(ri")].
Then we will form the same for the volumes c'2, <';;, ..., v,, the quantities §'^, ^'^, . . . ., éF)^. We will then have
(5) .r= #1 + J'2 + ---+^/r
Suppose that the volume element Vi is part of a body of which A' is the density at the point of coordinates {x, z'). Suppose further that the electricity is spread inside the body and that p' is its solid density at the point {x, z'). We will have, by virtue of equality (4),
♦
,fi= Wi / / / àJ oi^r) dx' dy dz' -^ Çi f I f ^' '^ {r) dx' dy' dz'
\^^JJj ?' X{'') dx' dy dz' .
CHAP. H. - THERMODYNAMIC POTENTIAL OF AN ELECTRIC SYSTEM. 353
Each of the three integrals
\] ^ Ç f Ç \'o{r)dx' dy'dz', (6) i \ = f Ç f i^' <^ {r) dx' dy' dz\
f W = f fp'y.ir) dx' dy' dz\
in which /- denotes the distance from the point [x',y',z') to a point of the element <,, must exist-, which requires that each of these integrals, extended to any uniquely small volume including a point M in its interior tends to o when this volume somehow vanishes at point M.
Once this has been done, let us come to the determination of the quantities y*),y2, ...,y", which appear in equality (i).
The quantity fx does not depend on the Jorme of the surface that limits the volume t^j.
To prove this first proposition, we will rely on a lemma that follows immediately from the proposition we have just established.
Let us imagine that we divide the element t'i in any way into an infinity of other elements ", b, . . . , /, of volumes "a, Ubi - - - 1 ui. Let/rt be what becomes the quantity y, for the element a. Let Ua, \ai Wa be the values, at a point of the element a, of the integrals given by the equalities (6), these integrals extending to the volume V). It is easy to see that we will have
-^ qaya-\- qb^ b-'r ... .-^ qi\ t-^ 7 (<7a\Va + C'A Wô-t- ... -+- 5-/ W/) ;
but
Ua, U,, ... U/,
V", V,, ... V/, W", W/" ... W/
are, according to what has just been demonstrated, infinite quantitiesD. - I. "3
354 BOOK IV. - THE THERMODYNAMIC POTENTIAL.
small as <). We can therefore write
(7) /i = /a + /^. + ... + //.
With this lemma, let us prove the stated proposition.
The distribution of the charge q^ on the volume V is indifferent, so let's suppose it is uniform and let's divide the volume V into A" small cubes ", ^, . . . , / equal to each other and to a cube taken in advance as a type. These cubes will be identical. We will have then
and, by virtue of equality (7),
The number k depends on the volume of the element (^,, but not on its shape; the same is true of /"; therefore, the same is true ofy*,, which had to be demonstrated.
The quantity f^ is a linear and homogeneous function of the mass mt and the load q^ of the clothing p,.
To prove this proposition, we divide in some way the element C) into k other elements a, ^, ...,/, having masses m^, m^, . . . , m/ and carrying charges ^a, qi,^ . . . , qi. Equality (7) will give us, by only highlighting, among the parameters on which y, depends, those which vary from one element to another,
/i(m,, qx)^fx{ma, qa)^fï{r)ib, Qh) ^ - . .-^ fi{>n,, g/).
But we'll note:
i" That the masses m", mi, . . . are subject to the only condition that we have
0.° That the distribution of the charge ^, on the element ç, being arbitrary, we can always dispose of the charges qa-, qb, - - - -, qi-, subjecting them to this single condition
<7a-+- ^6-*-----H qi- q\ Hence, the resulting equality demonstrates the previously stated proposition.
CHAP. II. - THERMODYNAMIC POTENTIAL OF AN ELECTRIC SYSTEM. 355
In summary, we have
(8) /i= niiFi-hqiGu
F,, G) depending on the temperature of the element c, its density and the parameters that determine its physical and chemical state.
The set of equalities (i), (4), (s), (8) determines the form of the thermodynamic potential of an electrified system.
§ 2 - Introduction of the fundamental hypothesis of compressibility.
We have seen that, if we move the various parts that make up a system relative to each other without altering the state of each of these parts, the forces given inside the system perform a work d^i related to the variation of the internal thermodynamic potential by the equality
(9) d(Bi^ - Brf.
Let us apply this theorem here. Let us displace with respect to each other the volume elements Pj, (^25 - ■ --, ^n into which we have decomposed an electrified system, without changing the state of these various elements. We will have, from the equalities (i), (4), (5),
8^ = 8#'i -t- S^'j + . . . -f- 8 j; and
2 L '''■lî ^'13 ^''in J
Equality (9) then leads to the following consequence:
The given internal forces acting in a system whose various parts are electrified fall into three classes:
\Two material masses /??,, m^, electrified or not, exert an attraction on each other
/"i nii ' .
orn
356 BOOK IV. - THE THERMODYNAMIC POTENTIAL.
proportional to the product of the two masses, depending on their distance and their physical and chemical state; this force is subject to the rule of Inequality of action and reaction.
1° Two electrified masses, carrying charges ^,, ^o, exert V one on Vautre an attraction
<(Ji - -,
proportional to the product of the two charges, depending on their distance and their physical and chemical state; this force is subject to the rule of equality between action and reaction.
3° A material mass m^ exerts on an element of volume carrying an electric charge , an attraction
proportional to the product of the mass by the electric charge, depending on the distance of the two elements and their physical and chemical state; this force is subject to the rule of equality between action and reaction.
Let us consider, on the one hand, the volume element p,, and, on the other hand, all the bodies which compose the universe, minus this volume element.
It is obvious that, in order to completely define the properties of the element v^ at a given moment, it is necessary to know completely the state and the relative position of all the bodies which compose the universe.
But, in Thermodynamics, it is generally accepted that, for the density of a volume element to be determined, it is sufficient to know :
i" The other parameters that define the state of this element supposedly isolated by thought from all the other bodies that make up the universe;
2° The given forces that the other bodies of the universe exert on it;
3° The connections imposed on it by the bodies that surround it.
CHAP. II. - THERMODYNAMIC POTENTIAL OF AN ELECTRIC SYSTEM. 357
This being said, let us imagine that the element (^i, which, for the sake of brevity, we will assume to be in a neutral state, and a certain number of elements v-i, ..., Çfi which surround it form a system; the parts of the universe excluded from this system exert external forces on it. Let us imagine that the element f, expands by ùVi. The other elements of the system are displaced without the volume and the state of each of them undergoing any variation. The density A, at a point (the element k, varies by oA,. The internal thermod^^namic potential of the system increases by
tj = nix - ^f^-^ Sa,
,.
Now this pressure is given by the equality
nii
.P.,..),,
nii
{-'t-
771 1
i^'t--'
oAi
SA] = G.
For it to have the same value as in the previous case, it is necessary and sufficient that we have
'"^Ir-^^^/AT^"'
whatever the quantities m^ and ^2" Lies two functions 'fi^,
(^7-12
the function cp,2 being able to change form with the physical and chemical nature and with the temperature of the two molecules m, and mi^ but not being able to depend on their density.
1° A material mass nit and an element carrying an electric charge q^ exert a mutual attraction
the i2i carrying electric charges Çi, q-y, are located at a distance /■^2 greater than a certain very small length \ which we will name the radius of molecular activity, their mutual attraction reduces to
As a result, when /"ja surpasses )>, the quantity ^(2 differs only from
by a term independent of /(o, and this term is zero, since d/,2 must cancel when /(o becomes infinite. We have thus
Xl2(''l2)= ;r-®i2(/-12),
'12
the quantities Si2i ^a, '^{2 depending on the physical and chemical state, on the temperature of the elements Ci, Co, but not on their density in the case where we admit the hypothesis indicated in the preceding paragraph. These quantities become equal to o when /'la exceeds 1.
Finally, let us add this hypothesis:
The sum
©12(/'l2)'/2-i-
''12 ''13 f'in
the sums being extended to all the elements of the system other than the element Vt. Let us then pose
(i2) Pi= 2Fi + £i(rn)m2-^£n{ri3)ni3-i-...-h£in{rin)>nn,
(i3) 01= Gi-+- t{^i(ri2)m2-4-i3(/'i3)/"3-t-----i-4'i"(''i")"^""
the sums extending to all elements located in a sphere
of radius A having its center in a point of the element (^|). We
will have, by virtue of equalities (i), (4), (5), (8),
£ = - - ( mi iii -f-"12 i22 + - - -+" "^n^n )
, , , + -(g-i Vi-(-72 V2-+-...+ ^,i V")
(>4) i -^
+ - ( mi Pi H- /?l2 P2 + . . . -r- nin P/i )
Each of the four summations in the second member extends to the entire system.
Each of the four terms in the second member has a remarkable meaning.
The internal actions that are exerted in an electrified system are, under the assumptions made, of four kinds:
1° Two material masses exert on each other an attraction given by the law of universal gravitation; the potential of these actions is the term
(niiQi -\- m^Qi-h . . . + mn^n) The study of this term constitutes Celestial Mechanics.
362 BOOK IV. - THE THERMODYNAMIC POTENTIAL.
2° Two material masses, located at a distance less than the radius of molecular activity, exert on each other an attraction that depends on their physical and chemical state and their temperature. These forces, called capillary forces, have for potential the term
- (miPi -\- m2P2 + . .-I- /n "P").
2
The study of this term constitutes the study of capillarity, in the general sense that we can give to this word today. We have sketched elsewhere (^) how Thermodynamics could bring out from this starting point the whole study of capillarity.
3" Two electrified masses exert on each other a repulsion given by Coulomb's law. These forces have for potential the term
-(g-iVi-^- <72V2-t-...H-gr„V„).
4° Two masses "^^, m^, carrying electric charges ^,, ^2? and located at a distance smaller than the radius of molecular activity, exert on each other an attraction
()4^i2(ri2) to<]t.2i{ri2 + ...-+-/n "a") -I- - (m, Pi -i- m,P,+...H-m "P").
On the other hand, let W be the electrostatic potential of the system; we have
With these notations, equality (i4) becomes (i5) i = E(r - TS)H-\v+e,^,+ e2^2 + -.- + Q "7". ■
This is the form of the internal thermodynamic potential of an electrified system which we had already arrived at in 1886 (^).
The formulas (9) and (10) of the previous chapter immediately give us the internal energy and the entropy of an electrified system. If we put
AT
(-7) K^e-T^,
(' ) II. Hklmholtz, Ueber die Erhaltung der Kraft, p. 4? (Berlin, 1847Helmholtz, Wissenschaftliche Abhandlutïgen, t. I, p. 48). (' ) P. DuHEM, Le potentiel thermodynamique et ses applications, p. 209.
364 BOOK IV. - I,K THERMODYNAMIC POTENTIAL.
we will have
(i8) ES =ES-i-IIi^,-f-Il2 5'2H--..+ H"^",
(19) EU= Er + W-4- K,^i-i-K2^2 + ...-(- K"^".
In what follows, we will often have to use these formulas (i5), (18), (19).
§ 4 - On the continuity of the quantity e.
By focusing on the principles we have just explained, we will arrive, in the next chapter, at this consequence:
The quantity
(i3) e, = Gi+ 4/12(^12) m2-+-4'i3(n3) "î3-)-. ■. + ^\n{rxn)inn
varies continuously with the coordinates x^^ j^,, z^ of a point of Vêlement v^^ even when this point crosses the surface of discontinuity which separates two bodies of different nature.
Let us see, right now, what the correctness of a similar ])roposition implies about the quantities G and <^^^.
We have seen (§ 1) that, if A' is the density of the element dx'dy'dz'^ the integral
///
A' (^ ( /') dx' dy dz' ,
where /- denotes the distance from a point of the element dx' dy' dz' k to a point of the element (',,, and which is extended to an infinitesimally small volume comprising the element Vi, must be infinitesimally small regardless of the shape of that volume.
Based on this lemma, we can easily show that the previous proposition is indeed true if we admit the two following hypotheses:
1** The quantity G, is an absolute constant which does not depend on any particular property of the element Vi.
1" The quantity (|>,2, while depending on the physical and chemical state of the temperature of the mass m2, does not depend on any of the properties of Vt which carries the mass q^ .
CHAP. II. - THERMODYNAMIC POTENTIAL OF AN ELECTRIC SYSTEM. 365
We will admit these hypotheses, the second of which is in accordance with Hehnholtz's way of seeing as explained at the end of the previous paragraph.
We will also admit, in what follows :
i" That the first order partial derivatives of quantity 6 vary continuously in all space;
2° That the second order derivatives of the quantity 6 only experience discontinuity at the points where the matter is discontinuous.
It is easy to show that one can make assumptions about the function (]/(2) which lead to the two results we have just stated ( ' ); we will not dwell on this.
(') P. DuHEM, Sur la pression électrique et les phénomènes électrocapillaires. i" Partie : De la pression électrique, Chap. I, § II {Annales de l'École Normale supérieure, 3" série, t. V, p. io3; 1888).
BOOK V.
ELECTRICAL BALANCE AND PERMANENT CURRENTS ON METALLIC CONDUCTORS.
CHAPTER ONE.
FUNDAMENTAL LAWS OF ELECTRICAL BALANCE ON METALLIC CONDUCTORS.
§ 1 - Condition of electrical equilibrium.
The internal thermodynamic potential of an electrified system is given by the equality (i 5) of the previous chapter
(0 ,f = E(r - TS) + w + 0ig',+ e25r2-f-... + e"^".
Let us suppose that we have the disposition of the various parts of the system, their physical and chemical state, and let us propose to know according to which law a given electric charge will be distributed on a similar system.
Let's imagine that we vary the electrical distribution without moving the various parts of the system; since such a modification does not generate any external work, we can see that we will obtain the condition of electrical equilibrium by writing that, for any modification of this kind,
that is
(■j.) E8(r - T2)-+-8W -1-0(01 5-1-1-02 5f2 + ...+ e "gr"):= o.
Let's suppose that the system is formed by conductors on which we can modify the electrical distribution without modifying the physical and chemical state of any of them, which we will express briefly by saying that it is formed by conductors
368 BOOK V. - METALLIC CONDUCTORS.
metal. Then, in any change of electrical distribution, we will be able to suppose invariable the physical and chemical state of the various parts of the system, which will give us
8(r - TS) = o,
801 = O, §02 = 0, ..., O0" = 0,
and reduce the equality (2)3
(3) 8W-+- ©1 og-i-i- 02 8g'2-^-- .-HÔraSçr,, = o.
Let us first imagine the following modification: The charge ùq passes from the element C) to a point of the element t'2. We will have
(4) 8^1= - ogr, oq^ = ^q\ 8^3 = 0, ..., 85-,^ = o.
Let's calculate 8W.
The initial value of W is
L \''12 ''13 n/i/
?.(?+---?"
The final value is
L \ ^^12
''13 ''!"
.^■23 "2rt
+ .
, Qn
1n-\n'\n
We have therefore
8W = -s^^+sS^^^ + il+...4--il
''12 ''21 ''23 ''in
Let V|, Vo be the respective potential levels of the elements
CHAP. I. - THE KLECTRK EQUILIBRIUM
^-1
P2- We will have
'"12 fl" end
'-21 ''23 ^2/1
369
which will give, neglecting the infinitesimals of the second order,
(5) 8W = £(V,-V,)S^.
By virtue of equalities (4) and (5), equality (3) becomes
The particular modification we have just considered thus leads to the following consequence:
For V electricity to be in equilibrium on a metallic conductor whose all points are at the same temperature, the quantity (eV-f-B) must have the same value at all points of the conductor.
We will now show that this condition is sufficient, i.e., if it is satisfied, equality (3) takes place for any variation in electrical distribution.
To prove this, let us first note that, in any modification of the electrical distribution, we have
(6) 8W = £(Vi ^^ -H V2 05-2 + - .+ V" 8^"),
equality which can be proved as we proved equality (5). Equality (3) then becomes
(s V, + e, ) 8^1 -t- (s V2-i- 62) 8^2 -^ . . + (e V" 4- e") Sgr,, :^ o.
Suppose that the elements c, ..., vi, form a first conductor C, at any point of which (e V-j- 0) has the same value A ; that the elements i'A+i, - - ., ^-t form a second conductor C, isolated from the first, at any point of which (s V + 0) has the same value A', etc.
The previous equality will become
A(85ri-f-. . . -{- Zqk) -+- A'(8^A-(-i-f-. . .-H og-/) -H. . - o. D. - I. 24
370 BOOK V. - METALLIC CONDUCTORS.
Now, according to the law of conservation of electricity (Liv. I, Chap. 1), we have
8g'i + - - --+- ^qk = 0,
the previous equality is therefore proved. Thus the equality
(7) £ V -t- 6 = consl.
is the necessary and sufficient condition for the electrical equilibrium on a metallic conductor whose all points are at the same temperature.
Is it possible to find an electric distribution having at the various points of the conductors a finite solid density, except on certain surfaces at all points of which it would have a finite surface density, and satisfying this condition? This would be impossible if the function 8 were not a continuous function in the whole extent of a given conductor; if, moreover, it were not regular in the whole space occupied by this conductor, except perhaps on certain surfaces. We are thus led to make the first of the hypotheses about the quantity indicated in § 4 of the preceding chapter.
If we admit, on the quantity 0, the hypotheses indicated in the paragraph in question, we will see without difficulty that equality (7) leads to the following consequences:
1° Inside a conductor on which the electrical equilibrium is established, there is no electrified surface;
2° At any point inside such a conductor, V electricity has a finite solid density given by V equality
(8) ■ p=7^Ae.
This density depends only on the constitution of the conductor in the vicinity of the point considered; it varies in a discontinuous way only if this point crosses a surface of discontinuity of the conductive substance.
cnAP. I. - Electrical balance. 871
Consider a region of a conductor filled with a homogeneous substance. will then have the same value at any point whose distance from the boundaries of this region is greater than the molecular activity ratio. Equalities (y) and (8) will then become
V = const., p = o.
It is therefore only in a region formed by a homogeneous substance that the potential function has a constant value and the electric density a zero value. We thus find the fundamental laws of Poisson's theory, but we find them surrounded by restrictions that Poisson's theory could not have predicted.
§ 2 - Stability of the electrical equilibrium.
Is the electrical equilibrium always stable? In other words, does an electrical distribution that satisfies the condition {'j) correspond to a minimum of internal thermodynamic potential? To solve this question, let us look for the sign of 8-J.
We have
As
fine wire r^s
V ,j = - - - -1 1- . . . H ; '
fnl rni l'n-in
(9) { ■ '^^..3
372 BOOK V. - THE MtTALLIQUE CONDUCTS.
So we can easily find the equality
l L Vn2 f'is ri,J
'2,1 /
>'ll-}ll .1
Let us imagine that we place, in the element r, a charge 5^, ; in the element Ç21 wa charge 0^25 - - -, in the element (^", a charge oq,i ; this electric distribution will admit an electrostatic potential ^, and we will see without difficulty that equality (9) can be written
Now we have seen [Liv. I, Ghap. IX] that the electrostatic potential of any electrified system is positive; the quantity o^W is therefore positive; so is t'^§^ and, since any equilibrium distribution necessarily makes the internal thermodynamic potential a minimum, the electrical equilibrium on a system of metallic conductors whose points are all at the same temperature is certainly a stable equilibrium.
CIIAP. II. - HOMOGENOUS CONDUCT. StS
CHAPTER II.
ELECTRICAL EQUILIBRIUM ON HOMOGENEOUS CONDUCTORS. LAWS OF ELECTRICAL DISCHARGE.
§ 1 - Electrical balance on homogeneous conductors.
We will not understand here by homogeneous conductor a conductor having the same physical and chemical constitution and the same density at all its points. We shall admit that bodies which appear to us to be homogeneous have substantially the same density at all points whose distance from the terminal surfaces is greater than a quantity [x which is of the order of the radius of molecular activity; but if a point M lies within such a body A at a distance /, less than [jl, from the surface which separates body A from another body B, homogeneous in appearance, we shall admit that the density at point M depends :
i" Of the nature of the body A, of its temperature, of the density (|u'it presents far from the terminal surfaces;
2" The nature of the body B, its temperature, the density (|u'it presents far from the terminal surfaces;
?)" From the distance / from point M to the surface that separates the two bodies A and B.
These are the assumptions that Poisson introduced in i83o in the theory of capillary action.
We will propose to investigate the laws of electrical equilibrium on a system of metallic conductors constituted as we have just indicated. In the present chapter, we will limit ourselves to the case where all the conductors are made of the same metal i and confined to the same insulator o.
Let a point M, situated at a distance /, less than()v -f- jx), of the surface which limits a similar conductor. It is easy to see, by referring to the definition of the quantity 0, that will have at this point a value 6 (/), variable with /. The form of the function
374 LlVnE V. - METALLIC CONDUCTORS.
lion S(l) depends on the nature of the conductor i and the insulator o, their densities, their temperature.
According to equality (8) of the previous chapter, we will have at this point a solid electric density given by the formula
(,) p=:^A0(O.
According to this equality, p is a function of / which cancels when / becomes greater than ()^+ [x) and which depends on the nature of the conductor (i) and the insulator (o) without depending on the electrical actions to which they are subjected. We thus arrive at the following propositions:
When the electrical equilibrium is established on a homogeneous conductor, there is no electricity at the points inside this conductor whose distance to the terminal surfaces exceeds {\ -\- [x).
The points whose distance to the end surfaces is less than ().-f-[JL) are electrified. The electrical density is the same at all points on a surface parallel to the end surface.
The law according to which Vélectrisation varies, when Von passes from a surface parallel to the terminal surface to another similar surface, depends only on the nature and state of the conductor and insulator away from the terminal surfaces; but not on their shape nor on the particular electrical actions to which the conductor is subjected.
Let's take a point M on the surface that separates the conductor from the insulator; at this point, the normal to the surface towards the inside of the conductor has the direction N, and the normal towards the outside has the direction Ne. If V is the potential function, the surface density at point M has the value
\ ( d\ àN\ _ y de(o) I dN
If we put
CHAP. II. - HOMOGENEOUS CONDUCTORS. 375
the density at point M will have the value
(4) (T = 2 + A.
The surface electric density at a point on the surface of a homogeneous conductor in equilibrium is the sum :
i" Of a density S, given by V equality (2), which depends only on the nature that the conductor and the insulator present away from the terminal surfaces, but not on their shape and the particular electrical actions to which the conductor is subjected ;
1° A density, given by the equality (3), which depends on the electrical actions which are exerted in the system and on the shape of the conductors which compose it.
We will give to the whole of the layer of density 2 spread in a uniform way on the surface of the conductor and of the electrified layers which follow one another in the interior of this conductor up to a distance (). + {x) from the surface the name to^ natural electricity of the conductor. The electric layer of density A will be Vélectricilé communicated.
Let's study the properties of the natural electricity of the conductor.
Let us take, on the surface S, an element rfSH-Se(o)i - V)-(-M'(?-V')]
and, therefore,
E Au= ^[MCC? - V) + M'(9- V')].
Since the exciter has never carried more than light loads, a very small amount of external work was enough to set it in motion. Moreover, its living force is null at the beginning and at the end of the modification. If therefore Q is the quantity of heat released in the modification in question, we will have
(20) EQ= ^[M(V-\'^) + M'(V'--(:;)].
It is the same as the amount of heat that would be given by Glausius' theorem, if we suppose that the system had remained motionless and that the electricity had passed directly from one of the conductors A, A' to the other.
Second remark. - It is assumed that one of the two conductors is formed by a solid part A (Jig-. 72), connected by an extremely fm wire F, to a ball of small dimensions B. One end of the exciter is placed in contact with the conductor A', and the other with the ball B. The experiment
384 BOOK V. - THE METAL CONDLCTELKS.
shows that the heat released by the decliff is almost exclusively given up to the medium surrounding the wire F. This fact is related to Joule's law, which we will study in a later chapter.
Fig. 7:^.
It allows to determine experimentally the amount of heat released in the discharge.
11 It is sufficient to use a kind of air calorimeter
Fig. 73.
invented by Snow Harris in 1837 (' ). The wire F {fig- 73) is enclosed in a glass envelope terminated by a manometer at
( ' ) Snow Harris, On the relative powers of varions metallic substances as conductors of electricity { Philosophical Transactions, t. CLVIII, p. 18; 1827).
CHAP. H. - HOMOGENEOUS CONDUCTORS. 385
free air. At the moment of discharge, the wire, initially heated, quickly reaches a temperature equilibrium with the air contained in the glass envelope. If we denote by Tq the initial temperature of the wire and of the air mass; by T, their final temperature; by m the reduced weight in water of the wire and of the air mass, the quantity of heat released has the value ci(T, - Tq). Thanks to the presence of the manometer, the apparatus forms an air thermometer with an approximately constant volume. If we denote by (P, - P^) the variation of pressure, we will have
P.-Po=aPo(T,-To),
a being the coefficient of expansion of the air; we will thus have
The quantity m remaining constant in the various experiments, this equality will make us know numbers proportional to the quantities of heat released in these experiments.
Riess ('), who perfected the Snow Harris manometer, used it to determine, under various circumstances, the amount of heat released by the electric discharge. Clausius (2) compared the results of Riess' experiments with the consequences of the theory. We shall indicate here some of these experiments.
§ 4 - Complete discharge of a capacitor. Experiments of Riess.
A capacitor is charged; the inner armature has received an amount of electricity a; the outer armature carries a communicated charge - b. The potential level of the communicated distribution is V on the first conductor and o on the second. Moreover (Liv. II, Chap. XI, § 2)
b = ma,
m being the first Gaugain coefficient.
Let's put the two armatures in communication with each other;
(') Riess, Lehre der Reibungs-Elektricitàt; Berlin, i863, (') Clausius, loc. cit.
D. - I. 25
386 BOOK V. - METALLIC CONDUCTORS.
a discharge will occur giving off a quantity of heat Q which, according to formula (20), is given by the equality
EQ= i[aV + (a - 6)t;;]
= ia[V + (i_m)t;;].
In particular, let's assume that the outer frame completely envelops the inner frame. Then, according to the principles explained in Book TI, Chapter V, we will have
a = 6, m =1 and therefore
(21) EQ==îaV.
Let us imagine a battery formed of n identical bottles, arranged in any manner, each having its internal armature in communication with the source and its external armature on the ground. According to the principles laid down in Book III, Chapter V, each of these bottles exerts a null electrostatic action on the outside; consequently, each of them will electrify as if it existed alone. If we discharge the battery, the amount of heat released will be
Moreover, we can write
"= £GV,
C being a constant (capacity of the cylinder) which depends on the construction of the cylinder. We will thus have, by designating by A the total load, equal to /la, of the internal armatures,
"I t A2
(-) Q=^G7r The amount of heat released by the discharge of a battery of identical jars is proportional to the square of the total load taken by the inner fittings, and inversely proportional to the number of jars.
Riess verified this law by the following experiments. He has first of all
CH.VP. II. - HOMOGENEOUS CONDUCTORS. 887
On board took a battery of five jars, to which he communicated variable charges; he then compared the number D of divisions by which the liquid of the thermometer rose, as a result of the different discharges, with the number D' by which it should rise according to the formula (22); he found the following results :
A. D (observed). D' (calculated).
3 1,5 1 ,6
4 3,0 2,8
5 4,5 4,4
6 6,5 6,3
7 8,8 8,6
8 11,3 11,3
9 i4,3 14,3
10 16,7 17,6
11 then varied the number n of bottles while keeping the internal armature of the battery at a constant charge A, and found the following results:
n. D (observed). D' (calculated).
2 i3,4 i5,8
3 9,7 10,6
4 7,3 7,9
5 6,5 6,3
6 5,5 5,5
We see that the agreement of formula (22) with experience is, in general, very satisfactory, and of such a nature as to leave no doubt as to the accuracy of the law we have just stated.
We5 have assumed only, so far, that the outer frame of the Lejde bottle completely enveloped the inner frame. Let us now assume, as we have already done (Liv. ni, Ghap. IV, § 4), that the inner surface of the outer frame is a level surface of the inner frame.
In this case, the charge a of the internal armature has the value [quote, equation (17)].
V being the potential level to which the internal armature would be brought if we distributed on this armature, taken in isolation, an electric charge equal to the unit; u being the value that would have the function
388 BOOK V. - METALLIC CONDUCTORS.
of this distribution at any point on the inner face of the outer frame.
Let us suppose that the two armatures are very close; let M be a point of the internal armature; let N^ be the normal at this point to the internal armature directed towards the outside of this conductor; let a be the density that an electric charge equal to the unit distributed on the internal armature takes at this point; let S be the very small distance between the two armatures; let dS be an element, drawn around the point M, of the surface S of the internal armature. We will have the various equalities
that give
I
dv
a -
"4^
dN,
S'
T^S:
= I,
di^
u
- V
at
I
l
c
dS
Let A V be the average thickness of the insulating layer between the two reinforcements, given by the equality
Ë - C -
The previous equality will become
1 Ë
471 Â'
Equality (28) will then become
I S
[\-K A
and equality (21),
(24) EQ = 2Tr£A^'.
Ley de bottles, formed as we have just indicated, in which the two armatures are at the same very small average distance^ give, whatever their shape, when they are discharged, a quantity of electricity proportional to the square of the charge of V internal armature, and in inverse proportion to the surface of this armature.
CHAP. II. - HOMOGENEOUS CONDUCTORS. 889
This law was found experimentally by Riess ( * )- M. Helmholtz (^) and Clausius (^) established it theoretically. The previous demonstration is due to M. J. Moutier (^).
§ 5 - Discharge of a capacitor by successive sparks.
A capacitor can be discharged by successive sparks: it is enough to isolate the capacitor and to put in communication with the ground, first the internal armature, then the external armature, then again the internal armature, and so on.
This discharge did not give rise to measurement experiments; the theory is very simple; it was given by Mr. J. Moutier (5).
Let's look at the amounts of electricity that flow into the ground with each spark.
Let A be the inner frame and B the outer frame.
Initially, conductor A has an electric charge a and conductor B an electric charge - b. When we put conductor A on the ground, with conductor B remaining insulated, we carry out Gaugain's third experiment (Liv. II, Chap. III, § II). After this first discharge, conductor A remains charged with the quantity at= mm! a of positive electricity. The first spark therefore brings to the ground a quantity
<7i = (i - mm')a of positive electricity.
When we then isolate the conductor A, carrier of the charge a, = mm' a, and we put the conductor B in communication with the ground, we carry out the second experiment of Gaugain, but assuming j the charge of the body A reduced in
(') Riess, Ueber die Erwàrmung in Schliessungsbogen der electrisclien Batterie {Poggendorff's Annalen, Bd. XLIII, p. 47; i838). Lehre der ReibungsElektricitât; Berlin, i853.
(") H. Helmholtz, Ueber die Erhaltung der Kraft, p. f\Z (Berlin, 1847. - Helmholtz wissenschaftliche Abhandlungen, l. I, p. l\b).
(' ) R. Clausius, On the mechanical equivalent of an electric discharge and the heating it produces in the conducting wire ( Mechanical theory of heat. Translation Folie, t. II, p. 60).
(♦) J. Moutier, Cours de Physique, t. I, p. 499; Paris, i883.
(') J. Moutier, Cours de Physique, t. I, p. 4^9 and p. 486; Paris, i883.
3jO BOOK V. - METALLIC CONDUCTORS.
the ratio mm! A similar reduction must be made in the charge of the conductor B, which thus contains, after the second spark, a negative charge bi = mm'b. The second spark therefore brings to the ground a quantity of electricity
^2 = m(i - mm') a of negative electricity.
At the moment of producing the third spark, we are in a state similar to the initial state, but where all the charges have been reduced in the ratio mm' . The third spark therefore brings to the ground a quantity of positive electricity
^3= mm' {i - mm') a.
We will see that the spark of order (2/^-1-1) will bring to the ground a positive quantity of electricity
qin+\ - m"^m'"(i - mm') a,
while the spark of order (2/^ + 2) takes to the ground a quantity of negative electricity
Çin+i = m"^+^ m''^(i - mm' )a.
Let's look for the amount of heat that each of these sparks would give off in the Riess thermometer.
Before the first spark, the electrostatic potential has the value
Wo- -aY. 2
After the first spark, the conductor B is at potential level \'; the electrostatic potential has the value W,, and we have
Wi = --6V'. 2
The Gaussian identity, applied to the initial and final states, gives, noting that ", is the final charge of conductor A,
aiY = - bY'. Thus we have
Wo- Wi= -(i - m/n')aW.
CHAP. II. - HOMOGENEOUS CONDUCTORS. 89 1
The amount of heat released by the first spark is
value
Q,= -^(i - mm')a\.
To find the amount of heat released in the second spark, we note that before this spark, conductor A is at potential level o and carries a charge "i; conductor B is at potential level V and carries a charge - b. The electrostatic potential has the value
W, = --6V'. 2
After the discharge, conductor A carries charge a, and is at potential level V". Conductor B carries a charge - b^ and is at potential level o.
The quantity ^, is given by
bi - mm' ù. The electrostatic potential has the value
W2=-aiV". Moreover, the Gaussian identity gives
So we have
and
W2 = --6,V' 2
W,-W2 = - J(^'-^',)V'
= - m7n'{i - m.m')aV.
The amount of heat released by the second spark has the value
Q2 = - =; mm' ( I - mm' ) a V = mw' Qi .
2ll>
We would also find that the amount of heat released by the third spark has the value
Q3=m2m'2Q,.
392 BOOK V. - METALLIC CONDUCTORS.
The quantities of heat released by the successive sparks decrease in geometric progression of reason mm' .
§6.
Cascaded batteries.
Let us suppose that we have {n-\-} bottles, identical to each other, and far enough apart from each other to have no influence on each other. The internal armor Ao of the
Soil
The first {Jig- 74) 6st is connected to the source at the potential level Vo- The external armature Bo is connected to the armature A of the second, and so on. The ar
CHAP. II. - HOMOGENEOUS CONDUCTORS. SgS
The last one is externally mature on the ground. We have thus a battery charged in cascade.
The theory of cascade loading was the subject of research by Green ('), Clausius and Béer. M. J. Moutier has given this theory a very elegant form (-), in the case where the two surfaces S', S" which limit the external framework of each bottle are level surfaces of the internal framework S.
Let Vo, V,, V2, ..., V" be the potential levels of the armatures Ao, A,, A2, ..., A".
Let's consider first the last bottle.
It carries no electric charge on the surface S"; on the surface S' it carries a charge - a,i and on the surface S a charge On- '
An electric charge equal to unity, in equilibrium on the surface S, would bring this surface to the potential level v; a charge equal to unity, distributed on the surface 2', would bring any point inside this surface to the potential level v' (^); finally a charge equal to unity, distributed on the surface S", would bring any point inside the surface 2" to the potential level v" .
With these notations, we can already write
For the [n - iy<'"" bottle, the inner armature carries a charge a,i-\; the surface 2' a charge - ""-4, and as the two bodies B"_i and A" form an insulated conductor carrying a total charge of zero, the surface 2" carries a charge (""_( - ""). The principles laid down in Liv. III, Chap. IV, easily give
a"-iv - a"_i (>'-+- (a"-i - a")t>" = V"_i. We will thus arrive from one step to another to have the two series
(') Green, An essay of the application of mathematical analysis to the theories of Electricity and Magnetism, Art. 8 (Nottingham, 1828, Green's mathematical Papers, p. 47 )-
(') J. Moutier, Cours de Physique, t. I, p. 491; Paris, i883.
(' ) To make the formulas we are about to write coincide with those obtained in Book III, Chap. IV, we would have to pose v' = u, v" = u'.
394 BOOK V. - METALLIC CONDUCTORS
of equations
ttnV - anV' =Yn,
aa-i{v - v'-\-v") - a,iv" = V,,-, cin-iiv - v'-\- v") - a,i-\ v"= V"_2,
(25)
a2{v - v'-{-v") - a^v" =¥2, ai{v - v'-\-v") - anv" = Vi, a^iv - v' -^ v") - a^v" = Vq.
(a"_i - a")p" =^V", (a,j_2 - ""~i)<^" = V"_i,
(26)
("1 - a<))v" = V2, ("0-"i)^'"= Vf
1 "1
1 "1 -F- "3
=
v"
V ~
- t"' -r
- IV
V
- t^'-r
■IV"'
Between the equalities (aS) and (26), we can eliminate the potential functions; we then have
i( ao - - ai ) p" = "1 ( c - v' -4- p" ) - "2 ^'''j (ai - "2)*^"= "2(^' - c'h-p") -- "3"^",
or
(28)
The charges of the internal armatures of the bottles are subject to the following law: If V we take three consecutive bottles, the sum of the charges of the internal armatures of the two extreme bottles is in a constant ratio to the charge of V internal armature of the average bottle.
The equations (aS) give
Vo-f- V2 = (ao-H a2)((^ - t^'-i- v") - {a^t-^ a^)v\
or, according to the equalities (28),
aip"(p - v' -\- v") a^v"'^
V0+V2
= [ai(v - v'-hv") - a^v"]
V - V -{- IV
(î9)
CIIAP. II. - HOMOGENEOUS CONDUCTORS. SgS
According to the equalities (20), this one can be written
V,
The potential levels of the internal armatures thus follow the same law as the charges.
Let us add member by member the last of the equations (aS) and the equations (27). We will find
(3o) \q = (uo-h ai-h. . .+ an)iv -v').
This equality, compared to equality (aS), highlights the following theorem:
The sum of the charges of the internal armatures of a cascaded battery is equal to the charge that the internal armature of the first cylinder would take if it were charged alone.
(^this beautiful theorem is due to Green.
Let's suppose that we leave the external armature of the last cylinder on the ground and that Ton puts the internal armature of the first cylinder in communication with the ground. The battery will be brought back to the neutral state. If W is the initial electrostatic potential of the system, we will have, for expression of the heat released by this discharge,
EQ = W.
However, the general formula
w=i2.
gives easily
W= -aoVo 2
The heat released has therefore the value
(3i) Q:^-^aoVo.
If we had charged only the first cylinder, and discharged it as we have just done for the battery, the discharge would have, according to equalities (21), (23) and (3o), released a
396 BOOK V. - METALLIC CONDUCTORS.
amount of heat
By comparing this equality with equality (3i), one sees that the heat released by the complete discharge of a battery charged in cascade is all the smaller as the number of the bottles is larger.
In the particular case where the two surfaces S', S" differ very little, as it happens if the external armature is formed by a simple metal sheet, we have approximately
We then easily find the equalities
(3-2)
(33)
If the external armatures are very thin, the loads of the internal armatures and their potential levels decrease in geometric progression.
Béer (^) erroneously stated that, in this case, the charges of the internal armatures were all equal to each other and that their potential levels decreased in arithmetic progression.
In this case, the amount of heat released by the complete discharge of the battery has the value
Q = -^. - - - - "^
"0 CTi
""-1
V
"1 "2
a,i
v'
Vi v^--'
V
v'
A = ao -T- "1 -I- . 4- a".
Let's put
We will have
(34) A= -, an.
We can therefore write, by designating by Q' the quantity of heat that would be released by the complete discharge of a cylinder
C) Béer, Einleitung in die Elektrostatik ..., p. 102 (Brunswick, i865).
CHAP. II. - HOMOGENEOUS CONDUCTORS. 897
unique,
(35) V ' ^^
Q V - 1>' v"^
In the particular case where the two armatures are very close to each other, we can easily find that this equality becomes
Q'=(n + i)Q.
If Von forms a cascade battery by means of bottles whose outer frame is very thin and whose two frames are very close together, the amount of heat released by the complete discharge is inversely related to the number of bottles.
This law was found experimentally by Riess [loc. cit.] and theoretically by Glausius [loc. cit.].
398 BOOK V. - METALLIC CONDUCTORS.
CHAPTER m.
THE INTENSITY OF THE CURRENTS.
§ 1 - Currents flowing in the mass of a conductor.
If the state of a conductor at time t were entirely determined by the electrical distribution it carries at time t^ the properties of this body would have to remain the same, either if this electrical distribution remained after time t what it is at time ?, or if it underwent changes after time t. But this is not the case. The conductor on which the distribution varies will exert certain actions on a magnet that a conductor on which the distribution is invariable does not exert, whatever this distribution may be.
It is therefore necessary, in order to define an electrified system, to use a more complicated representation than the one we have used up to now; to add new variables to the variables which determine the electrical distribution, these last ones disappearing in the particular case where, on the conductor, the distribution remains independent of time.
The definition of these new variables is mainly due to G. -S. Ohm (' ), Smaasen (2) and G. Kirchhoff (3).
(' ) G.-S. Ohm, Die galvanische Kette, mathematisch behandelt {Berlin, 1827. - Translated into French by Gaugain; Paris, 1860).
(") Smaasen, Vom dynamischen Gleichgewicht der Elektricitàt in einer Ebene oder in einem Kôrper {Pogg. Annalen, Bd. LXIX, p. i6r; 1846). - Vom dynamischen Gleichgewicht der Electricitàt in einem Kôrper und in unbegrânztem Raume{Pogg. Annal, Bd. LXXII, p. /|35; 1847).
(^) G. KiRCHHOFF, Ueber den Durchgang eines élektrischen Stromes durch eine Ebene, insbesondere durch eine kreisformige ( Pogg. Ann. Bd. LXIV, p. 497; 1845. - Kirchhoff's Abhandlungen, p. i). - Nachtrag zu dem vorigen Auf
CHAP. III. - the intensity of the currents. 899
These ph3'sicians were led to the definition of the variables in question by the comparison of the motion of electricity with the motion of fluids; a comparison of this kind had already led Fourier to the definition of the principal quantities which appear in the theory of the propagation of heat.
Let M be a point inside a conductor. We will suppose that this point M corresponds to a geometrical quantity F which we will name \electric flux at point M at time t. This quantity is related to the variation of the electric distribution inside the conductor by the following convention:
Around the point M {fig. 70), let us trace in the mass of the con Fig. 7,5.
Let AB = i/w be an element. Let N be the normal to this element in a given direction. Let F be the flux at the point M. The variation undergone by the electric distribution on the conductor in time dt is the same as if, during this time, the element doi had been crossed, in the direction of the normal N, by a quantity of positive electricity dq, given in magnitude and in sign by
([) dq = Fcos(F,N)diodt.
This equality (i) can be written in a slightly different way.
satze {Pogg. Ann. Bd LXVII, p. 344; i846. - A'. Abhandl., p. 17). - Ueber die Aujlôsung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Strômen gefiihrt wird {Pogg. Ann, Bd. LXXII, p. 497; 1847. - A". Abhandl., p. 22). - Ueber die Anwendbarkeit der Formeln fur die Intensitàten der galvanischen Strômen in eineni Système linearer Leiter auf Système, die zum Theil aus nicht linearen Leitern bestehen {Pogg. Ann, Bd. LXXV, p. 189; 1848. - K. Abhandl., p. 33). - Ueber eine Ableitung der Ohm'schen Gesetze, welche sich an die Théorie der Elektrostatik anschliesst {Pogg. Ann. Bd. LXXVIII, p. 5o6; 1849. - K. Abhandl., p. ^9). - Ueber die stationàren elektrischen Stromungen in einer gekriimmten leitenden Fldche {Monatsber. der Akad. der Wissenschaften zu Berlin, 19 July 1875. - K. Abhandl., p. .56).
400 BOOK V. - METALLIC CONDUCTORS.
Let us take three axes with rectangular coordinates; let m, ç, w be the components of the flow F along these three axes. We can write
(2) dq ~ [acos(N, x) -t- v cos(N,j) -t- w cos(N, z)] dia dt.
This equality (2) will lead us to a formula that will clearly highlight the relationship between the (lux and the change in electrical distribution.
Let us draw, inside the conductor, a closed surface S. Let M be a point of this surface and N,- the normal to this surface at the point M; this normal is directed towards the interior of the space limited by the surface S.
The change of electric distribution on the conductor is the same as if each of the elements ûfS of the surface S let, in the time dt^ penetrate inside this surface a quantity of electricity
dq = Yu cos(N/, x) -\- V cos(N/,j') -h v cos( Nj, z)]"iS dt.
The change in electrical distribution on the conductor must therefore have the effect, during the time dt., of increasing by
dt V [m cos(N/, x)-^ V cos(N/, J-) -+- "'cos(N,, z)dS
the amount of positive electricity contained in the surface S.
But, on the other hand, if we denote by p the electric density at the "point" (x, y, z) at time t, a density that we assume to be finite at any point inside the surface S, the change of electric distribution during time dt will increase the total electric charge enclosed in the surface S by
dt I I i Yfdx dy dz,
the Integration extending to all the space inside the surface S. We must therefore have, whatever the shape of the surface S,
V [acos(Nt-,a7)-f-(^cos(N;,jK)-H"^cos(I\;,^)]â?S - / / j-dx dy dz =^ 0.
This equality can be transformed. Suppose that the quantities^ M, V., w are continuous as well as their partial derivatives of the first order in all the space enclosed by the surface S; that it
CHAP. lit. - the intensity of the currents. 4oi
due
be the same for p and -r^". The previous equality will become
Under the hypotheses made, it cannot take place for any surface S, unless we have, at any point where the indicated conditions are verified,
(3)
of dx
at
dw
ùt
This equality does not apply to theK various points of a surface along which the quantities m, p, w, p may be discontinuous. Let us consider this case, assuming that this surface can carry a variable surface electrification of density n.
A similar surface S {fig- 76) separates two regions i and -i
Fig. 76.
/
/
■§
J
x^
- \
of the conductor. On this surface let us take an area and, by the contour AB of this area, lead straight lines normal to the surface S. We limit these lines by two surfaces Si, S2, parallel to the surface S, situated one in the region i, the other in the region 2, both infinitely close to the surface S. Let A, B,, AoB^ be the areas cut on these two surfaces by the considered ruled surface.
To the nearest terms of the order of A,Ao, the amount of positive electricity that enters the closed surface A/BiAoBo during the time dt can be written
- c?/ X [ Wi cos(Ni, a?) -+- t^i cos(Ni, y) -t- (V| cos(N,, 3) KJxn
D. - I.
Ui cos(N2, a:)H- Vi cos(N2, j) -f- w^ cos(ÎN2, 3)] rfS
26
402 BOOK V. - METALLIC CONDUCTORS.
and also
dt^ $dS.
dt
By equating these two quantities, we can easily see that we must have, at any point of the discontinuity surface S,
/ i(iCos(Ni, x) ■-!- i'i cos(Ni, j) -H Wi cos(N], z) ^^' I + "2 cos(N2, a?) -i- P2 cos(N2,jk) -I- M^a cos(N2, -z) = -7- -
Equalities (3) and (4) show us to what extent the fluxes on the one hand and the electric densities on the other hand can be regarded as independent variables. One can always, for a particular value of t, arbitrarily give oneself the magnitude and direction of the electric flux at each point of the conductor, and the magnitude of the solid or surface density of electricity at each point of the conductor. But, for later values of time t, it is no longer permissible to give oneself arbitrarily anything other than the magnitude and direction of the electric flux at each point of the conductor; for the densities, both solid and superficial, are then determined, for all values of t, by the equalities (3) and (4).
At the surface that separates the conductor from the insulating medium that surrounds it, we have, according to equality (4),
(5) u cos(N/, x) -h V cos{Ni,y) -+- w cos(N/, -z) = - -r- Some authors have admitted that we always have, in this case,
u cos(N;, x) -^ V cos(i\;,^) -\- w cos(Nj, ^) - o. -
But then no current could vary the electrical distribution on the surface of a conductor; it could never change, which is inadmissible.
§ 2 - Uniform currents.
If the electric flux is zero at any point of a conductive body, we say that V electrical equilibrium is established on this conductive body.
According to the equations (3), (4) and (5), the solid electrical density
CHAP. III. - the intensity of the currents. 4^3
OR surface then maintains a value independent of time at any point of the conductor.
But the solid or surface electric density can maintain, at any point of the conductor, a value independent of time without the electric flux being zero at any point. Indeed, it is enough for the currents not to cause any variation of distribution on the surface of the conductor or in its interior, that we have at any time, for any point inside the conductor,
-from ()(' ôiv
dx dy dz '
for any point on the surface separating the conductor from the insulator,
(7) u cos(N,-, x) -h V cos(N/,^) -r- w cos(N/. z) - o;
finally, for any point of a discontinuity surface of the conductor,
( Ml cos(Ni, ip) -H (^1 cos(Ni,_7) -4- (iPi cos(Ni, ^)
( -4-"2 cos(N2, x) -T- t'2 cos(N2,_y) -f- "'2 cos(N2, z) = o.
Such a current, which at each instant brings as much electricity to any point inside the conductor or on its surface as it carries away, is said to be a uniform current.
When at any point of a conductor the electric flow is independent of time, the current is said to be constant.
A current that is both uniform and constant is called permanent.
§ 3 - Linear currents.
Suppose that an infinitesimally small plane area A {Jig. 77), will Kig. 77.
(8)
It moves in such a way as to remain constantly normal to a line LL'. It sweeps a volume which we will suppose filled with conductive matter and which we will name
4o4 BOOK V. - METALLIC CONDUCTORS.
Let's take a wire. If this wire is traversed by any electrical flux, we will say that it is traversed by a linear current.
Let N be the normal to the area A, on a given side of this area. It is at the same time the tangent at L to the curve LL'. The area A is crossed, in the direction indicated by the normal N, during the time dt, by a quantity of electricity
c/Q = ^^C [mcos(N, 37)4- pcos(N,7)-+-"'cos(N,3)]£/A.
If we put
(9) J = V [a cos(N, x) -\- V cos(N,^) -\~ w cos(N, z)\dk,
we will have
^Q = J dt.
We then say that J is Vintensity at point L of the linear current.
Let us assume, first of all, that the wire LL' has no discontinuity surface between the points L and L'. It is easy to see that, under these conditions, the intensity J is continuous between the two points L, L'. Between two sections of the conductor, comprising between them a length ds of the curve LL', there accumulates during the time dt a total quantity of electricity
z- as dt.
ds
Let us now assume that the wire has a surface of discontinuity, and, in order not to complicate our reasoning too much, let us imagine that this surface coincides with a normal section A. This section divides the wire into two regions i and 2.
When we go from the first to the second, the intensity changes abruptly from the value Ji to the value Jo. In the time dt, a quantity of electricity accumulates on the surface A
(Ji -i2)dt.
If S is the average surface density over the area A, we have
(.0, g= ■(,,_,,).
CHAP. III. - the intensity of the currents. 4o5
Let us consider, on the other hand, a portion LL' of the wire, along which there is no surface of discontinuity. Let J, J' be the values of the intensity at the origin and at the end of this segment, in time dt, this segment acquires a quantity of electricity
{i - y)dt.
Let A be the average cross-section of the LL segment; G the average perimeter of this cross-section; a- the average surface electric density; p the average solid density. We will have
If we admit that the densities p and o- are, in general, of the same order of magnitude, A being negligible before G, we will have
(-0
Let us compare the equalities (lo) and (i i). Since A is negligible in front of G, if we want the electric densities S and ds = ^ ^' doi,
\ AUA'B'
and the previous equality can be written
(2) ^Q = [X') cos(rt, x) -+- \'> cos(rt, y )\ ds dl.
The quantities "O and <>, defined by the equalities (i), are what we will call the components of the surface flux at the point As
On the upper side of the plate, draw a small rectangle
Fig. 83.
y/ / ^
D
7
/ a//'~"
-s^-\
-JD'
--■^i'
ABCD i^fig. 83), whose sides AB ;= dx and AC ^=-dy are pa
CHAP. V. - MOVEMENT OF ELECTRICITY IN A PLATE. 4^1
parallel to the axes Ox, Oy. By the contour of this small rectangle, let us lead normals to the plate, which cut out a parallelepiped ABCDA'B'C'D'. This small parallelepiped contains a total quantity ^ of electricity spread inside it, or on its faces ABCD, A'B'C'D'. We will pose
and we will name P the electric density at a point A of the plate.
It is easy to see that we have
(3)
Oy
~ôi'
This equality shows that, if the currents are uniform, we must have
(1)
Ox Oy
Let ABA'B' be a rectangular element between two generatrices AA', BB' of the plate edge {Jig- 84)- Let Ng be the
normal to this element towards the outside of the plate. Let ds be the length AB. Let ^ds be the quantity of electricity it contains. We can easily find
(5)
X) coë (rie, ^) -+■ "->? cos(ne, y) =
dt
If the currents are uniform, the second member will be zero, and we will have, by designating by /?/ the direction opposite to ne-,
(6) XD cos(",-, x) -+- V cos(n,-, y) - o.
The surface Jlux lines must be tangent to the edge of the plate.
4'22 BOOK V. - METALLIC CONDUCTORS.
Let us assume the homogeneous plate and the permanent currents. Let àl be the specific resistance of the substance which forms the plate. Let V be the potential function. We will have, according to Ohm's law,
__ ^ dY __ ^ ^^
" ~ ~ ^ 5^ ' ^ ~ ~ Jl 6"/ ■
Equalities (i) will then become
AU A' 15'
^■^■"=-iS
dy
A 1!A' B'
d(M.
d\ d\ .
But the quantities -^j -- - vary in a continuous way, not
^ ox ôy
not only inside the plate, but also at the crossing of its faces. Each of these quantities will have, at any point of the element ABA'B', approximately the same value as at point A. We can therefore replace the previous equalities by
- ) - denoting here the values that these quantities take at
point A. - The quantity
(7) ^-f
is what we will call the specific resistance of the plate at point A; the introduction of this quantity allows us to write
Let us now assume that the plate has a constant thickness, so that the quantity Itt has the same value at every point. Equality (4) will become
it must take place at any point of the plate, while at any point of the
I
CIIAP. V. - MOVEMENT OF THE ELEOTRICIAN IN A PLATE. 4^3
point of the edge of the plate we must have, according to the equality (()),
Let us therefore consider a plate with L as its edge (Fig. 85). On this plate, two areas, one bounded by the line Lo, the other bounded by the line L,, are maintained respectively at the potential levels Vo and V,. To determine the perma
The first step is to find a function V which verifies the partial differential equation (9) at any point of the area between the lines L, Lo, L(, taking the value Vq at any point of the line Lo> 1^ value V, at any point of the line L, and satisfying the condition (10) at any point of the line L. The components of the flux will then be determined at each point by the equalities (8).
The problem thus posed is in the same form as the problem of the permanent motion of the eleclricle in a three-dimensional conductor. But we have methods to solve it which have no equivalent in the case of three variables. These methods, which we cannot detail here (*), are essentially based on a few theorems that we will indicate.
Consider a complex variable
z - X -^^ iy. Let
Z = F(a7 4- iy)
any analytic function of this variable. This function
(' ) See, for the exposition of these methods, G. Kirchhoff, Voiiesungen iibei mathematische Pliysik-Mechanik, Lesson XXI (Leipzig, 1877).
424 BOOK V. - METALLIC CONDUCTORS.
lion can always be put in the form of
Z = X + t Y,
X and Y being two real analytic functions of x and y. The first is the real part of Z; we will denote it by IHZ. The second is the coefficient of the imaginary part of Z. We will often denote it by Jf Z.
The quantity Z should depend on ^ and jK only })ar (jr H- /j-),
we will have
iTL _ dZ iïL dx i ôy 0:
equality which becomes
.dY _ .dX dx dy
^ <^y
or
dX
dY
dx
^ ¥'
dY
dY
'^
dx
From this, we can easily deduce
(>i)
l d'-X \ dx-^ \ d^Y \ dx*
d'-X dy'd'-Y
àXdYdXdY_ dx dx ày dy
The equalities (i i) show (jue, if we denote by F(^ + iy) any function of the imaginary variable [x 4- iy), the expressions
(i3) ^ ^%V{x^iy),
(i3 bis) V = J V{x-\-iy)
represent two integrals of V partial differential equation
^9^ 'dlc^^-^-dy^ =^ Equality (12) shows that both equations
( îl Y {x + iv) = const, (i4) \ \ Ji
( J Y{x H- ly) = const.
CHAP. V. - MOVEMENT OF ELECTRICITY IN A PLATE. 4^5
represent two families of lines that intersect orthogonally. If one of these families represents equipolential lines, V other represents dejlux lines and vice versa.
These beautiful propositions show that all the important theorems of the theory of functions of imaginary *'ariables will find an image in the study of the permanent motion of electricity in a plate (').
Any function of the complex variable {x - "jk), conjugate of the previous one, will have analogous properties. We can prove the following proposition:
Let F(x -'r iv), G{x - if) be any two analytic functions of two complex conjugate variables; the symbol
tl F( jr - t>) -4- -" G(^ - i"
represents V general integral of V equation
The relations of the theory of functions of imaginary variables with the theory of conformal representation lead to the following theorem, which we will only state:
Given two plates P and P', of which V one is the conformal representation of Vautre, which means that any point on one can be matched to any point on the other so that :
1° Any point of the contour of the plate P corresponds to a point of the contour of the plate P' and vice versa;
2" Any point on the electrode contour of plate P corresponds to a point on the electrode contour of plate P' and vice versa;
3° If a point M of the plate P tends towards a point m of the contour of the plate, the point M', which corresponds to M on the plate P', tends uniformly towards the point /?/ which corresponds to m on the <;ontour of the plate P' ;
(' ) See V. Klein, Ueber Hiemann's Théorie der algebraischen Funktionen und ihrer Intégrale (Leipzig, 1882).
426 BOOK V. - METALLIC CONDUCTORS.
4" Any infinitely small figure drawn on plate P corresponds to a similar figure drawn on plate P' and vice versa;'
If we know how to find the permanent motion of electricity in plate P, we know how to find it in plate P' and vice versa. The equipotential lines and the lines of jlux of one of the plates have respectively for representation the equipotential lines and the lines of Jlux of the other plate.
Let us apply the first of these theorems to an interesting special case.
The two electrodes are reduced to two points: one M, {fig. 86),
Fig. m.
of coordinates ( <, ^,), the other Mo, of coordinates {a->, b^)- The potential function at these two points will not be given, nor will it be required to have a finite value. We will only give the intensity J of the current brought by the first electrode and carried by the second. Let's put
Cl = "1 -1- ibi, C2 = "2-1- '^25 Z ^^ X -+- ij,
"^ z - c^
Let's find the real part and the coefficient of / in the imaginary part of Z. If we put
oc - "1 = /'i COSGi, 37 - "2 = r^ COS62,
y - ^1 = /'i sin6i, y - b^=^ r it is necessary and sufficient that we have
(18) - -f = consl.,
in any way that the two points M and M' are located on the plate.
Here is how G. Ivirchhoff verified the accuracy of this relationship:
In the wire of the galvanometer was inserted a thermoelectric cell, maintaining a constant potential difference between the two ends of the wire. In order for the galvanometer to show no current, it was necessary, as shown by the study of thermoelectric currents, that the two ends of the wire touch two points M, M' of the plate, whose potential level difference is equal and of opposite sign to the one that the thermoelectric cell maintains between the two ends of the wire. For two pairs of such points, whatever their position on the plates, the equality (i8) must be verified.
G. Kirchhoff first studied pairs of points located on the line M" Mo. We had, in this case,
n + ^2 = r\ -H r; = Ml M, = iij,
relation which, joined to equality (i8), allowed to calculate /', when /"j was known. G. Kirchhoff compared the values of r\ thus calculated with the observed values of r\, and he found the following results:
/*i 5 10 i5 20 25 3o
/-', obs 10.4 17.3 22.8 28 3i.5 34.4
r\ obs - r\ wedge -i-o,4 - o,[ - 0,4 -+-0,2 0,0 - 0,2
G. Kirchhoff then studied the couples located on a circle of
43o BOOK V. - METALLIC CONDUCTORS.
5 inches of radius, passing through the two points M', M2, whose distance was, in this experiment, i inch. He found:
10
20
3o
40
5o
60 70
80
25,4
48,3
62,5
70,9
78,7
84 88,75
9^
i- 0,2
-+- 0,3
- 0,4
- i,i
0,0
- 0,3 0,6
0,0
/-l
r\ obs
r\ obs. - r\ cale. . .
These experiments, as we can see, provide a very complete and precise verification of Ohm's law.
The problem of the permanent motion of electricity in a plate can be solved for a large number of cases (*). Mr. Quincke and Mr. Adams have given a large number of experimental verifications of the theoretical results obtained (^). Ohm's law is, by these verifications, placed beyond all dispute.
§ 2 - Currents in a curved blade.
We can study the permanent motion of electricity in a conductor bounded by two very close curved surfaces, as we studied the motion of electricity in a plate.
Let us consider the upper face of the plate and, on its surface, let us draw a system of orthogonal curvilinear coordinates. This system is formed by two families of lines: the lines [3, that
represents the equation
a = const,
and the a-lines, as represented by the equation
P = const.
If ds denotes the distance from point (a, (3) to point
(a + doc, p -+- d'^), we have
(19) ds^-=zA'idoc'^-{-B^dp,
A and B being two positive functions of a, [3.
(') G. KiRCHHOFF, Vorlesungen iiber niathematische Physik-Mechanik , XXI" Lesson (Leipzig, 1877). - E.Mathieu, Théorie de l'Électrodynamique, Chap. IV et V. Paris, 1888.
(") Quincke, Ueber die Verbreitung eines electrischen Stromes in Metall-platten {Poggendorff's Annalen, t. XCVII, p. 882; i856) ^ Adams, Proceedings of the royal Society of London, Bakerian Lecture, t. XXIV, p. i; 1875.
CIIAP. V. - MOVEMENT OF ELECTRICITY IN A PLATE. 43l
The small rectangle, whose vertices are the points
(a, P), (a + ^/a,P), (a, ? + rfp), {% + doL, ^ -^ d^ has area (io) diù^ hBd%d^.
The surface fluxes, in a similar blade, are defined as in a plate. At the point (a, ^), we will denote by/ the component of the flow along the line a, and by "- the component of the flow along the line ^.
Let P be the average density of electricity at the point (a, ^) of the plate. We can easily find that
(it) AB -,- = - ( - ^ -f- -,"- If S is the average density at the edge of the plate and n/ is the tangent line to the plate, normal to the edge, and directed towards the interior of the plate, we have
(-22) /cos(/î/, a)-+-^cos(/i/, p)=- -^-
If the currents are permanent, we must, according to equality (21), have, at any point of the plate,
and, according to equality (22), have, at any point of the edge,
(24) fcos(ni, a)-i- ffcosi/ii, ^) = o.
11 It is easy to see that this last equality can still be written, designating by dl an element of the edge of the plate,
(^5) ^f^-^^^dl='' Let us assume not only that the currents are permanent, but also that the plate is homogeneous-, let ^H. be the specific resistance of the material that forms it; let 8 be its thickness at the point (a, ^) Let us assume
(26) tl-î
432 BOOK V. - METALLIC CONDUCTORS.
Ohm's law will give us, as we can easily see,
^ -^ îl A da '
(^7) ■ - r dV
Assume that the plate has the same thickness at all points; % will be independent of (a, ^). By virtue of equations (2-), equation (23) will become
d /B dV\ d /A dV
while equality (25) will become
^'^-^^ A You II ^" B â^ di ~ "■
If we add to these equations the condition of taking given values along the contour of the electrodes, we will have obtained all the conditions that determine the potential function V. Once this function has been determined, the equalities (27) will give the components of the surface flvix at each point of the blade.
The determination of the function V becomes much simpler if we can find a curvilinear coordinate system on the surface of the plate for which A = B is constant. Such a system is called an isothermal system. It is also called an isometric system, which is due to M. O. Bonnet and which recalls the following fact: If Von agrees to always take d(^,jk), whose rectangular coordinates :r, r have for respective values
37 = a, j' = p.
To any infinitely small figure drawn on the surface this law causes a similar infinitely small figure drawn on the plane to correspond. It is said that such a correspondence provides a conformal representation or a geographic plot of the surface on the plane. The angle of two lines drawn on the surface is equal to the angle of the two lines that represent them on the geographic plot.
Conversely, if we have the geographical layout of a surface on a plane, any rectilinear and rectangular coordinate system on this plane is the representation of an isothermal system of the surface.
These proposals show that the search for an iso
(') Gauss, Allgenieiiie Auflosung der Aufgabe die Tlieile eiiier gegebeneii
Flàche auf einer andern gegebenen Flàche so abziibilden dass die Abbildung
dem Abgebildeten in den kleinsten Theilen àhnlich wird (Gauss, Werke,
Bd. IV, p. 193). - G. Darboux, Leçons sur la théorie des surfaces, t. I, p. 146.
D. - I. 28
434 BOOK V. - METALLIC CONDUCTORS.
therm on a surface is equivalent to the search for a conformal representation of this surface on a plane.
If, therefore, on the surface of the slide in which we want to study the permanent movement of electricity, we know how to find an isothermal system, we know, by this very fact, how to make the geographical trace of the slide on a plane. This tracing will draw a plate, whose edge and electrodes will be the images of the edge and electrodes of the studied slide.
We can always suppose that the correspondence between the blade and its representation has been obtained by equating respectively k X and y the parameters a, ^ of the isothermal system taken, on the blade, for coordinate system. The function V(a, [3) will then be transformed into a function Ni^x^y')^ which will have the following properties:
1° By virtue of equality (3o), at any point on the plate, image of the blade, we have
2° At any point of the contour of the electrodes of the plate, images of the electrodes of the blade, one will have, in virtue of the equalities (Sa),
3'^ At any point on the edge of the plate, image of the edge of the blade, we will have, by virtue of equality (3i),
^-="'
N,- being the normal to the edge of the plate towards the interior of this plate.
Thus, the function V(a,[^), which solves the problem of the permanent motion of electricity on the blade, is identical to the function V(^, y) which solves the same problem for the plate, a conformal representation of the blade, and we can state the following proposition:
When we know how to make on a plane the geographical trace of a curved blade and to find the permanent movement of V electricity in a plate cast on this trace, we know how to find
CHAI". V. - MOVEMENT OF' ELECTRICITY IN A PLATE. 435
see the permanent movement of electricity in the curved blade.
These theorems, whose inventor it would be difficult to name, since they are so closely related to a host of questions of analysis, were given explicitly by G. Rirchhoff(' ).
Let us prove again, to finish this study, this beautiful theorem:
When V electricity moves in a blade of permanent motion, the lines of equal potential level and the lines of flux always form an isothermal system.
These two families of lines, equipolential lines and flux lines, always form an orthogonal system. We can therefore assume that we have taken these lines as a system of curvilinear coordinates on the surface of the blade, the a lines coinciding with the equipotential lines, and the ^ lines with the flux lines.
We must always have equality
( ^\ "> /B ^\ _^ /A dW
But, for the equipotential lines to coincide with the lines
P = const.,
it is necessary that V depends only on a, which reduces equation (28) to
d /B dY\
ôx \A. doL J
Thus, denoting by W(^) a certain function of [i,
A d% ^^^
(*) G. KmcHHOFF, Ueber die stationàren elektrischen Strômungen in einer gekriimmten leitenden Flàche {Monatsber. der Akademie der Wissenschaften zu Berlin, 19 July 1875. - Kirchhoff's Abhandlungen, p. 56). - On the relation of these theorems to the theory of analytic functions, see the already quoted work of M- F. Klein, Ueber Rieniann's Théorie der algebraischen Functionen und ihrer Intégrale (Leipzig, 1882).
436 BOOK V. - METALLIC CONDUCTORS.
dV Moreover, -^ depends only on a; if we denote this function of a by $(a), we will have
and the equality (19) will become
"?52= X2(a,[B)[2(a)^a2H- W2(P)^j32].
Now let's make the following change of variables, a change that will not change the coordinate lines:
a'= / ^(a) da,
The function ).(a, [i) will turn into a function 4^(a', |i'), and we will have
The line system
a'=const, [B'=const.
thus forms an isothermal system; and, as this system coincides with the system
a = cons!., p = const.,
the stated theorem is proved.
M. Boltzmann ("), G. Kirchhofï (2), É. Mathieu (3) have completely solved, by means of these principles, the problem of the motion of electricity in certain surfaces. We refer the reader to their works, and in particular to the Treatise of É. Mathieu.
(') Boltzmann, Ueber die Bewegung der Elektricitàt in krummen Flâchen {Sitzungsber. der kaiserlichen Akademie der Wissenschaften zu Wien, t. LU, p. 2i4; i855).
(' ) G. KiRCHHOFF, loC.cit.
(' ) É. Mathieu, Théorie de l'Électrodynamique. Paris, 1888.
CHAP. VI. - JOULE'S LAW. 437
CHAPTER YI.
JOULE'S LAW.
When a linear conductor is crossed by a permanent current, it heats up; the laws of this heating have been determined experimentally, starting in 1840, by Joule (' ), Lenz (2) and M. Edmond Becquerel (^). The result they arrived at can be summarized in the following law, which is called Joule's law:
When a linear and homogeneous conductor, having at any point the same temperature, is traversed by a permanent current, each element ds of this wire becomes a heat source. In time dt^ this element gives off a quantity of heat whose value dQ is given by P equality
(i) Ed()=-Ri'^dsdt,
E being the mechanical equivalent of heat, Kds the resistance of r element ds, J the current intensity.
This law can also be put in another form.
Let V be the value of the potential function at the origin of the element ds ^\ (^ + ir ^^) ^^ value at the end of the same element; Ohm's law gives us
RJ o?i- = - £ - - ds. as
(') Joule, On the heat evolved by nietallic conductors of electricity, and in the cells of a battery during electrolysis ( Proceedings of the Royal Society, 17 December i84o; Philosophical Magazine, 3' series, t. XIX, p. 260; 1841, and t. XX, p. 204; 1843).
(') Lenz, Ueber die Gezetze der Wârme-Entwickelung durch den galvanischen Strom (Poggendorff's Annalen, t. LXI, p. 44; i844) (') Edmond Becquerel, Des lois du dégagement de la chaleur pendant le passage des courants électriques à travers les corps solides et liquides {Annales de Chimie et de Physique, 3' série, t. IX, p. 21; i843)
438 BOOK V. - METALLIC CONDUCTORS.
Equality (i) can then be replaced by the following:
dV (■2) ^dÇl = - z-T-idsdt.
G. Kirclihofr(<) has shown how Joule's law can be extended to permanent currents flowing in an extended conductor in all dimensions.
Let us consider, inside a homogeneous conductor through which permanent currents flow, an infinitely loose channel whose walls are generated by a series of flux lines {fi g- 87).
Fig. 87.
It is natural to assimilate such a channel to a linear conductor through which a permanent current flows.
Let us cut this channel by two normal sections co, to', whose infinitely small distance, MM', is equal to ds. If we assimilate our small channel to a linear conductor, the intensity of the current which crosses it at M will have the value
J = iiù,
i being the electric flux at point M; the resistance of the element MM will be
R f/5 = - ds,
Si. being the specific resistance of the substance which forms the conductor. We are therefore led, by applying formula (i) to the segment between the two sections o), w', to suppose that this segment gives off, in the time dt, a quantity of heat given by the equality
E(iQ =Sltùi^dsdt.
(') G. KiRCiiHOFF, Ueber die Anwendbarkeit der Formeln fur die Intensitâten der galvanischen Strôme in einem System Unearer Leiter auf Système, die zum Theil aus nicht linearen Leitern bestehen {Poggendorff's Annalen, Bd. LXXV, p. 189; 1888. - Kirchhoff's Abhandlungen, p. 33).
CHAP. VI. - L.\ JOULE'S LAW. 439
But, on the other hand, if we denote by u, v^ iv the components of the electric flux at point M, we have
J^a quantity iii ds is nothing else than the volume of the small segment between the sections to and w'. We thus arrive at the following statement, which is the extension of Joule's law to a conductor of finite extent in all dimensions:
When a homogeneous conductor, whose points are all at the same temperature, is traversed by a permanent current, each element of volume dx dy dz of this conductor releases, in the time dt^ a quantity of heat o?Q given by the formula
(3) Ed<^ = S^{u'^-^v'^^w^)dxdydzdt.
This equality can be transformed as we have transformed equality (i); indeed, Ohm's law gives
ox
S\,v = - £--,
at
^ (V = - s -r- .
dz According to these relations, equality (3) can be written :
/c)V d\ ôW \
(4) ^d^l^-zi^^u-^ -v-^^w'jdxdydzdt.
R. Clausius (') gave a statement of equations (2) and (4) which is interesting to know.
The electrostatic potential of a system having the expression
expression in which the summation extends to all loads
(-) R. Clausius, Ueber die bei einem stationâren elektrischen Strome in dem Leiter gethane Arbeit und erzeiigte Wàrme {Poggendorff's Annalen, Bd. LXXXVII, p. 4i5; i852. - Mémoires sur la Théorie mécanique de la chaleur, translated by Folie, t. II, p. ii4).
44o BOOK V. - METALLIC CONDUCTORS.
of the system, when the electrical distribution undergoes an infinitely small variation, this potential experiences a variation
(5) o\y = z\ègèg,
ùq being the variation of the electric charge at the point where the potential function has the value V.
This being said, let us first consider a segment MM' of linear and homogeneous conductor, crossed from M to M' by a permanent current of intensity J. In time dl, this segment releases, according to equality (2), a quantity of heat c/Q given by
ErfQ = - z(y-\)idf.
Let us imagine that, during time dt, the segment MM' had been traversed by the uniform current of intensity J, but that the electricity had remained at rest on the rest of the system. The electrical distribution on the system would then have undergone a change in time dt; the charge at point M would have decreased by J dt, and the charge at point M' would have increased by the same amount. According to equality (5), the electrostatic potential of the system would have increased by
8W = e(V'- V)J^;.
The comparison of the two equalities we have just written leads to the following proposition:
Let MM' be a linear, homogeneous segment of a conductor through which a permanent current flows. It gives off, in time dt, the quantity of heat dQ which can be calculated in the following way: let us suppose that, in time dt, the segment MM' has been traversed by the permanent current which in reality traverses it, while the rest of the system has not been traversed by any current. The electric distribution on the system would have undergone a certain change, resulting in a certain variation oW of the electrostatic potential of the system, and Von would have
(G) E^Q=- ûW.
This theorem can be extended to three-dimensional conductors as follows:
CHAP. VI. - I.A JOULE'S LAW. 44'
Inside a homogeneous conductor, having at any point the same temperature, through which permanent currents flow, let us trace a closed surface S (^fig- 88). The part of the conductor if
The heat generated inside the conductor releases, in the time dt^ a quantity of heat û^Q, and we have, according to the equality (4),
ErfQ =-srf/yyyy(-^"H- ^.-^ ^\.) dxdydz,
the triple integral extending to the volume bounded by the surface S.
Let JN/ be the normal to the surface S towards the interior of this surface. An integration by parts gives :
= - V V[?f cos(N/, x)-^ V cos(N/,_7) -+- w cos(N;, z)dS
If we notice that, the currents being permanent, we have, at any point inside the surface S,
du ôv ôw
T" -^ T" + X" = ^' ox oy az
we see that we can write
(7) Efi?Q = £C^^ C V["cos(N,-, ar)4-pcos(N/, 7)-r- tv cos(N/, z)] rfS.
Let us imagine, on the other hand, that the space inside the surface S remains crossed, during the time dt^ by^ the permanent currents which cross it in reality, while the electricity would be in rest on the rest of the system. The electrical distribution on the
44" BOOK V. - METALLIC CONDUCTORS.
system would undergo a certain change that we can determine.
The electric charge at a point outside the surface S would not change, since there would be no current at that point.
The electric charge at a point inside the surface S would not change either, since at this point the current would be uniform.
But it would not be the same at a point of the surface S. The element c/S of the surface S would acquire, in the time dt^ a charge
0^ = - dt\u cos(N/, a-) -+- f cos(N/, y) -t- w cos(N,-, z)\ dS.
This change in electrical distribution would lead to a change in electrostatic potential, having as value, according to the formula (5),
(8) SW ^- - zdt C V[i
the quantity - is in general finite, but that the quantity - ^ is
always extremely small. Finally, let us observe that
Vo[cos(r, n/)] = i, Vi[cos(r, n/)] = cos (r,n/),
454 BOOK V. - METALLIC CONDUCTORS.
and we will have
Taking into account equality (i), if we observe that
cos(r, rii) _ /■
/'■^ Ofii
if we finally put
(2) OIl^S -- C p / rf^-,
we will have
V = D1L - ^S. (Jrii
If it is a question of calculating the potential function, not of the small element A, but of the whole layer, we will have for expression of this potential function
ô^
(3) Y=^DrL~dS.
This expression is valid only as long as the point M, to which the function V refers, is at a large distance from the surface S with respect to (). H- jx).
DÏL will be named [ intensity of the double layer at a point of the element dS.
Let us assume that the double layer considered is the one that is found, in the state of equilibrium, at the separation surface of a conducting body A and an insulator o. In equality (2), we must make
dif = dl dS, P = p(A,o, /),
and 0 will be defined by the equality
01I(A,o) = / /?(A,o, l)dl,
or
(4) D\iiX,o)---= -- / lAe(k,o,l)dl.
4Tr£ J^
CHAP. VII. - POTENTIAL DIFFERENCE IN CONTACT. .05
This equality shows us that the quantity DTL(A, o) is completely defined when we know the nature of the metal k. and of V insulator o.
Let us consider in the same way the double layer which forms, in the state of electrical equilibrium, at the separating surface of two metals A and B. Let us suppose that the direction, designated in the foregoing by /i, is the normal "^ towards the interior of the metal. We will have again
dv = dS dl;
l will have to vary from - ()v -+- (jl) to H- (X -h a ); / varying from - (^ + [J"-) to o, p will take the determination p(B, A, - ■ l); on the contrary, / varying from o to ()v -f- [x), p will take the determination p(A, B, /). We will thus have
OTL=:/ lp{B,X, - l)dl-i- lp{X,B,t)dl,
or
i lp{k,B,l)dl- lp(B,A,l)dl,
what we can still write
(7) 3rt(A,B)--l:
4710
J^X,+ [JL ^X+[i "I
f l^e{X,B,l)dl- lù.e{B,A,l)dl\
^^0 .1
This equality shows us again that the quantity 31L(A, B) is fully defined when the nature of the two metals A and B is known.
These two quantities, 311 (A, o), 31L(A, B), are still susceptible of another expression.
Consider first the quantity D1I/(iV, o). The quantity varies
very quickly, in the direction of the normal to the surface S; at
on the contrary, its variations are very slow following a parallel to the
tangent plane to this surface. We can therefore reduce significantly
ACk/ A A ' <^2e(A,o, /) Ae(A, o, /)a ^^^
We will thus have
^'^■°>=4^eX
"/. *-
456 BOOK V. - METALLIC CONDUCTORS.
An integration by parts transforms this equality into
4TrE L (^t Jo 4to Jq àl
If we observe that
and if we integrate the last term, we find
(8) OrL(A,o)= -^[e(A,o,o)-0(A,o,X-h[x)].
In a similar way, equality (7) can be transformed into
3K(A.B)= ■ M''(A.B.^, _,ç "e(g^y. 4^£ L"^ "' Jo
If we observe that
rc)e(A ,B, /) 1 ^^
peçB, A, /) ] _
0(A, B, o) = e(B, A,o), the previous equality will become
(9) 01I( A, B) = -1- [e(B, A, X + fx) - 0(A, B, X -^ ^)\.
4 ir£
Let us return to the expression of the potential function of any double layer, given by equality (5).
Suppose that the surface S is closed and that the double layer has the same constitution at all points of this surface. The quantity Oit will then have the same value at all points of the surface S, and the equality (5) can be written
But Gauss's lemmas teach us that at any point M,
CHAP. VU. - POTENTIAL DIFFERENCE AT CONTACT. 4'^7
outside the surface S, we have
at
and that at any point M, inside the surface S, we have
We have therefore at any point M, outside the surface S, and whose distance to this surface is large compared to X,
(10) V = o
and at any point M, inside the surface S, and whose distance to this surface is large compared to ),
(11) V=:47rDR.
Let us make some applications of these equalities (lo) and (i i).
i" Consider a homogeneous conductor. The natural distribution forms on this conductor a homogeneous double layer {see p. 3-6) for which OÏL is given by one of the equalities (6) or (8). The potential function of this natural distribution will have the value o outside the conductor and the value
4TtDlL(A, o)= i[e(A, o, G) - e(A,o, X^ (x)].
This is what we had already found previously [Chap. II, equalities (5) and (6)].
2° Let us consider a heterogeneous conductor, formed by two metals A and B. If we denote by 8^^,07 Sgo the two portions of closed surface which limit the conductor, and by S^g the contact surface of the two metals, the equality (5) can be written more explicitly
d'- d^ d' V = D1I(A, o) j^ ^ dSA,o^ 3n,(B, o) g ^c?SB,o-i- 31I(A,B) g ^ ^Sa.b This expression can be simplified, by means of the equalities (8)
and (9).
V
4Tr£
458 BOOK V. - METALLIC CONDUCTORS.
If one obsei've that one has
e(A,o, ^) = 0(A, B,). -t- IX) = e(A),
e(B, o, X -f- [ji) = e(B, kA-v n.) = e(B),
we can write
-l-[e(A,o,o)§-i S r!i< '^"1 ■
Let's put
e(A,o, o) - e(B,o, o)
I-- -,
and
^1^ ^ e(A,o, o)- p. ^ [X - e(B, o, o) ^
47:2 i^-Kt
If we observe that, for any point outside the conductor, we have
) =0,
CllAP. VII. - POTENTIAL DIFFERENCE AT CONTACT. 4^9
we can write
= 3'^S3)
VCMa)4-£0(A) =
/(TTô
u
dl
Similarly, for a point Mj, inside the metal B, it gives
(i3 bis)
V(MB) + £e(B)
iT.Z
d'
0(A, o, o) V - c?Sa,o
e(B,o,o)S-^£^SB,"
]
The second members of these equalities are obviously not constant. To be sure of this, it is sufficient to note that V - - dSj^^^
and
V-- t/Seo are the angles under which, from the point M, we see
the inner faces of the surfaces S^^o, Suo- Now, for the natural distribution on the two metals A and B in contact to be an equilibrium distribution, we would have to have
V(MA)+£e(A) =V(MBJ-+-£e(B) = G,
G being a constant.
The two metals in contact must therefore be electrified. If we observe that
S5^/^'^'""^S^-^'^'"=^°'
CHVP. VII. - DIFFERENCE IN POTENTIAL ON CONTACT. î6l
it is easy to see that V eleclriché ■ conifnunrq née will have to be distributed in such a way that we have, inside the conductor,
e(A,o, o) - e(B,o,o)
à'
Sâ;::-^
c.
4 710
The study of this distribution, made by any means, can never make known that the quantity
e(A,o,o) - e(B, G, o), not the quantity
e(A) - e(B).
§ 3 Permanent currents in metallic conductors
heterogeneous.
The preceding considerations apply only to the case where the electrical equilibrium is established on heterogeneous metallic conductors. We will now study the case where these conductors are crossed by permanent currents.
In order to obtain the fundamental laws of permanent motion of electricity in heterogeneous metallic conductors, we will extend to them the first fundamental assumption concerning the steady state, an assumption which, in the case of homogeneous conductors, is only another form of Ohm's law [Chap. IV, equality (2)].
Let us consider the set of variables, other than the electric fluxes, which define the state of the system at time t (what we call the system supposed to be currentless). To this system supposed to be currentless we can apply the consequences of Carnot's principle; any virtual transformation of this system corresponds to a certain amount of uncompensated work.
Let {x,y, s) be a point in the system and(;z: -f- ox,y-^ Sy, z -{- ùz) an adjacent point. If an electric charge dq passed from the first point to the second, some uncompensated work would be generated in the system. Let us denote it by d'z. Let us denote by C^, ^y, Czthe components of the electromotive force at the point (cc,y,z). By hypothesis, we have
( l4) d-z = (C.c ox -i- Cy8y -i- Cz 0^) dq,
whatever Sx, 0JK5 ^^
46-2 BOOK V. - METALLIC CONDUCTORS.
Now we can calculate d-z. The system assumed without current has an internal thermodynamic potential §. The change of position of the electric charge dq does not move any part of the system and, consequently, does not cause any external work. We have therefore
(i5) d-^ - li.
Moreover, we have [Liv. IV, Ghap, II, equality (i5)]
the various letters that appear in this formula having a meaning that has been explained in the place quoted. Since the displacement of the charge dq does not cause any change in the physical or chemical state of the various conductors of the system, we see that the variation undergone by the internal thermodynamic potential is reduced to
(i6) / \ -^
l ' \dx dx "^ ~^ dx " ) ^'
The equalities (i4), (i5) and (i6) having to take place, whatever ox, hy, 8^, give
dx âx '
('7) ^y--^ -^)j, ^^
Cz = - £
dv of
dx âx dY from dy d\ __ from dz toz
These equalities give the components of the electromotive force at a point of a heterogeneous metallic conductor whose constitution varies in a continuous way and which is crossed by
T dB de de
permanent currents. The terms r- ? -j-, - -r- represent
r dz dy dx '
the correction that must be made to Ohm's law in the case where
the driver is not homogeneous.
If we denote by Si. the specific resistance of the conductor
at the point with coordinates x, y, z and by ", p, w the components
CHAP. VU. - POTENTIAL DIFFERENCE AT CONTACT. 463
of the electric flux at this point, the equalities (17) will become
(.8)
Let's see how electricity is distributed inside a similar conductor.
Let's divide the two members of each of the equations (18) by cîR; let's differentiate the two members of the first one with respect to x, the two members of the second one with respect to y^ the two members of the third one with respect to z^ and let's add member by member the results obtained, observing that
[^-=-^Tx
dx
1 dV
from
from
dz'
du dv div dx dy dz
We find
i^(sAV^Ae)
'9)
to- d- to-
o.
T
/ c)A dSi. dcfl
M -T \-V - h tV --
4t:£
\ dx dy . dz
\ dx dx dy dy dz dz
If denotes the electric density at the point (^, J', -s), we have
AV = - 4itp, and equalities (18) and (19) give
(20) o = -^Ae
- 4lTS
while, in the equilibrium state, the electric density at the same point would have the value
p = -^ Ae.
In the interior of a heterogeneous metallic conductor carrying permanent currents, the density does not have the same value as in the steady state. It is only in the interior of the conductor that the
464 rjvKE V. - metallic conductors.
The two values of the homogeneous conductor become equal to each other, and equal to o.
Let us now consider a discontinuity surface separating two regions a and b where the conductor has different properties
("■- 92)- _ ■ _ _
We know that at any point M of this surface we must have
(-M)
Ua COS(N", X) -+- Va. COS( Na, y) -+■ Wa COs(Na, z)
uo cos(N6, op) -+- v/, cos(N6, y) -\- w/, cos(Nft, z) - o,
or, by designating 'by ia the component following N^ of the flux at
Fig. 93.
inside the conductor a at a point infinitely close to the point M; by i^ the component along N^ of the flux inside the conductor b at a point infinitely close to the point M,
{-21 bis) ia-T- ib= ()■
The normal component of the flux has the same value on both sides of the surface.
Is the same true for tangential components?
Through the point M, let us lead two directions T, T', rectangular between them and located in the tangent plane to the surface S. At a point of the conductor rt close to the point M, the flux /"" has as components, along MT, MT', ta and t'^; at a point of the conductor b close to the point M, the flux fh has as components, along the
niAP. seen. - POTENTIAL DIFFERENCE AT CONTACT. 465
same lines, ti, and ^^. Equalities (i8) give
The well known properties of the first derivatives of the potential function give
dT dT' dT dT'
On the other hand, by hypothesis, the first order partial derivatives of the function B vary in a continuous way, even at the passage of a discontinuity surface, which gives
So we have
(22)
The electric jiux on either side of a discontinuity surface are in the same plane as the normal to the discontinuity surface. The projections of these two flows on the tangent plane to the surface have the same direction.
From the comparison of equalities (21a) and (22) we deduce the equality
S^a tang(/a, Na ) -+- iKi, tang(//" N^) = o,
which indicates how the direction of the electric flux varies when passing a discontinuity surface.
The set of propositions that we have just demonstrated has received from Maxwell (' ) the name of the laws of the refaction of the electric flux at the passage of a surface of discontinuity.
(') Maxwell, Traité d'Éleclricité et de Magnétisme, translated by G. Seligmann-Lui, t. 1, p. l\<.yi. Paris, i885.
D. - I. 3o
d^a dt
dT'
dSa dSb dT dT
l J^a ta -
■ Slt>t/j= 0,
{ ^at'a~
■ài'/J',,--- 0.
466 BOOK V. - METALLIC CONDUCTORS.
Returning to equation (21), qvii can be written, by virtue of the equalities (18),
OR
J_ / àV_ àY_\ f d(da , àei,\ I \ \ d
If we denote by t the surface density at the point M of the discontinuity surface, we have
- 4Tra,
inN'a
to&a
^ àeo
to^a
àNh
_ - (sV + 0) - - <îR/,4 - <^64, and the previous equality will become
Thus, at the separating surface of two metallic conductors a and h^ through which permanent currents flow, there is a surface electrical distribution whose intensity at each point is proportional to the excess of the resistance of conductor a over the resistance of conductor b and to the component, along the normal to the suif ace, of the flux that penetrates conductor a.
Similar considerations apply to the separation surface of a conductor a and the insulator i. In this case, the electric flux is zero at any point of the insulator i, and moreover we have, at any point of the separation surface,
tfaCOS(Na, X) H- Ç>aCOS(Na,7) -f- "'aCOS(Na, s) = O,
which gives, according to the equalities (18),
The electrical density at a point on the surface separating a conductor from an insulator is therefore, if we denote by N^ the normal
CflAP. VII. - POTENTIAL DIFFERENCE AT CONTACT. 4^7
external to the driver,
L ^X _^ _i_ -^
When the constitution of the conductor is known, equation (19) becomes a partial differential equation which, together with the boundary conditions (aS) and (sS), determines the function V. Once this function has been determined, the equalities (i8) determine the components of the electric flux at each point.
Let us apply the preceding considerations to a conductor through which permanent currents flow, in which two metals a and b, homogeneous up to a small distance from the end surfaces, are in contact with each other.
1° At a distance greater than (7^+ [x) from the surfaces that bound each of the two metals, there is no free electricity spread inside these metals; the equations of motion of electricity are those given by Ohm's law.
2° In the vicinity of the surface that separates the two metals a and b^ there is electricity spread inside each of the two metavixes. The distribution of this electricity depends on the intensity of the current flowing through the conductor.
3" On the surface itself, there is a surface distribution whose density has the value
(27) G= ■-^[Aa(0)-Sih{0)]ia,
Sla{i), ^^b{i) being the specific resistances of metals a and 6, at a distance / from their contact surface.
The electricity thus distributed on the contact surface of two metals through which permanent currents flow no longer forms a double layer, as in the steady state.
To prove this theorem, let us agree to represent by 0(a, Z/, /), <îfl(rt, ^, /), f(rt, 6, /) the values of ©"j^l^, ia^ at a point of the conductor a located at the distance / from the surface that separates the conductor from the conductor b. At this point, the electric density
4()8 BOOK V. - METALLIC CONDUCTORS.
has the value, according to the equalities (20) and (18),
('^8)
p(",è, l) ^ - !- Ae(a,6./)
^i_ I J dSl(a,b,l) d[ z\-^e(a,b,l)]
4TrE S{.{a, b, L) \ dx dx
dSl(a,b,l) d\zY ^%{a,b,l)\ dy
^ dSi{a,bJ) d{sY -^e (a,b, l)] dz
Let us lead two surfaces parallel to the surface S {/ig'- g'i), which separates the two metals a and b, these two surfaces being located at
Fig- 93AL _Tr"
a distance slightly greater than (). - [*■) from the surface S, one, S^, inside. the conductor "; the other, S^, inside the conductor b. On the surface S, let us take an element MN, of area dS. Through all the points of the contour of this element, let us lead normals to the surface S. These normals detach on the surface S^ an element M^N.i and on the surface Sô an element M/, Ni . Let us find the quantity of electricity contained in the closed surface M^N^MiNj, neglecting in this calculation the ratio of Çk -\- u.) to the radii of curvature of the surfaces S", S^.
The element dS carries a quantity of electricity which has the value, according to equality (27),
0,^=-' - [A{a,'b, o)~éfl(b, a,o)]i(a, b, o).
The MNAI^Na surface contains, inside, a quantity of electricity
i p(a,b,l)dl.
CH.VP. VII. - CONTACT POTENTIAL DIFFERENCE. 4^9
The MNMôNô surface contains inside a quantity of electricity
Qb=^dS ç>{b,a,l)dL
The quantity we want to calculate has the value
If we refer to equality (28), we can easily see that we can write
p(a, b, l)^^ ~ Ue(", b, l) -f- i{a. b, l) '^i'Î^^lM]]
and also
So we have
Q==-f^ / [Ae(". 6, /)-4-Ae(6,a, 0]^^
,"/ /, Nv / ^ r^'''^dSl{a,b,l) ., , ,, -^ S{.{a, o, o)i{a, b, o) -^- 1 -jj - - - i{a, b, i) al
-~S\.{b,a, o)i{b, a,o)-+- j t(6, a, l) dl\ .
The quantity
represents the amount of electricity that would be present on the MaNaM^Ni surface, if no current flowed through the system. In this case, the electricity distributed near the contact surface of the two metals would form a double layer. The previous quantity is therefore equal to o.
An integration by parts gives us
f i (a, b, l)
al
âl
9
= -/" ' Sl{a,b,l)'
dl
dl -r- i{a, è, X -f- \x)S{.{a, 6, X h- [j.) - i(a, b, o) Sl(a, b, o).
But it is easy to see that the integral in the second member is of the order of (X-h[x) and can, therefore, be
470 BOOK V. - METALLIC CONDUCTORS.
neglected. We have therefore
J"/H-U. f t(") b, l)
dl
and, similarly,
i{b,a,o) ël{o,a,o) -T- I i{b,a,l) ^^ -
= i{b. "(X -f- [x) S\.{b, a, X -+- p.). If we notice that
i(a, 6, X -i- jx) = i{a, b, o), i( b, a,A -h [i) = i(b, a, o),
to quantities of the order of (X + [x), and that, moreover, according to
equality (21 bis),
i{a, b, o) -r- i{b, a, o) = o,
we see that we will have
dS
('9)
( Q = ^~ [Siia, b, l -^- [i) - A {b, a, l-r- [i)]i{a, b,o) <
= - - [^ib, a, X -r- [x) - cR(a, 6, X -f- [x)]t(6, ", o).
If the electric flux is not tangent to the discontinuity surface, and if the specific resistance is not the same inside the two conductors, this quantity cannot be equal to o.
It is interesting to compare the results we have just obtained, by studying a heterogeneous metallic conductor carrying permanent currents, with the results we obtained in § 1, by studying the electrical equilibrium on a similar conductor.
When the electrical equilibrium is established on the system that encloses the two metals a and è, the electrical density at a point on the contact surface has the value
ff = 0,
and the amount of electricity contained in the volume MaN^MôNô a
for value
Q = o.
When the system is traversed by uniform currents, one
CHAP. VII. - POTENTIAL DIFFERENCE AT THE CONTACT. 47 1
a, by virtue of equalities (17) and (19),
!T = - - [.^(a, 6, o) - ifl(6, a, o)]f(", 6, o),
Q = 7 - [<îR.(a, 6, X -t- a) - Jl(6, a, X -i- ijL)]t(", 6, o).
47î£
The analogy of the expressions obtained for a and Q in the first case is found in the second.
Suppose that ABCD i^fig. 94) is a homogeneous metal a
than at the distance (X -i- |x) of the surfaces which terminate it. By the face AB, this metal borders on another metallic conductor, formed by a metal b^ which maintains all the points of the surface AB at the same potential level V,. Likewise, by the face CD, it borders on another metallic conductor, formed by a metal c, which maintains all the points of the surface CD at the same potential level Vo. The surface ABCD confines with the insulator. Let us propose to find the distribution of the uniform currents that flow through this conductor. This is a problem that we have already dealt with, assuming that the conductor is homogeneous up to the terminal surfaces.
Let us lead a surface A'B'C'D', located inside the conductor a, at a distance i^-^[a) from the surface ABCD.
Let M be a point on AB. Through this point, let us lead a normal N^ to the surface AB towards the interior of the body a. It meets at M' the surface A'B'. Let V'^ be the potential level at M', a level that we propose to determine.
At a point between M and M', at a distance / from the point M, we have, keeping the notations we just used
Sl(a,b, l)i{a,b,l) = - t-jj- - ■ -^
472 BOOK V. - MKTAI.LtQUES DRIVERS.
and, therefore,
£(Vi - V;)-4-e(a,6, o) - e(",6, X-^[x) = -- / ^(",6, l)i{a, b, l)clL
The second member is a quantity of the order of Q^-i-'^)', it is, therefore, negligible; therefore all points of the surface A'B' are at the same potential level
V'i = Vi -+- ^ [&{a, b, o) - e(a, b,\-^ [Ji)].
Similarly, all points on the surface CD' are at the same potential level
V'j =^ Vs-h - [6(a, c, o) - 0(a, c, X --t-[x)].
Let P be a point on the surface AC, through which the conductor a borders the insulator. Through this point, let us lead a normal PN' to the surface AC. It meets at P' the surface A'C
At point P, we have
u cos(N', x) -i- V cos(N', y) -^ w cos(N', z) - o.
Because of the continuity of the components of the electric flux, at the point P', the quantities u' , v\ w' will differ from ", c, w only by quantities of the order of (\-'\- ]x). One will have thus appreciably
u! cos(N', x) -f- v' cos(N', y) -- w' cos(N', z) - o.
But the point P' is in the homogeneous region of the conductor where Ohm's law becomes applicable, so that the previous relation can be written
M' = ""■
This equality will be verified at any point on the surface A'C'B'D'.
Finally, inside the closed surface A'C'B'D', the conductor
being homogeneous, we have
AV =-: o.
The function V is therefore harmonic at any point inside the closed surface A'C'D'B'; it takes, on the surfaces A'B', CD', given values Vj, V'^ -, on the surface A'C'B'D', it verifies the equality
m' = ^
cn\P. VII. - POTENTIAL DIFFERENCE AT CONTACT. 47^
It can be seen, therefore, that the determination of the currents flowing in our conductor at a distance greater than {\ -h jj.) from the terminal surfaces will be obtained precisely by the analytical method formed in Chapter IV, for determining the currents flowing in a homogeneous conductor up to the terminal surfaces. Moreover, once the components of the electric fluid are known up to a distance (1+ pi) from the terminal surfaces, it will be easy, by continuity, to know their values jto the terminal surfaces.
The result we have just obtained shows that the consequences to which, in Chapters IV and V, we were led by applying Ohm's law to absolutely homogeneous conductors, remain valid for conductors which lose their homogeneity at a very small distance from the terminal surfaces.
§ 4 Method of M. Pellat to determine the potential level differences of two metals in contact.
The study of electrical equilibrium and of the permanent motion of electricity on heterogeneous metallic conductors would still involve the examination of a great number of questions which it is not possible to deal with here. We shall therefore confine ourselves to the principles set forth in the preceding paragraphs, and for an examination of the general consequences of these principles, we shall refer to a Memorandum that we have published on this subject (' ).
Nevertheless, before abandoning this order of questions, we will expose the method followed by M. Pellat (2) to determine the potential level dilTerence of two metallic conductors in contact.
Let's take the two concentric spherical conductors that we have already considered in § 1, and that, in practice, we
(') P. DuuEM, Sur la pression électrique et les phénomènes électrocapillaires; première Partie, De la pression électrique {Annales de l'École Normale supérieure, 3" série, t. V, p. 97; 1888); seconde Parlie, Des phénomènes électrocapillaires {/bid., 3° série, t. VI, p. iS3; i88()).
(') H. Pellat, Difference of potential of the electric layers which cover two metals in contact {Annales de Chimie et de Physique, 5" série, t. XXIV, p. 5; 1881).
474 BOOK V. - METALLIC CONDUCTORS.
replace them with two parallel plates. Let us imagine that a wire MN with a bend in it is added to these spheres {fig. po). From the point m of this wire, which is made of any metal, another wire /??/>, also made of any metal, leaves and rejoins the
The conductive sphere 2, formed by the substance a, is connected at p. Similarly, the conductive sphere 4? formed by the substance h is connected by a wire of any kind p' n to the point 11 of the wire MN.
Let's assume that the steady state is established on the system.
First, I say that in this steady state, there is no current outside the MN wire.
If there were a current in the wire mp whose intensity differs from o, it would enter, during the time dt in the part of the system formed by the wire mp and the conductor 2, a quantity of electricity jdt which would not leave it. It would be necessary, therefore, that the electrical distribution be modified on the considered part of the system. Thus, the wire mp is not crossed by any current, and it is the same for the wire np' .
Can it happen that permanent currents exist either in the ground of conductor 2 or in the ground of conductor 4?
As we have just seen, these conductors can be considered as isolated.
Let M, ç", w be the components of the electric flux at a point of an isolated system formed by homogeneous or heterogeneous metallic conductors.
CHAP. VII. - POTENTIAL DIFFERENCE AT CONTACT. 47^
INoiis will have, according to the equalities ('j),
which can be written
[^ ox oy oz
d(£V-4-e) , ()(£V-T-e) , _àizY-^6y
Let's integrate the two members of this equation for the space occupied by the set of conductors 2 and 4" and we will have
I j jSi{u^--^v^'i-w^-)dxdydz
rrr\ d{z\-^e) (^(ev-^e) ()(£V4-e)i , , ,
An integration by parts allows to transform the second member into
- C (ôV-H 0)[mcos(N/, x)-^v cos(N,,jKj -4- w cos(N/, z)] dZ.
The first sign of integration extending to all the metallic parts whose nature varies from one point to another in a continuous way, and the second to all the surfaces which limit these various parts, one deduces easily from there
I j I Si(u^~v^-^w^-)dxdydz
^^-ffJ{t\-^e)^^dxdydz-^{zY-^e)~d^.
In the second member, the first summation extends to all electrified volumes and the second to all electrified surfaces.
But, if the steady state is established, we have, at any point,
dp di
476 BOOK V. - METALLIC CONDUCTORS.
and the previous equality requires that we also have, at any point,
u - o, (^ = o, w - o.
Thus, on an isolated system, formed exclusively of homogeneous or heterogeneous metals all at the same temperature, there can be no other permanent regime than V electrical equilibrium.
If we apply this general theorem to the case at hand, we see that the conductors (2) and (4) are in electrical equilibrium.
Let m! be a point of the conductor MN infinitesimally close to the derivation m; let n' be a point of the same conductor closely following the derivation n; let U(m'), U(n'), 0(m'), ©("') be the values of the potential function and of the function at these points; let R be the resistance of the segment m' n' and J the intensity of the current flowing through this segment. We will have
RJ = sU(m') -+- e(m') - eU(w') - <è{n').
Inside the conductor 2, we have
EV2-^-0(a, i, À-f-[Ji) = £U(7n')--t-6("2'), and inside conductor 4 we will have
sV^H- 0(Z>, î, À -4- ]x) = £U(rt') + e(n'): hence the relation
£(V; - V'4'i-^e(a, t, \-^ p.) - 0(è, i, X -f- [Ji) - RJ.
For our two spheres not to carry electricity, it is necessary and sufficient, as we saw in § 1, that the two surfaces S2 and S;) be at the same potential level T, i.e. that we have both
sV'a -4- 6(a, i, X -+- fz) - sT - "-- 0(a, ï, o). eV'j -t- 0(6, i, X -H p.) = eT -f- 0(", J, o)
and, therefore, according to the previous equality
(20) 0(", i, o) - 0(6, J, o) -- RJ.
If we therefore adjust the resistance R in such a way that, in the previous experiment, the inner sphere takes no load
CHAP. VII. - POTENTIAL DIFFERENCE ON CONTACT. 4/7
(which an electroscope will always allow to see), the determination of this resistance R and the current J will allow to determine the difference
(-)(", f, o) --0(6, i, <)).
This procedure does not allow, any more than the one indicated in § 1, to determine the difference
e(a, i, 1 -r- [x) - 0(6, i, y^ -+- [J-) The experimental precautions that must be taken when using this process can be found in M. Pellat's Memorandum.
478 BOOK V. - METAL CONDUCT.
CHAPTER VIII.
THE PELTIER EFFECT.
§ I. - The Peltier effect.
We have said that the study of the permanent motion of electricity follows entirely from two fundamental hypotheses: one, which is arrived at by generalizing Ohm's law, has been indicated in Chapter IV; the other, which is obtained by generalizing Joule's law, has been indicated in Chapter VI.
In the previous chapter, we applied the first of these hypotheses to a conductor made of different metals that all have the same temperature, and we deduced a series of consequences on the laws of permanent currents in a similar conductor. We will now apply the second hypothesis to this same conductor.
Let us consider a closed surface S traced inside a heterogeneous metallic conductor, all the points of which are at the same temperature, and which is crossed by permanent currents. In the time dl^ the part of the conductor that is enclosed by this surface gives off a quantity of heat ûfQ that we want to calculate.
For this, under the assumption indicated in Chapter VI, we can operate as follows.
We will assume that, during time dt^ the currents that flow through the part of the system enclosed within the surface S are maintained, but the currents that flow through the rest of the system are reduced to o. This would result, during the time dt, in a change of electrical distribution on the system. This change consists, as is easily seen, in bringing onto the = - £
If then, in accordance with the notations indicated elsewhere [Book IV, Chapter II, equalities (16) and (19)], we pose
(3) Kr^e+HT,
T being the absolute temperature, we can replace the equality (2) by the equality
/ E o?Q -r ^("2 _4- t.2 ^- w2 ) dx dy dz dt
i - T M \-v-^ h w -- ] dx dy dz dt .
\ dx dy dz / -^
When a heterogeneous metallic conductor, whose points are all at the same temperature, is crossed by permanent currents, each element of this conductor gives off, in time dt, a quantity of cJialeur which is the sum of two parts:
dY
from
dx
dx
dV
d&
dy
dy
dV
dQ
dz
'dz
CHAP. Wine.
The first part
THE PELTIER EFFECT.
48 1
i3 .^(M^-i- t;2-|- w-)dx cly dz dt
is given by Joule's law; it is always positive; it is proportional to the square of the electric flux at a point of Vêlement and to the specific resistance at this point. The second part
dx
Oy
- 1 dx dy dz dt
dz j -^
would be equal to o if the element were homogeneous; it changes sign if the direction of the electric flux is reversed at a point of the element; it is, in size, proportional to this flux; it depends on its direction.
This second release of heat is called Peltier's phenomenon or à^efj'et Peltier. It is in fact Peltier (') who, in 834, was the first to demonstrate the particular calorific phenomena that heterogeneous conductors present. Let us see how one can deduce from the preceding law consequences that can be verified experimentally.
Let us consider a conductor {fig. 96), formed by two metals a and ^, welded together along the surface AB by means of a metallic weld of any kind. To metal a is attached a
wire F, of any nature, carrying a uniform current of intensity J. This current is carried by a wire F', of any nature, fixed to the metal b.
_ ^ - - ^ ^ - ^ ^ ^ -
(') Peltieu, Nouvelles expériences sur la caloricité des courants électriques {Annales de Chimie et de PJi,ysique, i" série, t. LVf, p. 871; i834).
D. - î. 3i
I
(5)
4B'Jl BOOK V. - Li:S METAL CONDUCTORS.
Noti-e conductor is flowing with uniform currents.
Let us lead, inside this conductor, two surfaces normal to the flux lines: one MN, or S, inside metal <2, the other PQ, or S', inside metal b. Each of these surfaces has this property that (£V+ 0) has the same value at all its points. Each of them normally intersects the conductor surface.
Let us apply equality (i) to the closed surface S or MPQN, replacing K by (0+TH). The part of the conductor inside this closed surface will release, in time dt, a quantity of heat dÇ) given by
But, at any point on the surface where the conductor meets the insulator, we have
u cos(N;, x) -\- V cos(N/, y) -+- w cos(N/. z) - o.
The previous equality becomes
EdQ= dt^(z-he + TH)[ucos{Ni,x)-{-vcos{Ni,y)-h-wcos{Ni,z)]dS
+ dt ^ (£V+e-f-TH)[K cos(N,-, x)^v cos(N/, y)^w cos(N,', z )] dS'.
At all points on the surface S, (sV-i-©) has the same value, which we will denote by (eV+ 0); at all points on the surface S', (eV+0) has the same value, which we will denote by (£V'+ 0'). Moreover, we have
J - X [" cos(Nj, X) -\- V cos(N/, y) -^ w ces (IN/, z)] dS
== - ^[u cos(l\j, x) -{- V cos(N/, y)-^iv cos(Ni, z)] dS'. Thus we have
1 Q (sV-i- e)[u cos(N/, x)-i-v cos(N;, jk) -H iv cos(N/, z)] dS
(6)
\ -^ C (£V+0)[itcos(N,-, x) -+- PCOs(N/, y)-~ w cos(N,', z)] )TJ(/".
This is the expression of the amount of heat released by a uniform current in a section of conductor formed by two welded metals.
If the MNPQ section is taken large and short, its resistance R will be an extremely small quantity. If the current is not very high, we can neglect the term RJ- and simply write
(he) Ed(l^{ha-hb)'Y]dt.
Thus, when a current of not very great intensity crosses the weld of two metals, it gives off a quantity
I
484 BOOK V. - METALLIC CONDUCTORS.
of heat which depends only on the nature of the two metals at a certain distance from the end surfaces and does not depend on the shape and size of the contact surface or on the nature of the weld.
This calorific phenomenon is, in size, proportional to U current intensity. It changes sign if Von reverses the current.
This is the phenomenon first observed by Peltier by means of the following experiment, which can easily be repeated in the classroom.
The strongest effect is obtained with bismuth and antimony. The current gives off heat when it passes from antimony to bismuth and absorbs heat when it passes from bismuth to antimony.
Two antimony-bismuth solders are placed respectively in the two balls ", b of a Rumford ihermoscope {fig. 97).
Fii
97
The same current flows through both welds, but in ball a it passes from antimony to bismuth and from bismuth to antimony in ball b. A heat release takes place in ball a and an absorption in ball b. The two effects add up to move the index I of the thermoscope from ball a to ball b. If the current is reversed, the index moves in the opposite direction.
Let us consider three metals a, b, c; a current of intensity J, passing from metal a to metal 6, produces in time dt^ a release
ciiAP. VIII. - the peltier effect. 485
of heat o?Q, given by
ErfQ, = T(/ta- hb)i dt.
This same current, passing from metal b to metal c, produces, in the same time, a heat release C/Q2 given by
ErfQj^ T{hb-hc)i dt.
Finally, this same current, passing from metal a to metal c, produces, in the same time, a heat release c/Qa given by
'^dÇii^T{ha~hc)ldt.
It is easy to see that we have
This relation shows that, in the experimental study of the Peltier effect, it is not necessary to study the welds that can be formed with all the metals taken two by two; it is sufficient to study the welds that can be formed by joining successively each of the metals with the same standard metal. The preceding relation will then show the Peltier effect that occurs when any two metals are welded together.
The experimental study of the Peltier effect has given rise to many recheixhes that one will find exposed in the treaties of Physics.
§ 2 - Relation between the Peltier effect and the potential level difference between two metals in contact.
We will give the name of coe^Cf'e/i^ of V Peltier effect for the welding ", 6 to the quantity
U2j Ta-- jg- '
which represents the quantity of heat released during the unit of time by a current of intensity equal to the unit passing from metal a to metal b.
Let ©", B^ be the values of the quantity 6 inside the metals <7, 6, at a sufficient distance from the terminal surfaces. For electrical equilibrium to be established on a conductor containing
486 BOOK V. - METALLIC CONDUCTORS.
the two metals a and 6, it is necessary that the potential function has, within these two metals, constant values V,^, V^, linked by the relation
(£V«-h0a)-£(V/.-T-06).
The quantity
(i3) D^=- ^"-^^
is the potential level difference between the inner regions of the two metals a and b joined metallically.
But, on the other hand, according to the definition of the quantity H [Book II, Chap. II, equality (i6)], we have
_ to^n , _ of/,
Equalities (12) and (i3) thus give
Ut) *^" - E ^T
The coefficient of V Peltier effect at the welding of two metals a, b measures the product of the absolute temperature by the derivative with regard to the temperature of the fall that experiences, in the state of equilibrium, the potential level when one passes from the inside of the metal a to the inside of the metal b.
This fundamental relation was first obtained by M. Lorentz (' ).
This relationship cannot be verified by experiment, at least in a direct way. Indeed, we have
e^ = e(a, i, X -f-[x), e^ =e(è, i, X-t-[ji)
(') H. -A. LoR^yiz, Sur l'application aux phénomènes thermo-électriques de la seconde loi de la Théorie mécanique de la chaleur (Archives néerlandaises des Sciences exactes et naturelles, t. XX, p. 129; i885). - For the history, t>oi/' P. DuHEM, Sur la relation qui lie l'effet Peltier à la différence de niveau potentiel de deux métaux au contact {Annales de Chimie et de Physique, 6° série, t. XII, p. 433; 1887). - The principles of the theory set forth in this Chapter are indicated in the latter Memoir and in P. Duhem, Le Potentiel thermodynamique et ses applications, 3* Partie (Paris, 1886).
CHAP. VIII. - The PELTIEn effect. 4^7
and, consequently, equality (i4) can be written
Pa = ^^'Y [Q(^> ^ X -^ (ji) - 0(a, i, X -i- [!)].
The first member of this equality can be determined experimentally; but the same cannot be said of the second, because, as we pointed out in the previous chapter, the methods for determining the potential level difference in contact with two metals determine the quantity
6(6, i, 0) - 6(a, i, o) and not the quantity
e(b, i, X -r- |jl) - &(a, i, l -+- [x).
Clausius (' ) had admitted that the relation between the coefficient of the Peltier effect and the potential level drop was the following
He believed that this equality necessarily resulted from Thermodynamics. This relation, admitted by Clausius and by a great number of other phvsicians, must be replaced, in general, by the relation (i4)' . However, it is possible that, for certain particular metals, the relation (i4) may take the particular form ( 1 5 ) .
(') R. Clausius, Ueber Amvendung der mechaiiischcn Wàrtnetlieorie auf die thermo-elektrisclien Erscheinungen ("The Mechanical Theory of Heat", translated into French by Folie, vol. - Théorie mécanique de la chaleur, translated into French by Folie, t. II).
488 BOOK V. - METALLIC CONDUCTORS.
CHAPTER IX.
THERMOELECTRIC CURRENTS.
§ 1 - Conditions in which thermoelectric currents occur.
So far, we have studied systems formed by metals whose points are at the same temperature. We will now study the properties of systems formed by metallic conductors whose temperature varies from one point to another.
We will assume that the shape and position of the various conductors are given, as well as the physical and chemical constitution of each of the volume elements of which they are composed, so that, in order to know completely the state of a volume element of the system, it will suffice to give the coordinates x^ y, z of a point of this element, and the absolute temperature T at this point.
The internal energy U and the entropy S of the system will be given by the equalities [Book II, Chap. II, equalities (i8) and (19)]
(1) EU = Er + W + Kig',-4-K2^" + ...+ K"^",
{%) ES=ES +Hi^i-hH2<72-+-... + H "g'".
From what we have just said, the cpiantites H and K will be continuous functions of ^, jKi -2, T,
H = H(a7, jK, z, T), K= K{x,y,z,T).
Let us imagine that an electric charge oq passes from the point of coordinates (x, y, z) to the point of coordinates
(x -^ ox, y -i- oy, z -\- oz),
the internal energy and the entropy of the system undergo variations
CIIAP. IX. - THERMOELECTRIC CURRENTS. 489
ôU and 5S given by the equalities
L V ^■'^ ^^ ~^ OT dx ) d\ dK dT\
('
dj ' dy ' dT dy)^^
d\ dK dK dT\ ,
e 1 1 ■ - ' '
dz dz dT d.
-joz^dq,
ss =r
/dH dH dT'
\dx ^^ dT dx
/dH dH dT\,
-^(d^ -^d-T d-^r-^
/dH dH dT\. 1,
^(dJ -^dT d^j'^^J'S' The uncompensated work involved in this change is
value
ox = ETSS - EoU.
If we observe, moreover, that Ton has [Book II, Chap. II, equalities (16) and (17)]
K = e + TH,
we see that we will have
dV
from
of dT
' tox"^
" dx
■ dT d^
dV
from
of dT
dy '
dy
^ dT dy
dV
dJ-^
of dz '
of dT ■^ dT dz
H^
da^
(3) ( +(^"^.-E-^^+H^)o7
H - ) 02 c^fir. dz / J ^
This equality is fundamental; it provides the laws of equilibrium and permanent motion of electricity on metallic conductors whose temperature is not the same at all points.
The conditions of electrical equilibrium on a similar conductor are obtained by writing that the uncompensated work 8t, given by equality (3), is equal to o, whatever tx, ùy, 83; these conditions are thus
ày ^dS dx dx
Oy dy
dV _^dB
'dz '^ dz
of dT dT dx
of dT
u'^T
dT dy
-T-H - =0,
dy
of dT
u^'^
dT dz
-^H- =0.
dz
490 BOOK V. - METALLIC CONDUCTORS.
If the system has permanent currents, it follows from the fundamental hypothesis concerning permanent currents, which was obtained in Chapter IV, by generalizing Ohm's law, that we must have
Sx = ( Ca; 037 -I- Cy oy -i- CzOz) dq,
Cx, ^y, ^z being the components of the electromotive force at point (^, y, z). If SI is the specific resistance at the point (^, y, z), and II, V, w the components of the flux at this point, we have, according to equality (3),
/ to\ d& de àT "dT\
dx /
, c, "i, ..., /, each of which is homogeneous, except at a very small distance (X + pi) from the surfaces that terminate it.
With these metals, let's form a chain in such a way that each of them is in contact only with the one preceding it in the series and with the one following it.
Let us assume that each of the contact surfaces has the same temperature at all its points, this temperature not necessarily being the same for the different contact surfaces.
We will have an open thermoelectric chain.
If, moreover, we suppose the two metals a. and / put in contact by a surface having the same temperature in all its points, we will have a closed thermoelectric chain.
The theory, based on Thermodynamics, of thermoelectric chains, was given in 1862 by Sir W. Thomson (' ). The preceding principles easily provide the results obtained by Sir W. Thomson (2), and even allow to give them a more general form, which does not oblige to reduce the chain to a simple fd.
First law. - An open thermo-electric chain cannot be the seat of any sensible permanent current.
Let us consider an open chain, formed, for example, of four metals ", b, c, d {Jîg. 98), separated from each other by the surfaces S,, S2, S3, whose temperatures are T,, T2, T3.
If the metal has permanent currents flowing through it, it must receive as much electricity at any given moment as it lets out, and, as these exchanges can only take place through the in
(' ) Sir W. Thomson, On a mechanical theory of thermo-electric currents {Philosopliical Magazine, 4° series, t. III, p. 629; i852. - W. Thomson's Mathematical and physical papers, 1. 1, p. 3i6).
(") P. DuHEM, Applications de la Thermodynamique aux phénomènes thermoélectriques et pyro-électriques. F" Part: Thermoelectric phenomena {Annales de l'École Normale supérieure, 3" série, t. II, p.4o5; i885).
CHAP. IX. - THERMOELECTRIC CURRENTS. 49^
of the surface S, we see that we have
(a) V [u cos,(Na,x)-^vcos{]Sa,y) H- w cos(Na,-s)] dSi=: o.
The metal b must also receive at each moment as much electricity as it loses, which gives
V [u cos(N;!" x) -+- pcos(N/"j') -H iv cos(N6, z)dSi
-h V ["cos(N6, ce) -+- V cos{l^t "y) 4- tvcos(NA, z)] dS^- o.
But equality (a) gives
(?) ^ [Mcos(Ni, x) -i- V cos(N6,jk) -+- tv cos(N6, z)] dSi = o,
so that the previous equality becomes
(v) V [ucos{^/j,x)-h- i^cos(N6,7) -f- w cos(N/" z)] ^82 = o.
We will see that, from one to the other, each surface S(,
Fi g.
So, S3 delivers, at each instant, a total amount of electricity equal to o.
Gela posed, let us write, for the metal a, an equality analogous to equality (7), and let us transform it in the same way. Let J9a(T), ha(T), be the functions 3^(T), A(T), particular to metal a, we will have
///
3i{ii--t-v^-^w^)dxdydz = - ^[eV -{- e -}- jÇa(T)] [u cos(Na, x)^v cos(Na,jK) 4- w cos(Na, z)\ dSi -JfJ[Hix,y,z,T)-K{T)](u^^-^v^^-^".^^^^dxdjdz.
496 BOOK V. - METALLIC CONDUCTORS.
If we observe that the temperature T and, consequently, the function ^"(T) have the same value at any point of the surface S, and if we take into account the equality (a), we will have
///
Jl("2 -4- p2 _|_ (^,2 ) dx dy dz
= - X (sV-l- 0) [U COS(Nrt, X)-^V COS(Na,JK) -4- W COS(Na, Z)] fl?Si
The metals b, c, d give analogous equalities. By adding them member by member and noting that we have, at any point of S,,
u cos(]Na, x)-^ V COS(Na,JK) + W cos(Na, z) -\- u co?>{'^b,x) -h V cos(N6,jK,> ■+- w cos(N/j, z) - o,
and analogous equalities at any point of the surfaces So, S3, we find
I f IS{.(u'^^v^--r'W^')dx,dj,dz
■fU
[H(x,y,z, T) - Arf(T)] ("^-f-''j- + ^^) dxdydz.
The integration in the first member extends to the entire chain.
Since the second member is a very small quantity of the order of (X + {jl), the same must be true of the first; the currents can therefore have no appreciable intensity in any region of appreciable extent in the system.
The electrical balance will be established on the system. We will have, at any point, according to the equality (8),
H(a7, jK, -s, T) t/T = - i/(£V 4- 0).
Let us take a point M {fig. 99) inside the first metal a of the chain, and a point M/ inside the last metal d of the chain. Let T and T' be their temperatures. Let us join the two
ICVP. IX. - THERMOELECTRIC CURRENTS. 497
points M, M' by a line / meeting the surfaces S,, So, S3, in P, P2, P3. Equality (8) gives us
.M £V(M)-f-ea(T)-sV(M')-e,/(T')= / U(x,y,z,T)dT.
But the second member can be written
T,
I, -^T-^T-X -^'^^Jr, -^^'"A
.T ^T
f [H(^,r,^, T)-A^(T)]c/T+...-^ r [11(^,7, 2, T)-/irf(T)]f/T. If Ton elTects the integrations shown in the first four
Fig- 99
terms, and if we neglect the following terms, which are very small quantities of the order of (-+- [jl), we find
£V(M)-4-e"(T)-£V(M')-e,/(T')
(9) = i5a(T,)-i5a(T)+i5^(TO-i56(T,)
( -^i5,(T3)-i^,(T2)-f-i5rf(T')-;^rf(T3).
Let us assume, in particular, that the two metals a and d are identical; that the two temperatures T and T' are identical. Between the points M and M' there will be a potential level difference given by the equality
l £[V(M)4-V(M')]= i5"(TO-i^6(Ti) ("o) _ -^i5/,(Tj)-i9c(T2)
whereas this difference would have been equal to o, if the two metals had been joined by metallic conductors having the same temperature at all their points.
D. - I. 32
(II)
498 BOOK V. - METALLIC CONDUCTORS.
Second law. - A closed thermo-electric chain, in which all the welds are at the same temperature, cannot, whatever the distribution of temperatures between the welds, be traversed by sensitive permanent currents.
Let a closed electric chain be formed, for example, of three metals a, ^, c {/ig- 100); let S, Sa, S3 be the surfaces
which separate these three metals, and Ti, To, T3 their temperatures. Let us count as positive direction of travel the direction abcda. It is easy to show that, in time dt, each of the surfaces Si, S2, S3 is necessarily crossed in the positive direction by the same total quantity of electricity idt. If we reason as we did to establish the previous law, we can easily find
-JJJ[Yl{x,y, z, T) - A.(T)] (^^. _ + ,- + ". _j clx dy dz.
The last three terms of the second member are very small quantities of the order of()v+[i.); the first term of the second member disappears if we have
T, = T2 = T3.
CHAP. IX. - THERMOELECTRIC CURRENTS. 499
If, therefore, the solder joints of a closed thermoelectric chain are all at the same temperature, this chain cannot have a sensitive electric flux at any point in a region of sensitive extent.
Third law. - V electrical balance is, in general, impossible on a thermo-electric chain where all the welds are not at the same temperature.
Indeed, if the electrical equilibrium was established on a similar chain, we would have at any point
(8) d{z-^T)=-}{x,y,z,T)cn.
Let's draw a line that goes from metal a to metal 6, from metal b to metal c, and closes within metal a.
Integrate the previous equality along this closed line and we find
^H(^,jK, 2, T)^T = o.
/'
Let us transform this equality by a calculation analogous to the one that gave us equality (9) and we will see that it becomes, neglecting the small quantities of the order Q^+ [J.),
(12) J^a(T,)-45«(T3)-^i56(T2)-;0/,(T,)-^i5^(T3)-iPc(T2)=o.
The necessary condition for electrical equilibrium to persist on a thermoelectric chain is that the temperatures of the welds verify equality (12) to the nearest quantities of V order of {\ + [t-) This condition is at the same time sufficient, because, if it is verified, the second member of equality (11) is a very small quantity of the order of Çk -\- [jl) and, consequently, the chain can have no current of appreciable magnitude in a region of appreciable extent.
When the temperatures of the welds do not verify the equality (12), the thermoelectric chain becomes the seat of a permanent current whose laws we will study.
Fourth law. - On a thermo-electric chain running
500 BOOK V. - METALLIC CONDUCTORS.
jme by permanent currents, no flow line closes without having covered the whole chain.
In any limited system, with permanent currents, any line of flow is bound to close on itself.
Can it happen that a closed flux line appears, on our thermo-electric chain, as the line M PM'QM in fig. law ,
Fig. 101.
in such a way that it is possible to open the chain with a section S that does not cut this flow line?
Let us assume for a moment that this is possible. Our flux line can meet certain welds; let's imagine that it meets the weld S, at two points M, M'.
Let us take this line as the director of a small flow channel of section diù. Let /"be the flux at a point, and let us pose
The equalities (5) will give us
Çs^fds^-fd{zS-^Q)- CH{x,y,z,T)dT,
integrations extending to the closed MPM'QM line.
The quantity i having the same value at all points of this line, we will have
We will also have
/-
d{z\-he) = 0.
CHAP. IX. - THERMO-ELECTRIC SHORTS. 5oi
Finally we will find without difficulty
J'H{x,y,z,T)dT= J [H(.r,j',z,T)-A"(T)]crr
.M
+ / {n{x,y z,'ï)-h,{'ï)\dl.
The second member of this expression is a very
small of the order {\-\-[x). The same applies to i j -,-- which
which shows that no line of the indicated shape can be traversed by a sensitive flow.
Consider a small channel formed by flux lines. According to what we have just demonstrated, it runs through the whole thermoelectric chain as represented by ^^Jig- 102. Let's take for direction
Fis;. lori.
of this channel the direction abca. Let dio be its section at a point; soity the flow at this point. The quantity
i =f dui
has the same value all along this channel. Let us suppose, to fix the ideas, that it is possible to lead through the channel a section S, which normally meets all the flow lines once and only once. We will then easily find, with quantities of the order of Çk + ^), the equality
ij^= ^a(Ta)-i5a(T0
-^J0^(T,)-i56(T2)
--i5,(T2)-i5c(T3).
Let due be the element that our channel cuts on the surface S.
5o2 BOOK V. - METALLIC CONDUCTORS.
The quantity
J = ^idQ
will represent the total amount of electricity that, during the unit of time, crosses, in the positive direction, the surface S or any other section that opens the chain.
As we did in Chapter IV, let us define the total resistance R of the chain for the flux distribution we consider by the equality
i = S
from
IOC}
the sign V indicating a summation that extends to all elements
of the surface S, and the sign /a summation that extends to all
elements of a closed flow line coming from a point of the due element. We will have, according to this definition,
(i3) ^S>,{T,)-^,{T^)
This is the fundamental formula obtained by Sir W. Thomson. If we call the second member of this equality the electromotive force of the chain, we can state the following law:
Fifth law. - The electro-motive force of a thermo-electric chain depends exclusively on the nature of the metals which compose the chain and on the temperature of the welds.
The expression
(l4) C = i5a(T3)-i5a(T,) + i^6(Ti)-i56(T2)~-^,(T2)-i5.(T3)
of the electromotive force of a thermoelectric chain easily provides all the properties of such chains.
CHAP. IX. - THERMO-ELECTRIC CURRENTS. 5o3
§ 3 - Properties of bimetallic chains. The equation (14) can be written
C= i5"(T3)-i5a(T.) + i56(T,)-i^/,(T3)
The terms of the first line represent the electromotive force of a thermoelectric chain formed by the two metals a and b only, having its welds at temperatures T, and T3, and counted positively when it runs the current from metal " to metal b through the weld whose temperature is T, .
The terms in the second line represent the electromotive force of a thermoelectric chain formed by the two metals ^ and c only, having its welds at temperatures T2 andTs, and counted positively when it runs the current from metal b to metal c through the weld whose temperature is T2.
It is thus seen that the study of a thermoelectric chain formed of several metals can always be reduced to the study of a certain number of thermoelectric chains each formed of only two metals.
By means of this rule, obtained by A.-C. Becquerel ('), it is sufficient to study the properties of bimetallic thermoelectric chains.
This rule has been subjected to a very precise experimental verification by Messrs Chassagny and Abraham (2).
It is not necessary to study the properties of a bimetallic chain for all possible combinations of the temperatures of the two welds.
Let us suppose, in fact, that we want to know the electromotive force of a chain formed by the two metals a and b whose welds are at temperatures T and To. Let us designate this
(') A.-C. Becquerel, Du pouvoir thermo-électrique des métaux {Annales de Chimie et de Physique, 2" série, t. XLI, p. 353; 1839)1
{') Chassagny and Abraham, Recherches de Thermo-électricité {Comptes rendus, t. CXI, p. 4/7! 1890}.
5o4 BOOK V. - METALLIC CONDUCTORS.
electromotive force, counted posilively when it tends to make the current pass from metal a to metal b through the weld whose temperature is T, by C^(T\^ Ta). We will have
d:^(Ti,T2)=i5"(T2)-^"(T.) + i3/.(T,)-i96(T,.).
Let To be a third arbitrary temperature. We can write
i'â(Ti,T2)= i5"(T2)-i5a(To)^i56(To)-+-^/,(T2)
-[i9a(Ti)-i5"(To)-i|6(To)-i5^(Tijl = d:,'i(To, T,)-tf;(To,T,).
By means of this relation, due to A.-C. Becquerel ('), we see that it will be sufficient, by keeping the temperature Tq of the cold soldering fixed, to study the electromotive force of the bimetallic chain for all values T, of the temperature of the hot soldering.
To study the electromotive force "L'*(To, T,), counted positively when it passes the current from metal a to metal b through the weld at temperature To, we will operate as follows.
According to equality (6), we have
and, consequently, equality (i4) can be written
(i4') C'^{To,T,) = J[/t,(T)-A,(T)]
Let us choose a standard metal, always easy to reproduce identical to itself at the same temperature, lead, for example. Let us designate this metal by the index o. Once this metal is chosen, the
quantity
/î"(T)-Ao(T)
will be a quantity depending only on the temperature and the nature of the metal a. We will pose
(i5) jK"(T) = A,(T)-Ao(T),
(' ) A.-C. Becquerel, loc. cit.
CHAP. IX. - THERMO-KELECTRIC CURRENTS. 5o5
and we will say that jKa(T) is, at temperature T, the thermoelectric power of metal a related to lead. The equality which gives "L'*(To, T|) will then become
(.6) i:iJ(To,T,)= / [ya{'ï)-yi,{'ï)\dl.
It is thus seen that if, for each metal, one knows the thermoelectric power related to lead, and this at any temperature, one knows, under all circumstances, to predict the effects of any thermoelectric chain.
We shall see, in § 5, how one can determine the thermoelectric powers related to lead. For the moment, let us discuss the consequences of equality (i6).
On the x-axis of a rectangular coordinate system {Jig' io3), let us plot the absolute temperatures T. Let us plot the thermoelectric powers on the ordinate. Let AA', BB' be the
Fis. io3.
°i_A'
A
r
B-^
-^
curves representing the thermoelectric powers ^'"(T), j^6(T), of metals a and b.
Let us lead through the points of abscissa Tq, Tj, of the parallels Tq ^o^^o" T, ^)a,, to OY. It is easy to see that the area ^o^^o^ti '^\ represents the absolute value of the electromotive force C^(To,T,), and this force will be positive or negative depending on whether the line AA' will be above or below BB'.
To study the variations of the electromotive force £* (T©, ïi) with the temperature T<, it will be enough to leave invariable the line Toj^o^to and to study the variations that the area j^o^^o^il^i 1oï "S' undergoes that we move parallel to itself the ordinate T)j3,a(.
It can happen that, in the whole extent of the field where the experiments are carried out, the line BB' is constantly above or constantly below the line AA': the previous area, and consequently
5o6 LlVnE V. - METALLIC CONDUCTORS.
The absolute value of the torque increases as the difference between the temperature of the hot and cold weld increases.
Many couples present these simple phenomena; but
Fig. 104.
A_
■"^^^^
B'
w
t
B-^
/fp.
T" T^ S^ T
it may happen that the two lines AA', BB' intersect at a point C i^fig- io4), whose temperature 3 is included in the field of experiments.
Let us assume, to fix the ideas, that at temperatures lower than 2r the AA' curve is above the BB' curve. Let us take the temperature To lower than Sr.
1" The temperature T, starting from Tq, remains at first also lower than 2" {fig- io4). The electromotive force is then positive. The current goes from metal a to metal h through the cold probe. The electromotive force increases with the temperature of the hot solder, but slower the higher the temperature.
2" The temperature T, reaches, then exceeds the temperature 2r,
F'ig. io5.
which is called the/?om^ /zcM^/'e. From this moment (/""-. io5), the electromotive force stops growing, because the area Ca, (^2 must be
CHAP. IX. - THERMOELECTRIC CURRENTS. 5oy
counted negatively and subtracted from the Caol^Q area. However, the electromotive force remains positive at first.
3° But T, continuing to grow, can reach a value t {Jig. io6) for which the cah area is equal to the Gao I^q area. For
Fig. io6.
At this inversion temperature, the electromotive force cancels. When the temperature of the hot solder becomes higher than the inversion temperature, the area to be subtracted Ca, ^, becomes greater than the area Cao|3o from which it is subtracted. The electromotive force becomes negative, and the current flows from metal b to metal a through the cold solder.
At the moment when the temperature of the hot solder is equal to the inversion temperature, the electrical equilibrium is established on the system, because, it is easily seen, the equality (12) is then verified.
The temperature of the neutral point depends only on the nature of the two metals a and b; but it is not the same of the inversion temperature, which depends moreover on the temperature Tq of the cold junction, and is all the higher as the temperature of the cold junction is lower.
If, on two axes of rectangular coordinates, the temperatures T, of the hot weld are plotted on the abscissa and the electromotive forces (To)](T,-To)f2r-I^^^'y
This formula represents very exactly the results of the various experiments of Gaugain. It was first given by M. Avenarius ('). A few years later, it was found again by Mr. Tait (-). The method followed by Mr. Tait to reach it is reproduced in all the treatises of Physics. It could easily, by its form, mislead one into thinking that the formula (17) is a rigorously exact formula, whereas it is only an approximation.
The very precise research of Messrs. Chassagny and Abraham (' ) shows that the differences between the results of the experiment and the formula of Messrs. Avenarius and Tait exceed the limit of the errors of observation, at least if one admits that one can confuse the temperature read on the hydrogen thermometer with the absolute temperature.
If the neutral point is very far from the experimental field, the two lines AA', BB' can be assimilated to two parallel lines, and we have
t^(To,T,) = fjK"(To)-jK/.(To)](T,-To).
The electromotive force of the thermo-torque is proportional to the difference in the absolute temperatures of the two welds. Such a torque is called a steady-state torque. The antimony-bismuth couple has an appreciably regular operation.
§ 4 - Relation between thermoelectric phenomena and potential level differences at contact.
The electromotive force of a thermoelectric chain formed by the two metals a and ^ and whose two welds are at the
( ' ) Avenarius , Ueber electrische Differenzen der Metalle bei verschiedenen Temperaturen {Poggendorff's Annalen, t. CXXII, p. 198; 1864 ).
(") Tait, Ou tlienno-electricity {British Association Repertoriuni, vol. XLI, p. 48; 187.).
(') Chassaony and Abraham, /?ec/te/-c/ies de thermo-électricité {Comptes rendus, t. CXI, p. 4:7" 602 et 782; 1890).
5lO BOOK V. - METALLIC CONDUCTORS.
temperatures Tq el T) is represented by the formula
"^^'(To,T,)= /[haiT)-/t/AT)]dT.
Let 0a(T), 0ô(T) be the values of the quantity B at temperature T, inside the two metals a and b. It is known that [Book IV, Ch. Il, equality (i6)]
/'n( 1 ) = ^j- ' "/>( 1 ) ^ J^Tp 5
which gives
C^(To,T,) = [ea(T")-0/,(To)]-[0"(T,)-e/,(T,)].
But, on the other hand, when the two metals a and b are in contact, when all their points are at the same temperature T, when equilibrium is established on the conductor they form, the potential level inside conductor a exceeds the potential level inside conductor b by an amount
So we have
(i8) and (To,T,) = £[D;;'(T,) - D^'(To)].
This simple relation extends to a chain formed by any number of metals a, b, c, d. Let To, T, Ta, T3 be the temperatures of the welds (a, Z>), (6, c), (c, d), [d, a). We will have, by counting positively the electromotive force e when it tends to make the current run in the direction abcda
(19) t=_£[D^(To)+Dg(TO + D;f(T2)+D3(T3)].
Therefore, with the factor t, the electromotive force of a thermoelectric chain is equal and of opposite sign to the sum of the FALLS that the potential level experiences when crossing the layers of thickness 2(à-|- [jl) that are close to the welds, in the direction of travel where the electromotive force is counted positively.
CHAP. IX. - THERMO-ELECTRIC CURRENTS. 5ll
This is the very simple relation (') which links the thermoelectric phenomena to the potential level drops which characterize the equilibrium between two metals in contact.
The sign of the second member of this relation is somewhat paradoxical. Clausius(-), who was the first to look for a theoretical relationship between thermoelectric phenomena and the drop in potential level at the contact, was led to write a relationship which becomes, by using our notations
(20) c = £[D;;'(To) -+- d;,;(T, ) + D^^iTo -4- d^kt,)].
According to Clausius, the electromotive force of a thermoelectric chain is equal and opposite in sign to the sum of the increases that the potential level experiences when crossing the layers adjacent to the welds, in the direction of travel where the electromotive force is counted positively.
The Clausius formula would give an exact value for the thermoelectric emf, but a false sign. This remark is all the more important since a large number of authors take the relation (20) as a starting point for the theory of thermoelectric currents.
The relation (19) cannot be directly controlled by experiment. The electromotive force C is measurable, but the same cannot be said for the quantities in the second member. We have, in fact,
D*(To) = -^[e,(To)-e"(To)]
(') P. DuHEM, Sur la relation gui lie l'effet Peltier à la différence de niveau potentiel de deux métaux au contact {Annales de Chimie et de Physique, 6* série, t. XII, p. 433; 1887). In this work I have, by mistake, considered relation (19) to be identical to relation (20).
(") R. Clausius, Ueber die Anwendung der mechanischen Wàrmetheorie auf die thermoelektrischen Erscheinungeii , equation i5 {Poggendorffs Annalen der Physik und Chemie, t. XC, p. 5i3; i853. - Théorie mécanique de la chaleur, translated into French by F. Folie, t. II).
5£2 BOOK V. - METALLIC CONDUCTORS.
and we have seen that the methods used to determine the potential level differences in contact with two metals only provide a measure of the quantity
6(b, i, o,To) - 6(", i.o, To).
§ 5 - Relation between thermoelectric phenomena and the Peltier effect.
It is to Sir W. Thomson who gave, in the Memorandum we quoted at the beginning of § 2, the relation which links thermo-electric phenomena to the Peltier effect.
We have seen that the Peltier effect produced during the time dt^ by a current of intensity J which passes from metal a to metal 6, was given by the formulas [Chapter VIII, equality (12)]
( d(l=-- P'/Jdt,
^ ^ P'â = g[A"(T)-A,(T)].
i" Suppose we consider the lead o and the metal a, the second equality (21) will give, by virtue of equality (10),
(î2) y"(T) = -|p-(T).
If, therefore, at any temperature ï, we measure the Peltier effect produced by the passage of a current from lead to metal a, we will have, by V equality (22), the thermoelectric power of metal a related to lead.
M. Leroux (' ) used this method to determine the thermoelectric power of a large number of metals in relation to lead.
2" Equality (i4 bis)
,-T,
Câ(To,Ti)=/ [ha{'ï)-h,{T)]dt
(') F. -P. hv.Y^(iV\^ Recherches sur les courants thermo-électriques, Mémoire lu à l'Académie des Sciences le 20 avril i865 {Annales de Chimie et de Physique, 4' série, t. X, p. 201; 1867).
CHAP. IX. - THERMO-ELECTRIC CURRENTS.
becomes, according to the equalities ( i),
5i3
(23)
This remarkable relationship lends itself to experimental verification. For example, M. Bellati(') has studied with great care the thermo-electrical electromotive force of the iron-zinc couple whose cold solder is maintained at o "C. (Tq = 2^3), while the temperature of the hot solder joint varies from o° C. to i20°C. 11 found that this electromotive force could be represented by the formula
^(To,Ti) = 9'7.77(Ti -273)- i.9488(T, - 278)2.
For T, - 273= i3",8, this formula, joined to equality (23),
gives
P(T,) = o''", 006065.
The direct measurement of the Peltier effect at this temperature gives
P(Ti)=:0'=''', 005923.
The error is less than^ of the number to be measured; this agreement will appear very satisfactory if Ton takes into account the numerous experimental data necessary to carry out the comparison and the difficulty that the determination of each of them presents.
3" The diagrams {Jig'' 109) which represent by the curves
Fig. 109.
AA', BB' the thermoelectric powers of two metals with respect to lead provide immediately, at each temperature
(' ) Bellati, Atti del B. Istituto Veneto, 5* series, t. V; 1879.
D. - I. 33
5l i BOOK V. - METALLIC CONDUCTORS.
the Peltier refTel coefficient between these two metals. In elFet, the relations (i 5) and (21) give
Pfj(T)-|[7a(T)~jK,(T)J;
On inspection of fig. 109, we see that this relationship can be written
P,^(T) = -laire 6a "p,
Taire bartf^ being counted positively when point a is above point b.
This geometrical representation allows us to predict the direction of the thermal phenomena that will occur in the vicinity of the two welds of a bimetallic chain. We will assume that the temperature varies little in the regions surrounding the two welds, so that the laws relating to the Peltier phenomenon can be applied in these regions.
Let us first assume that curve AA' is constantly above curve BB' in the experimental field {^fig. io3). We know that, in this case, the current flows from metal a to metal h through the cold junction. Thus, according to the previous rule, the current heats the cold solder and cools the hot solder.
Let us now consider the case where the two curves AA', BB' intersect at a neutral point C, the curve AA' being above the curve BB' when the temperature is below the neutral point. Let's assume that the temperature Tq of the cold junction is maintained below the neutral point.
As long as the temperature T( of the hot solder is also below the neutral point {^fig- io4), the current heats the cold solder and cools the hot solder.
When the temperature T, i^fig- io5) is between the neutral point and the inversion temperature, the current heats up both welds.
When the temperature T, exceeds the inversion temperature (/fo-. 106), the current cools both welds.
the THOMSON effect. 5l5
CHAPTER X.
THE THOMSON EFFECT.
§ 1 - The electrical transport of heat.
The theory of permanent currents is based, in all cases, on two hypothetical principles, one of which is discussed in Chapter IV, by generalizing Ohm's law, and the other in Chapter VI, by generalizing Joule's law. The application of the first principle to metallic conductors whose temperature is not uniform has provided us with all the consequences explained in the previous chapter. Let us now apply our second principle to these same conductors.
Let us consider a metallic conductor whose points are not at the same temperature and, inside this conductor, let us trace a closed surface S. Let N/ be the normal at a point on this surface, normal directed towards the interior of the space enclosed by this surface.
Suppose that the uniform currents flowing through the system are maintained inside this surface for the time dt, but that all currents outside this surface are suppressed. In this hypothesis, the system would experience a change of electrical distribution during the time dt. Each element dS of the surface S would take a charge
- [k cos(N;, x)-h w cos(N/,^) 4- w cos(Ni, 5)] dS dt.
This change in electrical distribution would cause a variation oU in the internal energy of the system assumed without current; this system would release a quantity of heat û?Q'= - SU. The principle that we want to apply states that the volume that limits the surface releases in time dt a quantity of heat
5i6 BOOK V. - METALLIC CONDUCTORS.
which can still be written
^Q = -3U.
Equality (i) of the previous chapter allows us to calculate SU and we obtain
EdQ = dt^iiY -^ K)[u cos(Nj, x)-^v cos(N,-, y) -+- w cos{Ni,z)] dS.
If we notice that we have [Book IV, Ghap. II, equalities (i6)
and (17)]
K = 0-4-TH,
that we have in addition, at any point inside the surface S,
of the di> dw
àx dx dz '
it will be easy to see that we can write
. r r ru àY d& ^c
(0
dT dx ^ âl dx~^ àx)"' d\ ^ de dH
' ày'^ dj 'dy
dT toy ^ d^ dy^ dy) ^
of ^ dïi
■ - -f- - -t- 1 - dz dz dz
of dT dT 51
^dH dT dT\"1 , , ,
T -TT, -; h li^- ]w \dx dy dz.
di. dz"-2 / J"^
This equality, which must take place whatever the shape of the surface S, provides the expression of the amount of heat released in time dt by an element of volume of the conductor.
If we observe that we have [Chap. IX, equality (5)]
_ fdW de^ '^^T _^Y{-\ ~ \ dx dx dT dx ' dx j
I dy de de_dT_"^\
^ ~ dy dy dT dy dyj'
/ dT "dT
^"^^-y'Tz^Tz-^lîTz^^^y we can easily see that the amount of heat released during the
CHAP. X. - the thomson effect. 5l7
time dt^ by an element of a metallic condiicleur whose temperature is not uniform and which permanent currents run through, is given by the equality
Ec?Q= S^{u'^-^v'^-^Kv''-)dxdydzdt
(■'-) { \ ôx '^ ôy ' " ôz )
u 'i- V - - !- w ■ - j dx dy dz dl
ô^ I ffl dl dT
- 'Y -=r\u \- V 'x- w -- )dx dy dz dt.
dT \ ux dy dz J"^
The first term of the second member represents the heat release given by Joule's law.
The second term represents the Peltier effect. It disappears if the conductor material is homogeneous around the element dx dy dz-^ but it does not change its value depending on whether the temperature distribution is uniform or not around the element dx dy dz.
The third term, on the other hand, remains the same whether the conductor material is homogeneous or not around the element dxdydz] but it would vanish if the temperature were distributed in a uniform way around this element.
Let/ be the electric flux at point (^, y, s). Let F be the heat flow at the same point. If K is the heat conductivity of the substance which forms the conductor, this last flow will have as components
dl
T. ^T
. to the
- K - ,
- K - ,
- K -
ôx
<^y
dz
and the third term of Ee- We can write
(3) Eo(r - TS) - fl?5c=: - Crf^r,
o being a quantity that depends on the nature of the reaction that the positive electricity produces when passing from M to N, on the physical state of the various bodies that take part in this reaction and on the external forces that act on the system. The variation SW can be divided into two parts; a first part
528 BOOK VI. - ELECTROLYTES.
of this variation is represented by the term
^ (Y li - y a) dcj,
Va and Vjj being the potential levels inside the two electrodes A and B; a second part o',W comes from the displacement that the various electrified parts of the electroljte must have undergone in relation to each other. This part could not be calculated without a complete knowledge of the laws of electrical distribution on the electroljte. However, for the reasons indicated at the end of the preceding paragraph, we cannot hope to acquire this knowledge. We shall therefore take the decision to neglect 8',W, which, as we shall see, will not prevent the results obtained from being very much in conformity with experience for batteries and voltameters of the dimensions ordinarily employed. For the same reason, we will reduce the quantity
8(61 ^Ti -4- 02 5-2-1- ...-^S "q")
à
(eB-eA)rf^y,
neglecting the terms that can come from the change of state of the electrol}'te.
The uncompensated work dz, generated by the modification we have just considered, will have the value
(4) ck = [C-h{t\A-heA) - {s\n-^eK\]dq, and this modification will be impossible if we have
(5) C^{tYx^ex)-{tK + YB)io.
Let us now take another virtual modification: the passage of a positive electric quantity dq from point N to point M. We will have, in this case,
(W H- 01^1 -+- 025-2-+-. .+ e "gr") = [(£'a-+- Oa) - (£Vb4- Vb)] rfg-.
The passage of the electric charge produces some chemical reaction which may be different from that produced by transport in the opposite direction. This reaction causes E(r - TS) to vary by Eû'(r - TS); the external forces do d^^ work.
CHAP. 1 - I.\ PILE. 529
We will pose
(6) Eo'(V-Tl.)-d^', = -C'dq,
C' being a quantity analogous to C.
Our modification will give rise to an uncompensated work of z' having the value
(7) dz'=[C- (t\x-i-B x)-^{zYji^ en)] dq, and this modification will be impossible if we have
(8) C'- (£VA-4-eA) + (£VB+eB)^0.
For there to be equilibrium in the system, electricity must not be able to flow through Teletrolyte in either direction. It is necessary, therefore, that we can find two values V;^, V^, of the new potential satisfying both conditions (5) and (8).
Three cases are to be distinguished:
1" We have
(9) -C' o.
The inequality (y) will therefore be verified in the present case; it will be imD. - [. 34
53o BOOK VI. - MÎS ELECTROLYTES.
possible to find an electrical distribution on the system which prevents the chemical reaction from occurring. 2° We have
(lo) -OC.
In this case, the electrical equilibrium is ensured on the system for an infinite number of values of the potential level difference between the two electrodes,
In fact, for this balance to take place, it is sufficient to have
-C'5(£VB+-eB)-(£VA-i-eA)â*^,
or, in other words, that (Vu - V^) is at least equal to
z and at most equal to
-C'+eA- e"
Such a system is said to be polarizable; when equilibrium is established, the system is polarized.
Let's take an example.
Two platinum electrodes are immersed in acidulated water. Whichever way an electric charge passes from one to the other, the water is decomposed; hydrogen is released on one electrode and oxygen on the other. The reaction being the same in both directions, we have
Without electrical transport, under ordinary conditions, water cannot decompose into oxygen and hydrogen; it is in a stable state. The decomposition in question therefore corresponds to a negative uncompensated work; we have
E8(r - TS) - t/5e>o, or
"::
which would be established between these two electrodes if they were joined together, not by the electroljle, but by metals all having the same temperature.
This property is used to obtain standard potential level differences; the Daniell stack, the Latimer-Clark stack, are constantly used for this purpose.
§ 3 - Properties of a closed stack.
Let us now leave aside the systems of the first kind, on which the electrical equilibrium cannot be established, and let us study only the systems of the two other kinds: polarizable and non-polarizable.
Given one of these systems, let us join the two metals A. and B. by a chain formed by a single metal or several metals at the same temperature and let us ask ourselves if electrical equilibrium is possible on the system thus formed.
532 BOOK VI. - ELECTROLYTES.
To the equilibrium conditions already obtained, we must add the condition that we obtain by expressing that the internal thermodynamic potential does not vary when a charge dq passes from point M to point N or vice versa through the metal circuit. This condition is
(t2) £VA + eA = £VB+eB.
Let's see if it is compatible with the conditions found previously.
i*' Non-polarizable pairs.
The two conditions of equilibrium
tYx-^^K - ^^h
- 08+ C = O,
sVA-t-eA-sVB
-08 =0
are incompatible if you do not have
1^ = 0.
According to the definition of C given by equality (3), this equality cannot take place unless the reaction, of which the battery is the seat, supposedly carried out without any electrical phenomenon, is a reversible phenomenon. Let us exclude this particular case and we arrive at this proposition:
If Von closes a non-polarizable couple with a metal chain, V electrical balance will not be able to remain on the system.
2° Polarizable pairs.
For all polarizable pairs, we have
(lo) C>C.
We can then classify these couples into three classes:
a. Those for which we have
^. Those for which we have
CHAP. I. - THE PILE. 533
Y" Those for which we have
- C'>C>o. The two equilibrium conditions
(îVB + eB) - (£VA-4-eA) =
are only compatible for couples of the first class.
Thus, among the three categories of polarizable couples, the couples of category a are the only ones on which electrical equilibrium can be established if the circuit is closed. On the others, electrical equilibrium is impossible. The polarizable pair taken as an example in § 2 falls into this category a.
We will properly reserve the name of voltameters for polarizable couples of the category a.
Non-polarizable pairs and polarizable pairs of the categories p and Y are such that the electrical equilibrium, possible on such a pair in open circuit, becomes impossible in closed circuit. These couples are called voltaic elements.
If we close the circuit of a voltaic element, the system of which this element is part will become the seat of a permanent current. What will be the intensity of this current?
To answer this question, we will make use of the first of the two hypothetical principles which dominate the whole theory of permanent currents, the principle obtained in Chap. IV of Liv. V by generalizing Ohm's law.
Let's remember this principle:
At a point (x, j, z) of a conductor through which permanent currents flow, the electric flux has the components u, p, w, and the specific resistance has the value ,R. If an electric charge dg passes from the point (^,J)', ^) to the point
{x -+- dx, y -\- dy, z -\- dz),
in the system assumed without current, it would give rise to a change that would generate uncompensated work dz and we have, whatever dx^ dy^ dz.,
(t3) d-z = A{u dx -{- V dy -{- w dz) dq .
Now let's consider the closed circuit we want to study
534 BOOK VI. - ELECTROLYTES.
dier. Let us assume that all the flow lines that are to be closed close after running through the entire chain kfië- * ' 0- Consider an infinitely untied flow channel. Let d'co
Fig. III.
the area of its section at a point, /the flow at this point. The product
keeps the same value all along the flow channel; the flow has the same direction all along this channel.
The current can, in this channel, run in the MQNP direction or in the MPNQ direction. Let us discuss these two hypotheses:
i" The current flows in the MQNP direction.
Suppose a positive charge dq describes an element ds of this path. In the system assumed to be currentless, it would produce uncompensated work d-z^ and we would have, according to equality (i3),
dx = Si fdsdq,
= -j- idsdq. doi ■'
By integrating the two members along the entire closed line
MQNPM, we would have
rsi 01 = idq I -7- ds, J d(M
8t being the uncompensated work produced, in the system assumed without current, by the passage of a positive electricity charge dq through the closed line MQNPM.
Now this work consists of the uncompensated work produced by the load dq while traversing the path MQN, work which has the value, according to equality (4),
[C + sV(M) + Oa - £V(N) -+- QYi]dq
CHAP. I. - THE PILE. SS")
cl of the uncompensated work produced by the same load while travelling the NPM path, work which has the value
[zY {N )-+- Su -t\ (M) -ex]dq. So we have
oz = Cdq ,
cl, therefore, if the current in the considered flow channel runs in the direction MQNPM, we have
(,4) C = ij^ds.
2" The current flows in the MPNQ direction. We find then, in the same way,
M 5) C'=i'f-^ds.
The quantities in the second members of equalities (i4) and (i5) are all positive. Each of these equalities is therefore only possible if its first member is positive. Let us see the various cases that can arise:
i" Non-polarizable pairs.
We then have
C = - C.
If C is positive, equality (i5) is impossible; in all flow lines, the current flows in the MQNPM direction.
If C is negative, equality (i4) is impossible; in all flow lines, the current runs in the MPiNQM direction.
'>,' Polarizable pairs.
If the couple belongs to the class ^, we have
C o.
Equality (i4) is impossible; in all flow lines, the current runs in the MPNQM direction. If the couple belongs to the class y, we have
"!^>o, C'Jfi?" = -T82,
8S being the variation that the entropy of the system, supposed in the neutral state, would undergo by the reaction produced in the system during the time dt.
We also have
(1-2) CJ"=--E8(r-TS) + c?5e,
dQ>e being the work done, under the same conditions, by the external forces to which the system is subjected.
Let's assume that the system is subjected only to a normal, uniform and constant pressure P. We will then have
(i3) CJrff=-8[E(r-T2)-i-Pt^],
V being the volume of the system.
548 BOOK VI. - ELECTROLYTES.
On the other hand, we can easily find the relations
ES-- A[E(r-TS)-Pp]
so that the equalities (i i) and (i3) give us the relations
T dC ('4) "^-"Ëdî'
(,5) . ^-^idt^-^..
The first of these two relations, which assume the variables P and ï as independent variables, is due to M. H. von Helmholtz (').
This relationship leads to the following consequences:
If the chemical heat in a battery is greater than the voltaic heat, the electromotive force of the battery decreases when the temperature is increased under constant pressure.
If the chemical heat in a battery is less than the voltaic heat, the electromotive force of the battery increases with temperature, the pressure remaining constant.
If, in a battery, the chemical heat is equal to the voltaic heat, the electromotive force of the battery, under constant pressure, remains independent of temperature.
The Daniell element is in the latter case.
Mr. Siegfried Czapski (2), Mr. Hans Jahn (3), Mr. Lucien Poincaré (') have verified the accuracy of the relation of Mr. H. von Helm ( ') H. VON Helmholtz, Zur Thermodynamik chemischer Vorgdnge {Sitzungsberichte der Akademie der Wissenschaflen zu Berlin, 1882, p. 2).
(^) S. Czapski, Ueber die thermische Verànderlichkeit der electromotorischen Kraft galvanischer Elemente und ihrer Beziehung zur freien Energie derselben ( Wiedemann's Annalen, t. XXI, p. 209; 1884 ).
( ') H. Jahn, Ueber dieJEquivalenz von chemischer Energie und Stromenergie ( Wiedemann's Annalen, t. XXVIII, p. 491; 1886).
(*) L. PoiNCARÉ, Sur les piles à ëlectroly tes fondus et sur les forcées thermoélectriques à la surface de contact d'un métal et d'un sel fondu ( Comptes rendus, t. CX, p. SSg; 1890).
CHAP. II. - CHEMICAL HEAT AND VOLTAIC HEAT. 549
holtz for various batteries, containing dissolved or melted electroljtes.
Let us suppose that, in a battery, the difTéi'ence between chemical heat and voltaic heat is independent of temperature; or, which amounts to the same thing, according to the formula {i4)i that the electromotive force of the battery can be represented, as a function of temperature, by an expression of the form
C=KlogT + K'.
In this case, the relation (i i) gives
Now, if we designate by C the heat capacity under constant pressure of the system, we have
The previous equality becomes
8G = o.
Therefore, for the difference between chemical and voltaic heat to be independent of temperature, it is necessary and sufficient that the reaction that occurs in the system does not alter its heat capacity.
If, in particular, the chemical heat is constantly equal to the voltaic heat, a case in which the electromotive force is independent of temperature, the reaction of which the system is the seat does not vary its heat capacity. This proposition is due to M. Lippmann (' ). The reciprocal of this proposition is not exact.
We have, in the Daniell element, an example to which Mr. Lippmann's proposal applies. The Daniell element having, as we have seen, an electromotive force independent of temperature, the heat capacity of this element must not vary by the reaction of which it is the seat.
( ' ) Lippmann, De l'action de la chaleur sur les piles et de la loi de Kopp et Wœstyne {Comptes rendus, t. XCIX, p. Sgô; i88^).
55o BOOK VI. - THE ELECINOLYTES.
In fact, the molecular specific heat of copper is approximately equal, according to Dulong and Petit's law, to the molecular specific heat of zinc. The heat capacity of a solution of copper sulfate and zinc sulfate does not change when one weight of copper sulfate is replaced by an equivalent weight of zinc sulfate. Therefore, the reactions that occur in the Daniell element do not change its heat capacity.
Mr. L. Poincaré (' ) gave another example of a couple to which Mr. Lippmann's theorem applies.
Let us now examine relation (i5).
This relationship has the following consequences ("):
If the reaction in the cell is accompanied by an increase in volume, the electromotive force of the cell decreases as the pressure increases.
If, on the other hand, the reaction in the battery is accompanied by a decrease in volume, the electromotive force increases with the pressure.
If the reaction is not accompanied by a significant change in volume, the electromotive force is essentially independent of the pressure.
The batteries that give off gases, such as the Volta battery, are the seat of a chemical reaction accompanied by a considerable increase in volume; the electromotive force of such batteries is all the weaker as the battery operates under a more considerable pressure. On the contrary, gas cells are the seat of a chemical reaction accompanied by a great contraction; the electromotive force of these cells increases with the pressure. Finally, in batteries where no gaseous element is involved, such as the Daniell battery, the chemical reaction causes only a small change in volume; these batteries therefore have an electromotive force that is substantially independent of pressure.
These are the most general properties of the flight elements
(' ) L. Poincaré, loc. cit.
(^) P. DuHEM, Le potentiel thermodynamique et ses applications, p. 117 Paris, 1886.
CHAP. II. - CHEMICAL HEAT AND VOLTAIC HEAT. 55 1
laity. To penetrate more deeply into their study than we have done in the preceding Chapter and in this one, it would have been necessary to make an examination of the properties of dissolutions which would have led us too far.
§ 3 -- Two general consequences of the theory of currents
permanent.
The theory of permanent currents is based on two principles which we have constantly applied in the foregoing. These principles lead to certain consequences which, when generalized, will be of great use to us in the study of any current. We will end this Volume by stating these consequences.
The second of the two principles we used in the study of permanent currents is the following:
Let's draw a closed surface containing a part of the system. Let us imagine that the permanent currents remain what they are inside this closed surface and cancel out outside. The system would then experience, in time dt, a change that would cause the internal energy of the system to vary by 8U, assuming no current and giving rise to a diBe work of the external forces. The part of the system which is inside the surface S gives off, in time dl^ a quantity of heat û?Q given by
E c?Q = - E SU -r- d(^e This equation can be applied in particular to a closed surface containing the whole system inside. It then gives the following result:
The internal energy of a system crossed by permanent currents undergoes at each instant a variation equal to the variation of V energy of the system supposed without current.
This proposition can be seen as a special case of the following one, which will serve as a fundamental hypothesis in the study of electrodynamics:
When a system experiences a change in which
552 BOOK VI. - ELECTROLYTES.
the various conductors which enclose it do not change position and in which the electric flux in each point of these conductors remains invariable, its internal energy experiences a variation equal to that which this modification would cause to the internal energy of the same system supposed without current.
The first of the two principles that govern the study of permanent currents is the following:
Let (a:, y-, z) and (.r -h 5^, y + 5/, z 4- oz) be two neighboring points of a system through which permanent currents flow. Let us pass a charge dq from the first to the second. This transport would produce, in the system supposed to be currentless, an uncompensated work dx, and we would have
Cx, Cy, Cz being the components of the electromotive force at point £C,y, z.
Let us imagine that the various material masses which constitute the system do not undergo, in the transport of the charge dq, any displacement. Let U be the internal energy of the system supposed without current and ^ its internal thermodynamic potential. We will have
EU = # - T ^ , dr= - L^
and, therefore.
E8U = - rfx^T ^(dz). at
The transport of the charge dq from the point (x,y,z) to the point (x + 5^, y + 8/, z + 82) thus varies the internal energy of the system assumed without SU current and we have
This equality takes place regardless of the load dq and the direction of displacement.
Let us take in particular an infinitely untied channel formed by
CIUP. H. - CHEMICAL HEAT AND VOLTAIC HEAT. 553
flow lines. Let S, S' be two normal sections of this channel. Suppose that the point (^, y, z) belongs to the first one and the point (x -f- 8x, y + OjK, z -+- 8s) to the second. Let i be the flow from the first to the second. Let dio be the area of the section S. Let ds be the distance between the two sections.
Let us transport from the first to the second a quantity of electricity idiù dt. This is precisely the quantity of electricity that would be transported from one to the other if the currents remained what they are inside the small segment that the two sections cut into the flux tube, cancelling out everywhere inside this segment. We can therefore, in this case, replace ( - 8U) by the quantity of heat
Chapter III. - Green's theorem i(>
Chapter IV. - Gauss' Lemma. Attraction of a homogenous spherical layer 22
§ 1 - The three Gauss lemmas 22
§ 2 - Existence of the force of attraction at a point inside the
acting loads. Consequence of the Gauss lemmas 27
§ 3 Attraction exerted at a point by a homogeneous spherical layer 35
Chapter V. - Properties of the potential function at a point inside the acting charges 38
§ 1 Existence and continuity of the potential function at a point
inside the acting loads 38
§ 2 - Existence of first order partial derivatives of the potential function. Their relation with the components of the force 43
§ 3 Existence of second order partial derivatives of the function
potential at a point inside the acting charges 4^
§ 4 - Poisson equation 5i
§ 5 History 52
Chapter VI. - Criteria of the potential function of an electrified volume. Attraction of ellipsoids 57
§ 1 Criteria of the potential function of an electrified volume 67
§ 2 Potential function of a homogeneous ellipsoid 62
556 TABLE OF CONTENTS.
VH'ftt".
Chapter VII. - Electrostatic action and potential function of an electrified surface -ji
§ 1 - Study of the normal component of the action exerted at a point
by an electrified surface 71
§ 2 - Study of the tangential components of the action exerted in a
point by an electrified surface 79
§ 3 - Refraction of the force when passing an electrified surface 83
§ 4 Potential function of an electrified surface 88
§ 5 Criteria of the potential function of an electrified surface 92
§ 6 - History 94
Chapter VIII. - Reminder of some notions of Mechanics 99
§ 1 - The principle of virtual velocities 99
§ 2 - Statement of d'Alembert's principle. Fundamental formula of the
io3 dynamics
§ 3 - Various remarks on the connections. io5
§ 4. ^ Theorem of the living forces 108
§ 5 - From the potential T 1 10
§ 6 - Criterion of equilibrium stability 1 14
Chapter IX. - Of the electrostatic potential 117
BOOK II.
Electrical distribution on conducting bodies and the Lejeune-Dirichlet problem.
Chapter I. - Condition of electrical equilibrium. Electricity is resistive at
over/under conductive bodies i25
§ 1 Principles of Poisson theory 126
§ 2 - In the state of equilibrium, electricity resides on the surface of bodies
drivers 128
Chapter II. - // there is one and only one state of electrical equilibrium i33
§ 1 - Analytical translation of the Poisson principle i33
§ 2 - Is there a state of electrical equilibrium? i34
§ 3 - If there is an equilibrium distribution, there is only one
only i36
Chapter III. - The Gauss identity and the arithmetic mean theorem , i4o
§ 1 . - The Gauss identity i4o
§ 2 The arithmetic mean theorem 142
§ 3 Theorems on the variation of the potential function outside the
acting loads i43
Chapter IV. - Some theorems on the sign of the electric density
on the surface of a conductor i49
Chapter V. - The Lejeune-Dirichlet Problem i54
§ 1 - The problem of electrical distribution can be reduced to the Lejeune-Dirichlet problem i54
TABLE OF CONTENTS. 55^
Popes.
§ 2 Demonstration of the Lejeune-Dirichlet principle 169
§ 3 - Statement of a more general problem than that of Lejeune-Dirichlet i63
Chapter VI. - Ijx Green's function i65
§ 1 - Green's problem is equivalent to Lejeune-Dirichlet's problem ,65
§ 2 - Fundamental proprieties of the Green function 171
§ 3 Determination of the Green's function in some cases
simple , -',
CuAPiTRE VIL - Transformation , in any orthogonal coordinates, of the equation AV = o. Electrical distribution on an ellipsoid . 180
§ 1 - Transformation of the equation AV = o into any orthogonal coordinates 180
§ 2 - Transformation of the equation AV = o into geographic and elliptical coordinates 186
§ 3 - Electrical distribution on an ellipsoid removed from any influence 189
§ 4- - Special cases iq',
§ 5 - Geometric solution of the problem of the distribution on an isolated ellipsoid i()5
§ 6 - Electrical distribution on an ellipsoid subject to an influence
any 200
Chapter VIII. - The method of inversion of)3
§ 1 - The method of inversion or electrical images 2o3
§ 2 - Application. Electrical distribution on a spherical cap . 210
§ 3 - Liouville's and M. P. Painlevé's theorems 2i5
Chapter IX. - Cari Neumann's method 220
§ 1 The theorem of Mr. Vito Volterra 220
§ 2 The theorem of Mr. Axel Harnack 228
§ 3 - Some definitions 932
§ 4 Definition and properties of the fundamental function 232
§ 5 Definition and properties of subordinate functions 235
§ 6 - Solution of the Dirichlet problem for the interior space of a
second row surface, non-biased 239
§ 7 - Solution of the Dirichlet problem for the space outside
a second row surface, non-biased 240
Chapter X. - The natural distribution 243
§ 1 - How the solution of the Lejeune-Dirichlet problem for all bodies is reduced to the determination of the distribution
natural on all bodies 243
§ 2 - Method of Mr. G. Robin to determine the natural distribution on a 24 > conductor
Chapter XI. - The alternating process. 202
§ 1 - Extension of the Lejeune-Dirichlet problem 252
§ 2 - Extension of the arithmetic mean method 254
558 TABLE OF CONTENTS.
agcs.
§ 3 The alternating process; formation of the solution 256
§ 4 - Proof of the theorem just stated 269
§ 5 Successive applications of the alternating method 263
Chapter XII. - Murphy's Problem 264
§ 1 - Murphy's problem ... 265
§ 2 Fundamental laws of electrical condensation 269
§ 3 Murphy's method 270
§ 4 - The combinatorial method of Mr. Cari Neumann 2^3
BOOK III.
Experimental study of electrical distribution.
Chapter I. - The test body 279
§ 1 - Theory of the body of test 279
§ 2 - Use of the test body 283
Chapter II. - Open conductors 287
^5 1 - Electrical distribution on an open conductor subjected to a
any influence 287
§ 1 - Open conductor not subject to any influence 291
Chapter III. - Level surfaces and their orthogonal trajectories 297
§ 1 . - General Theorems 297
§ 2 - Experimental study of level 3o3 surfaces
Chapter IV. - The layers of level 3o5
§ 1 . - An identity of Green 3o5
§ 2 Fundamental properties of the level layers 309
§ 3 - Complete study of an electrical influence case 3i2
§ 4 - A special class of 3i4 capacitors
§ 5 - Absolute electrometer by Sir W. Thomson Thomson 3i8
Chapter V. - Green's problem and Faraday's theorems 32 1
§ 1 - Green's inner problem; Green's solution 32i
§ 2 - Gaussian solution 323
§ 2. ■ Lejeune-Dirichlet solution 328
§ 4. ^ - Application to questions of electrical influence; Faraday's first theorem 329
§ 5 - Green's external problem; Faraday's second theorem; electrical screens 33i
BOOK IV.
The internal thermodynamic potential of an electrified system.
Chapter I. - Some notions of Thermodynamics 337
§ 1 - Thermodynamic potential 337
§ 2 - Properties of displacements without change of state 34i
TABLE OF CONTENTS. SStJ
Pages. Chapter II. - Determination of the internal thermodynamic potential
of an electrified system 348
§ 1 - How, in the study of a system, one can take into account
the mutual disposition of its parts 348
§ 2 - Introduction of the fundamental assumption of compatibility 355
§ 3 - Introduction of the law of universal gravitation and the law
of Coulomb 359
§ 4 - On the continuity of quantity 8 364
BOOK V.
Electrical balance and permanent currents on metallic conductors.
Chapter I. - Fundamental laws of electrical balance on metallic conductors 867
§ 1 - Condition of electrical equilibrium 367
§ 2 - Stability of the electrical equilibrium . 371
Chapter II. - Electrical equilibrium on homogeneous conductors,
laws of electrical discharge 873
§ 1 - Electrical equilibrium on homogeneous conductors 878
§ 2 - Theory of electric discharge; Clausius theorem 878
§ 3 - Snow Harris 38i thermometer
§ 4 - Complete discharge of a capacitor; Riess experiments 385
55 5 Discharge of a capacitor by successive sparks 889
§ 6 Cascaded batteries 892
Chapter III. - The intensity of the currents 898
§ 1 - Currents flowing in the conductor's mass 898
§ 2 -^ Uniform currents 4o2
§ 3 - Linear currents 40-^
Chapter IV. - Ohm's law 4^8
§ 1 - Statement of Ohm's law 4o8
§ 2 - Statement of Ohm's law for linear currents 410
§ 3 - Permanent currents in the mass of a conductor
Chapter . - The permanent movement of electricity in a blade
metallic 4 M"
§ 1 - The permanent motion of electricity in a plane blade. 4'!}
§ 2 - Currents in a curved blade 43
Chapter VI. - Joule's law 4^7
Chapter VII. - The potential level difference of two metals in
contact 44-^
§ 1 - The electrical equilibrium on a heterogeneous metallic conductor. 44'' § 2 - Some theorems on the attraction of electric layers
doubles 4'^"-"
56o TABLE OF MATTER.
Pages.
§ 3 Permanent currents in heterogeneous metallic conductors 46'.
§ 4 - M. Pellat's method for determining level differences
potential of two metals in contact ^73
Chapter VIII. - The Peltier effect 47S
§ 1 - The Peltier effect 47S
§ 2 - Relation between the Peltier efl'et and the potential level difference
of two metals in contact 4^^
Chapter IX. - Thermoelectric currents 4*^'^
§ 1 - Conditions under which thermoelectric currents occur 4^8
§ 2 - Properties of thermoelectric chains 494
§ 3 - Properties of bimetallic chains 5o3
§ 4 Relationship between thermoelectric phenomena and potential level differences at contact 509
§ 5 - Relation between thermoelectric phenomena and the effect
Peltier 5i2
Chapter X. - The Thomson effect 5i5
§ 1 - The electrical transport of heat 5i5
§ 2 - Remarks on the theory of Clausius 020
BOOK VI.
Electrolytes.
Chapter I. - The electromotive force of a battery 525
.1 - Faraday's law 525
§ 2 - Properties of an open stack 527
§ 3 - Properties of a closed stack 53i
§ 4 - Experimental checks 538
Chapter II. - Chemical heat and voltaic heat 542
§ 1 - Distinction between chemical and voltaic heat... 5'|2
§ 2 - Relation of Helmhollz 547
§ 3 - Two general consequences of the theory of permanent currents 55 1
Table of Contents 555
END OF TABLE OF CONTENTS.
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