A Treatise on Electricity and Magnetism/Volume 1/Part 1/Chapter 4

by James Clerk Maxwell

CHAPTER IV.

GENERAL THEOREMS.

95.] In the preceding chapter we have calculated the potential function and investigated its properties on the hypothesis that there is a direct action at a distance between electrified bodies, which is the resultant of the direct actions between the various electrified parts of the bodies.

If we call this the direct method of investigation, the inverse method will consist in assuming that the potential is a function characterised by properties the same as those which we have already established, and investigating the form of the function.

In the direct method the potential is calculated from the distribution of electricity by a process of integration, and is found to satisfy certain partial differential equations. In the inverse method the partial differential equations are supposed given, and we have to find the potential and the distribution of electricity.

It is only in problems in which the distribution of electricity is given that the direct method can be used. When we have to find the distribution on a conductor we must make use of the inverse method.

We have now to shew that the inverse method leads in every case to a determinate result, and to establish certain general theorems deduced from Poisson's partial differential equation

${\frac {d^{2}V}{dx^{2}}}+{\frac {d^{2}V}{dy^{2}}}+{\frac {d^{2}V}{dz^{2}}}+4\pi \rho =0$

The mathematical ideas expressed by this equation are of a different kind from those expressed by the equation

$V=\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }{\frac {\rho }{r}}dx'\,dy'\,dz'$.

In the differential equation we express that the values of the second derivatives of V in the neighbourhood of any point, and the density at that point are related to each other in a certain manner, and no relation is expressed between the value of $V$ at that point and the value of $\rho$ at any point at a sensible distance from it.

In the second expression, on the other hand, the distance between the point $(x',\,y',\,z')$ at which $\rho$ exists from the point $(x,\,y,\,z)$ at which $V$ exists is denoted by $r$, and is distinctly recognised in the expression to be integrated.

The integral, therefore, is the appropriate mathematical expression for a theory of action between particles at a distance, whereas the differential equation is the appropriate expression for a theory of action exerted between contiguous parts of a medium.

We have seen that the result of the integration satisfies the differential equation. We have now to shew that it is the only solution of that equation fulfilling certain conditions.

We shall in this way not only establish the mathematical equi valence of the two expressions, but prepare our minds to pass from the theory of direct action at a distance to that of action between contiguous parts of a medium.

Characteristics of the Potential Function.

96.] The potential function $V$, considered as derived by integration from a known distribution of electricity either in the substance of bodies with the volume-density $\rho$ or on certain surfaces with the surface-density $\sigma ,\rho$ and $\sigma$ being everywhere finite, has been shewn to have the following characteristics:—

(1) $V$ is finite and continuous throughout all space.

(2) $V$ vanishes at the infinite distance from the electrified system.

(3) The first derivatives of $V$ are finite throughout all space, and continuous except at the electrified surfaces.

(4) At every point of space, except on the electrified surfaces, the equation of Poisson

${\frac {d^{2}V}{dx^{2}}}+{\frac {d^{2}V}{dy^{2}}}+{\frac {d^{2}V}{dz^{2}}}+4\pi \rho =0$

is satisfied. We shall refer to this equation as the General Characteristic equation.

At every point where there is no electrification this equation becomes the equation of Laplace,

${\frac {d^{2}V}{dx^{2}}}+{\frac {d^{2}V}{dy^{2}}}+{\frac {d^{2}V}{dz^{2}}}=0$.

(5) At any point of an electrified surface at which the surface-density is $\sigma$, the first derivative of $V$, taken with respect to the normal to the surface, changes its value abruptly at the surface, so that

${\frac {dV'}{dv'}}+{\frac {dV}{dv}}+4\pi \sigma =0$,

where $v$ and $v'$ are the normals on either side of the surface, and $V$ and $V'$ are the corresponding- potentials. We shall refer to this equation as the Superficial Characteristic equation.

(6) If $V$ denote the potential at a point whose distance from any fixed point in a finite electrical system is $r$, then the product $Vr$, when $r$ increases indefinitely, is ultimately equal to $E$, the total charge in the finite system.

97.] Lemma. Let V be any continuous function of $x,y,z,$ and let $u,v,w$ be functions of $x,y,z,$ subject to the general solenoidal condition

${\frac {du}{dx}}+{\frac {dv}{dy}}+{\frac {dw}{dz}}=0$,

(1)

where these functions are continuous, and to the superficial solenoidal condition

$l(u_{1}-u_{2})+m(v_{1}-v_{2})+n(w_{1}-w_{2})=0\,\!$,

(2)

at any surface at which these functions become discontinuous, $l,m,n$ being the direction-cosines of the normal to the surface, and $u_{1},v_{1},w_{1}$ and $u_{2},v_{2},w_{2}$ the values of the functions on opposite sides of the surface, then the triple integral

$M=\iiint (u{\frac {dV}{dx}}+v{\frac {dV}{dy}}+w{\frac {dV}{dz}})\,dx\,dy\,dz$

(3)

vanishes when the integration is extended over a space bounded by surfaces at which either $V$ is constant, or

$lu+mv+nw=0\,\!$,

(4)

$l,m,n,$ being the direction-cosines of the surface.

Before proceeding to prove this theorem analytically we may observe, that if $u,v,w$ be taken to represent the components of the velocity of a homogeneous incompressible fluid of density unity, and if $V$ be taken to represent the potential at any point of space of forces acting on the fluid, then the general and superficial equations of continuity ((1) and (2)) indicate that every part of the space is, and continues to be, full of the fluid, and equation (4) is the condition to be fulfilled at a surface through which the fluid does not pass.

The integral $M$ represents the work done by the fluid against the forces acting on it in unit of time.

Now, since the forces which act on the fluid are derived from the potential function $V$, the work which they do is subject to the law of conservation of energy, and the work done on the whole fluid within a certain space may be found if we know the potential at the points where each line of flow enters the space and where it issues from it. The excess of the second of these potentials over the first, multiplied by the quantity of fluid which is transmitted along each line of flow, will give the work done by that portion of the fluid, and the sum of all such products will give the whole work.

Now, if the space be bounded by a surface for which $V=C$, a constant quantity, the potential will be the same at the place where any line of flow enters the space and where it issues from it, so that in this case no work will be done by the forces on the fluid within the space, and $M=0$.

Secondly, if the space be bounded in whole or in part by a surface satisfying equation (4), no fluid will enter or leave the space through this surface, so that no part of the value of $M$ can depend on this part of the surface.

The quantity $M$ is therefore zero for a space bounded externally by the closed surface $V=C$, and it remains zero though any part of this space be cut off from the rest by surfaces fulfilling the condition (4).

The analytical expression of the process by which we deduce the work done in the interior of the space from that which takes place at the bounding surface is contained in the following method of integration by parts.

Taking the first term of the integral $M$,

$\iiint u{\frac {dV}{dx}}\,dx\,dy\,dz=\iint \Sigma (uV)\,dy\,dz-\iiint V{\frac {du}{dx}}\,dx\,dy\,dz$,

where

$\Sigma (uV)=u_{1}V_{1}-u_{2}V_{2}+u_{3}V_{3}-u_{4}V_{4}\,\!$ + &c.;

and where $u_{1}V_{1}$, $u_{2}V_{2}$, &c. are the values of $u$ and $v$ at the points whose coordinates are $(x_{1},y,z)$, $(x_{2},y,z)$, &c., $x_{1}$, $x_{2}$, &c. being the values of $x$ where the ordinate cuts the bounding surface or surfaces, arranged in descending order of magnitude.

Adding the two other terms of the integral $M$, we find

$M=\iint \Sigma (uV)\,dy\,dz+\iint \Sigma (uV)\,dz\,dx+\iint \Sigma (uV)\,dx\,dy$

$-\iiint V({\frac {du}{dx}}+{\frac {dv}{dy}}+{\frac {dw}{dz}})\,dx\,dy\,dz$.

If $l,m,n$ are the direction-cosines of the normal drawn inwards from the bounding surface at any point, and $dS$ an element of that surface, then we may write

$M=-\iint V(lu+mv+nw)dS-\iiint V({\frac {du}{dx}}+{\frac {dv}{dy}}+{\frac {dw}{dz}})\,dx\,dy\,dz$;

the integration of the first term being extended over the bounding surface, and that of the second throughout the entire space.

For all spaces within which $u,v,w$ are continuous, the second term vanishes in virtue of equation (1). If for any surface within the space $u,v,w$ are discontinuous but subject to equation (2), we find for the part of $M$ depending on this surface,

$M_{1}=\iint V_{1}(l_{1}u_{1}+m_{1}v_{1}+n_{1}w_{1})dS_{1}$,

$M_{2}=\iint V_{2}(l_{2}u_{2}+m_{2}v_{2}+n_{2}w_{2})dS_{2}$;

where the suffixes $_{1}$ and $_{2}$, applied to any symbol, indicate to which of the two spaces separated by the surface the symbol belongs.

Now, since $V$ is continuous, we have at every point of the surface,

$V_{1}=V_{2}=V\,\!$;

we have also

$dS_{l}=dS_{2}=dS\,\!$;

but since the normals are drawn in opposite directions, we have

$l_{1}=-l_{2}=l,{\color {White}xxxx}m_{1}=m_{2}=m,{\color {White}xxxx}n_{1}=-n_{2}=n;$

so that the total value of M, so far as it depends on the surface of discontinuity, is

$M_{1}+M_{2}=-\iint V(l(u_{1}-u_{2})+m(v_{1}-v_{2})+n(w_{1}-W_{2}))dS$.

The quantity under the integral sign vanishes at every point in virtue of the superficial solenoidal condition or characteristic (2).

Hence, in determining the value of $M$, we have only to consider the surface-integral over the actual bounding surface of the space considered, or

$M=-\iint V(lu+mv+nw)dS$.

Case 1. If V is constant over the whole surface and equal to $C$,

$M=-C\iint (lu+mv+nw)dS$.

The part of this expression under the sign of double integration represents the surface-integral of the flux whose components are $u,v,w,$ and by Art. 21 this surface-integral is zero for the closed surface in virtue of the general and superficial solenoidal conditions (1) and (2).

Hence $M=0$ for a space bounded by a single equipotential surface.

If the space is bounded externally by the surface V = C, and internally by the surfaces $V=C_{1}$, $V=C_{2}$, &c., then the total value of $M$ for the space so bounded will be

$M-M_{1}-M_{2}\,\!$ &c.,

where $M$ is the value of the integral for the whole space within the surface $V=C$, and $M_{l},M_{2},$ &c. are the values of the integral for the spaces within the internal surfaces. But we have seen that $M,M_{1},M_{2}$, &c. are each of them zero, so that the integral is zero also for the periphractic region between the surfaces.

Case 2. If $lu+mv+nw$ is zero over any part of the bounding surface, that part of the surface can contribute nothing to the value of $M$, because the quantity under the integral sign is everywhere zero. Hence $M$ will remain zero if a surface fulfilling this con dition is substituted for any part of the bounding surface, provided that the remainder of the surface is all at the same potential.

98.] We are now prepared to prove a theorem which we owe to Sir William Thomson ^[1].

As we shall require this theorem in various parts of our subject, I shall put it in a form capable of the necessary modifications.

Let $a,b,c$ be any functions of $x,y,z$ (we may call them the components of a flux) subject only to the condition

${\frac {da}{dx}}+{\frac {db}{dy}}+{\frac {dc}{dz}}+4\pi \rho =0$

(5)

where $\rho$ has given values within a certain space. This is the general characteristic of $a,b,c$.

Let us also suppose that at certain surfaces (S) $a,b$, and $c$ are discontinuous, but satisfy the condition

$l(a_{1}-a_{2})+m(b_{1}-b_{2})+n(c_{1}-c_{2})+4\pi \rho =0\,\!$;

(6)

where $l,m,n$ are the direction-cosines of the normal to the surface, $a_{1},b_{1},c_{1}$ the values of $a,b,c$ on the positive side of the surface, and $a_{2},b_{2},c_{2}$ those on the negative side, and $\sigma$ a quantity given for every point of the surface. This condition is the superficial characteristic of $a,b,c$.

Next, let us suppose that $V$ is a continuous function of $x,y,z$, which either vanishes at infinity or whose value at a certain point is given, and let $V$ satisfy the general characteristic equation

${\frac {d}{dx}}K{\frac {dV}{dx}}+{\frac {d}{dy}}K{\frac {dV}{dy}}+{\frac {d}{dz}}K{\frac {dV}{dz}}+4\pi \rho =0$;

(7)

and the superficial characteristic at the surfaces $S$,

$l(K_{1}{\frac {dV_{1}}{dx}}-K_{2}{\frac {dV_{2}}{dx}})+m(K_{1}{\frac {dV_{1}}{dy}}-K_{2}{\frac {dV_{2}}{dy}})+n(K_{1}{\frac {dV_{1}}{dz}}-K_{2}{\frac {dV_{2}}{dz}})+4\pi \rho =0$

(8)

$K$ being a quantity which may be positive or zero but not negative, given at every point of space.

Finally, let $8\pi Q$ represent the triple integral

$8\pi Q\iiint {\frac {1}{K}}(a^{2}+b^{2}+c^{2})\,dx\,dy\,dz$,

(9)

extended over a space bounded by surfaces, for each of which either

$V$ = constant,

or

$la+mb+nc=Kl{\frac {dV}{dx}}+Km{\frac {dV}{dy}}+Kn{\frac {dV}{dz}}=q$,

(10)

where the value of $q$ is given at every point of the surface; then, if $a,b,c$ be supposed to vary in any manner, subject to the above conditions, the value of $Q$ will be a unique minimum, when

$a=K{\frac {dV}{dx}},{\color {White}xxxx}b=K{\frac {dV}{dy}},{\color {White}xxxx},c=K{\frac {dV}{dz}}$.

(11)

Proof.

If we put for the general values of $a,b,c,$

$a=K{\frac {dV}{dx}}+u,{\color {White}xxxx}b=K{\frac {dV}{dy}}+v,{\color {White}xxxx},c=K{\frac {dV}{dz}}+w$;

(12)

then, by substituting these values in equations (5) and (7), we find that $u,v,w$ satisfy the general solenoidal condition

(1) ${\frac {du}{dx}}+{\frac {dv}{dy}}+{\frac {dw}{dz}}=0$.

We also find, by equations (6) and (8), that at the surfaces of discontinuity the values of $u_{1},v_{1},w_{1}$ and $u_{2},v_{2},w_{2}$ satisfy the superficial solenoidal condition

(2) $l(u_{l}-u_{2})+m(v_{1}-v_{2})+n(w_{1}-w_{2})=0\,\!$

.

The quantities $u,v,w$, therefore, satisfy at every point the solenoidal conditions as stated in the preceding lemma.

We may now express $Q$ in terms of $u,v,w$ and $V$,

$Q=\iiint K(\left.{\frac {\overline {dV}}{dx}}\right|^{2}+\left.{\frac {\overline {dV}}{dy}}\right|^{2}+\left.{\frac {\overline {dV}}{dz}}\right|^{2})dx\,dy\,dz+\iiint {\frac {1}{K}}(u^{2}+v^{2}+w^{2})dx\,dy\,dz$

$+2\iiint (u{\frac {dV}{dx}}+v{\frac {dV}{dy}}+w{\frac {dV}{dz}})dx\,dy\,dz$. (13)

The last term of $Q$ may be written $2M$, where $M$ is the quantity considered in the lemma, and which we proved to be zero when the space is bounded by surfaces, each of which is either equipotential or satisfies the condition of equation (10), which may be written

(4) $lu+mv+nw=0\,\!$.

$Q$ is therefore reduced to the sum of the first and second terms.

In each of these terms the quantity under the sign of integration consists of the sum of three squares, and is therefore essentially positive or zero. Hence the result of integration can only be positive or zero.

Let us suppose the function $V$ known, and let us find what values of $u,v,w$ will make $Q$ a minimum.

If we assume that at every point $u=0$, $v=0$, and $w=0$, these values fulfil the solenoidal conditions, and the second term of $Q$ is zero, and $Q$ is then a minimum as regards the variation of $u,v,w$.

For if any of these quantities had at any point values differing from zero, the second term of $Q$ would have a positive value, and $Q$ would be greater than in the case which we have assumed.

But if $u=0$, $v=0$, and $w=0$, then

(11) ${\color {White}xxx}a=K{\frac {dV}{dx}},{\color {White}xxx}b=K{\frac {dV}{dy}},{\color {White}xxx}c=K{\frac {dV}{dz}}$.

Hence these values of $a,b,c$ make $Q$ a minimum.

But the values of $a,b,c$, as expressed in equations (12), are perfectly general, and include all values of these quantities con sistent with the conditions of the theorem. Hence, no other values of $a,b,c$ can make $Q$ a minimum.

Again, $Q$ is a quantity essentially positive, and therefore $Q$ is always capable of a minimum value by the variation of $a,b,c$. Hence the values of $a,b,c$ which make $Q$ a minimum must have a real existence. It does not follow that our mathematical methods are sufficiently powerful to determine them.

Corollary I. If $a,b,c$ and $K$ are given at every point of space, and if we write

(12) ${\color {White}xxx}a=K{\frac {dV}{dx}}+u,{\color {White}xxx}b=K{\frac {dV}{dy}}+v,{\color {White}xxx}c=K{\frac {dV}{dz}}+w$.

with the condition (1)

${\frac {du}{dx}}+{\frac {dv}{dy}}+{\frac {dw}{dz}}=0$,

then $V,u,v,w$ can be found without ambiguity from these four equations.

Corollary II. The general characteristic equation

${\frac {d}{dx}}K{\frac {dV}{dx}}+{\frac {d}{dy}}K{\frac {dV}{dy}}+{\frac {d}{dz}}K{\frac {dV}{dz}}+4\pi \rho =0$

where $V$ a finite quantity of single value whose first derivatives are finite and continuous except at the surface $S$, and at that surface fulfil the superficial characteristic

$l(K_{1}{\frac {dV_{1}}{dx}}-K_{2}{\frac {dV_{2}}{dx}})+m(K_{1}{\frac {dV_{1}}{dy}}-K_{2}{\frac {dV_{2}}{dy}})+n(K_{1}{\frac {dV_{1}}{dz}}-K_{2}{\frac {dV_{2}}{dz}})+4\pi \rho =0$

can be satisfied by one value of $V$, and by one only, in the following cases.

Case 1. When the equations apply to the space within any closed surface at every point of which $V=C$.

For we have proved that in this case $a,b,c$ have real and unique values which determine the first derivatives of $V$, and hence, if different values of $V$ exist, they can only differ by a constant. But at the surface $V$ is given equal to $C$, and therefore $V$ is determinate throughout the space.

As a particular case, let us suppose a space within which $\rho =0$ bounded by a closed surface at which $V=C$. The characteristic equations are satisfied by making $V=C$ for every point within the space, and therefore $V=C$ is the only solution of the equations.

Case 2. When the equations apply to the space within any closed surface at every point of which $V$ is given.

For if in this case the characteristic equations could be satisfied by two different values of $V$, say $V$ and $V'$, put $U=V-V'$, then subtracting the characteristic equation in $V'$ from that in $V$, we find a characteristic equation in $U$. At the closed surface $U=0$ because at the surface $V=V'$, and within the surface the density is zero because $\rho =\rho '$. Hence, by Case 1, $U=0$ throughout the enclosed space, and therefore $V=V'$ throughout this space. Case 3. When the equations apply to a space bounded by a closed surface consisting of two parts, in one of which $V$ is given at every point, and in the other

$Kl{\frac {dV}{dx}}+Km{\frac {dV}{dy}}+Kn{\frac {dV}{dz}}=q$,

where $q$ is given at every point.

For if there are two values of $V$,let $U'$ represent, as before, their difference, then we shall have the equation fulfilled within a closed surface consisting of two parts, in one of which $U'=0$, and in the other

$l{\frac {dU'}{dx}}+m{\frac {dU'}{dy}}+n{\frac {dU'}{dz}}=0$;

and since $U'=0$ satisfies the equation it is the only solution, and therefore there is but one value of $V$ possible.

Note.—The function $V$ in this theorem is restricted to one value at each point of space. If multiple values are admitted, then, if the space considered is a cyclic space, the equations may be satisfied by values of $V$ containing terms with multiple values. Examples of this will occur in Electromagnetism.

99.] To apply this theorem to determine the distribution of electricity in an electrified system, we must make $K=1$ throughout the space occupied by air, and $K=\infty$ throughout the space occupied by conductors. If any part of the space is occupied by dielectrics whose inductive capacity differs from that of air, we must make K in that part of the space equal to the specific inductive capacity.

The value of $V$, determined so as to fulfil these conditions, will be the only possible value of the potential in the given system.

Green's Theorem shews that the quantity $Q$, when it has its minimum value corresponding to a given distribution of electricity, represents the potential energy of that distribution of electricity. See Art. 100, equation (11).

In the form in which $Q$ is expressed as the result of integration over every part of the field, it indicates that the energy due to the electrification of the bodies in the field may be considered as the result of the summation of a certain quantity which exists in every part of the field where electrical force is in action, whether elec trification be present or not in that part of the field.

The mathematical method, therefore, in which $Q$, the symbol of electrical energy, is made an object of study, instead of $\rho$, the symbol of electricity itself, corresponds to the method of physical speculation, in which we look for the seat of electrical action in every part of the field, instead of confining our attention to the electrified bodies.

The fact that $Q$ attains a minimum value when the components of the electric force are expressed in terms of the first derivatives of a potential, shews that, if it were possible for the electric force to be distributed in any other manner, a mechanical force would be brought into play tending to bring the distribution of force into its actual state. The actual state of the electric field is therefore a state of stable equilibrium, considered with reference to all variations of that state consistent with the actual distribution of free electricity.

Green's Theorem.

100.] The following remarkable theorem was given by George Green in his essay ‘On the Application of Mathematics to Electricity and Magnetism.’

I have made use of the coefficient $K$, introduced by Thomson, to give greater generality to the statement, and we shall find as we proceed that the theorem may be modified so as to apply to the most general constitution of crystallized media.

We shall suppose that $U$ and $V$ are two functions of $x,y,z$, which, with their first derivatives, are finite and continuous within the space bounded by the closed surface $S$.

We shall also put for conciseness

${\frac {d}{dx}}K{\frac {dU}{dx}}+{\frac {d}{dy}}K{\frac {dU}{dy}}+{\frac {d}{dz}}K{\frac {dU}{dz}}=-4\pi \rho$,

(1)

and

${\frac {d}{dx}}K{\frac {dV}{dx}}+{\frac {d}{dy}}K{\frac {dV}{dy}}+{\frac {d}{dz}}K{\frac {dV}{dz}}=-4\pi \rho '$,

(2)

where $K$ is a real quantity, given for each point of space, which may be positive or zero but not negative. The quantities $\rho$ and $\rho '$ correspond to volume-densities in the theory of potentials, but in this investigation they are to be considered simply as abbreviations for the functions of $U$ and $V$ to which they are here equated.

In the same way we may put

$lK{\frac {dU}{dx}}+mK{\frac {dU}{dy}}+nK{\frac {dU}{dz}}=4\pi \sigma$,

(3)

and

$lK{\frac {dV}{dx}}+mK{\frac {dV}{dy}}+nK{\frac {dV}{dz}}=4\pi \sigma '$,

(4)

where $l,m,n$ are the direction-cosines of the normal drawn inwards from the surface $S$. The quantities $\sigma$ and $\sigma '$ correspond to superficial densities, but at present we must consider them as defined by the above equations.

Green's Theorem is obtained by integrating by parts the expression

$4\pi M=\iiint K({\frac {dU}{dx}}{\frac {dV}{dx}}+{\frac {dU}{dy}}{\frac {dV}{dy}}+{\frac {dU}{dz}}{\frac {dV}{dz}})dx\,dy\,dz$

(5)

throughout the space within the surface $S$.

If we consider ${\tfrac {dV}{dx}}$ as a component of a force whose potential is $V$, and $K$ ${\tfrac {dU}{dx}}$ as a component of a flux, the expression will give the work done by the force on the flux.

If we apply the method of integration by parts, we find

$4\pi M=\iint VK(l{\frac {dU}{dx}}+m{\frac {dU}{dy}}+n{\frac {dU}{dz}})dS$

$-\iiint V({\frac {d}{dx}}K{\frac {dU}{dx}}+{\frac {d}{dy}}K{\frac {dU}{dy}}+{\frac {d}{dz}}K{\frac {dU}{dz}})dx\,dy\,dz$;

(6)

or

$4\pi M=\iint 4\pi \sigma 'VdS+\iiint 4\pi \rho 'V\,dx\,dy\,dz$.

(7)

In precisely the same manner by exchanging $V$ and $T$, we should find

$4\pi M=+\iint 4\pi \sigma \,U\,dS+\iiint 4\pi \rho \,U\,dx\,dy\,dz.$

(8)

The statement of Green's Theorem is that these three expressions for $M$ are identical, or that

$M=\iint \sigma '\,V\,dS+\iiint \rho '\,V\,dx\,dy\,dz=\iint \sigma \,U\,dS+\iiint \rho \,U\,dx\,dy\,dz$

$={\frac {1}{4\pi }}\iiint K({\frac {dU}{dx}}{\frac {dV}{dx}}+{\frac {dU}{dy}}{\frac {dV}{dy}}+{\frac {dU}{dz}}{\frac {dV}{dz}})dx\,dy\,dz.$

(9)

Correction of Green's Theorem for Cyclosis.

There are cases in which the resultant force at any point of a certain region fulfils the ordinary condition of having a potential, while the potential itself is a many-valued function of the coordinates. For instance, if

$X={\frac {y}{x^{2}+y^{2}}},Y=-{\frac {x}{x^{2}+y^{2}}},Z=0$

we find $V=\tan ^{-1}{\tfrac {y}{x}}$, a many-valued function of $x$ and $y$, the values of $V$ forming an arithmetical series whose common difference is $2\pi$, and in order to define which of these is to be taken in any particular case we must make some restriction as to the line along which we are to integrate the force from the point where $V=0$ to the required point.

In this case the region in which the condition of having a potential is fulfilled is the cyclic region surrounding the axis of z, this axis being a line in which the forces are infinite and therefore not itself included in the region.

The part of the infinite plane of $xz$ for which $x$ is positive may be taken as a diaphragm of this cyclic region. If we begin at a point close to the positive side of this diaphragm, and integrate along a line which is restricted from passing through the diaphragm, the line-integral will be restricted to that value of $V$ which is positive but less than $2\pi$.

Let us now suppose that the region bounded by the closed surface $S$ in Green s Theorem is a cyclic region of any number of cycles, and that the function $V$ is a many-valued function having any number of cyclic constants.

The quantities ${\frac {dV}{dx}}$, ${\frac {dV}{dy}}$, and ${\frac {dV}{dz}}$ will have definite values at all points within $S$, so that the volume-integral

$\iiint K\left({\frac {dU}{dx}}{\frac {dV}{dx}}+{\frac {dU}{dy}}{\frac {dV}{dy}}+{\frac {dU}{dz}}{\frac {dV}{dz}}\right)$

has a definite value, $\sigma$ and $p$ have also definite values, so that if $U$ is a single valued function, the expression

$\iint \sigma UdS+\iiint \rho Udxdydz$

has also a definite value.

The expression involving $V$ has no definite value as it stands, for $V$ is a many-valued function, and any expression containing it is many-valued unless some rule be given whereby we are directed to select one of the many values of V at each point of the region.

To make the value of $V$ definite in a region of $n$ cycles, we must conceive $n$ diaphragms or surfaces, each of which completely shuts one of the channels of communication between the parts of the cyclic region. Each of these diaphragms reduces the number of cycles by unity, and when n of them are drawn the region is still a connected region but acyclic, so that we can pass from any one point to any other without cutting a surface, but only by reconcileable paths.

Let $S_{1}$ be the first of these diaphragms, and let the line-integral of the force for a line drawn in the acyclic space from a point on the positive side of this surface to the contiguous point on the negative side be $\kappa _{1}$, then $\kappa _{1}$ is the first cyclic constant.

Let the other diaphragms, and their corresponding cyclic constants, be distinguished by suffixes from 1 to n, then, since the region is rendered acyclic by these diaphragms, we may apply to it the theorem in its original form.

We thus obtain for the complete expression for the first member of the equation

$\iiint \rho 'Vdx\,dy\,dz+\iint \sigma 'V\,dS+\iint \sigma _{1}'\kappa _{1}\,dS_{1}+\iint \sigma _{2}'\kappa _{2}\,dS_{2}+\mathrm {etc} .+\iint \sigma _{n}'\kappa _{n}\,dS_{n}$.

The addition of these terms to the expression of Green's Theorem, in the case of many-valued functions, was first shewn to be necessary by Helmholtz^[2], and was first applied to the theorem by Thomson^[3].

Physical Interpretation of Green’s Theorem.

The expressions $\sigma \,dS$ and $\rho \,dx\,dy\,dz$ denote the quantities of electricity existing on an element of the surface S and in an element of volume respectively. We may therefore write for either of these quantities the symbol e, denoting a quantity of electricity. We shall then express Green's Theorem as follows—

$M=\sum (Ve')=\sum (V'e)$;

where we have two systems of electrified bodies, e standing in succession for e₁, e₂, &c., any portions of the electrification of the first system, and V denoting the potential at any point due to all these portions, while e' stands in succession for e₁', e₂' , &c., portions of the second system, and V' denotes the potential at any point due to the second system.

Hence Ve' denotes the product of a quantity of electricity at a point belonging to the second system into the potential at that point due to the first system, and $\sum (Ve')$ denotes the sum of all such quantities, or in other words, $\sum (Ve')$ represents that part of the energy of the whole electrified system which is due to the action of the second system on the first.

In the same way $\sum (V'e)$ represents that part of the energy of the whole system which is due to the action of the first system on the second.

If we define V as $\sum \left({\tfrac {e}{r}}\right)$, where r is the distance of the quantity e of electricity from the given point, then the equality between these two values of M may be obtained as follows, without Green Theorem—

$\sum {'}(Ve')=\sum {'}\left(\sum \left({\frac {e}{r}}\right)e'\right)=\sum \sum \left({\frac {ee'}{r}}\right)=\sum \left(\sum {'}\left({\frac {e'}{r}}\right)e\right)=\sum (V'e)$.

This mode of regarding the question belongs to what we have called the direct method, in which we begin by considering certain portions of electricity, placed at certain points of space, and acting on one another in a way depending on the distances between these points, no account being taken of any intervening medium, or of any action supposed to take place in the intervening space.

Green's Theorem, on the other hand, belongs essentially to what we have called the inverse method. The potential is not supposed to arise from the electrification by a process of summation, but the electrification is supposed to be deduced from a perfectly arbitrary function called the potential by a process of differentiation.

In the direct method, the equation is a simple extension of the law that when any force acts directly between two bodies, action and reaction are equal and opposite.

In the inverse method the two quantities are not proved directly to be equal, but each is proved equal to a third quantity, a triple integral which we must endeavour to interpret.

If we write R for the resultant electromotive force due to the potential V, and l, m, n for the direction-cosines of R, then, by Art. 71,

$-{\frac {dV}{dx}}=Rl,\quad -{\frac {dV}{dy}}=Rm,\quad -{\frac {dV}{dz}}=Rn.$

If we also write R for the force due to the second system, and l, m, n for its direction-cosines,

$-{\frac {dV'}{dx}}=R'l',\quad -{\frac {dV'}{dy}}=R'm',\quad -{\frac {dV'}{dz}}=R'n';$

and the quantity M may be written

$M={\frac {1}{4\pi }}\iiint {(KRR'\cos \epsilon )\,dx\,dy\,dz},$

(10)

where

$\cos \epsilon =ll'+mm'+nn'\,$,

$\epsilon$ being the angle between the directions of R and $R'$.

Now if K is what we have called the coefficient of electric inductive capacity, then KR will be the electric displacement due to the electromotive force R, and the product $KRR'\cos \epsilon$ will represent the work done by the force $R'$ on account of the displacement caused by the force $R'$, or in other words, the amount of intrinsic energy in that part of the field due to the mutual action of $R$ and $R'$.

We therefore conclude that the physical interpretation of Green's theorem is as follows :

If the energy which is known to exist in an electrified system is due to actions which take place in all parts of the field, and not to direct action at a distance between the electrified bodies, then that part of the intrinsic energy of any part of the field upon which the mutual action of two electrified systems depends is $KRR'\cos \epsilon$ per unit of volume.

The energy of an electrified system due to its action on itself is, by Art. 85,

${\frac {1}{2}}\sum (eV)$,

which is by Green's theorem, putting U = V,

$Q={\frac {1}{8\pi }}\iiint {K\left({\frac {\overline {dV}}{dx}}{\Big |}^{2}+{\frac {\overline {dV}}{dy}}{\Big |}^{2}+{\frac {\overline {dV}}{dz}}{\Big |}^{2}\right)dx\,dy\,dz};$

(11)

and this is the unique minimum value of the integral considered in Thomson's theorem.

Green's Function.

101.] Let a closed surface S be maintained at potential zero. Let P and Q be two points on the positive side of the surface S (we may suppose either the inside or the outside positive), and let a small body charged with unit of electricity be placed at P; the potential at the point Q will consist of two parts, of which one is due to the direct action of the electricity on P, while the other is due to the action of the electricity induced on S by P. The latter part of the potential is called Green's Function, and is denoted by G_pq.

This quantity is a function of the positions of the two points P and Q, the form of which depends on that of the surface S. It has been determined in the case in which S is a sphere, and in a very few other cases. It denotes the potential at Q due to the electricity induced on S by unit of electricity at P.

The actual potential at any point $Q$ due to the electricity at $P$ and on $S$ is

${\frac {1}{r_{pq}}}+G_{pq},$

where $r_{pq}$ denotes the distance between $P$ and $Q$.

At the surface $S$ and at all points on the negative side of $S$, the potential is zero, therefore

$G_{pa}=-{\frac {1}{r_{pa}}},$

(1)

where the suffix $_{a}$ indicates that a point $A$ on the surface $S$ is taken instead of $Q$.

Let $\sigma _{pa'}$ denote the surface-density induced by $P$ at a point $A'$ of the surface $S$, then, since $G_{pq}$ is the potential at $Q$ due to the superficial distribution,

$G_{pq}=\iint {\frac {\sigma _{pa'}}{r_{qa'}}}dS',$

(2)

where $dS'$ is an element of the surface $S$ at $A'$, and the integration is to be extended over the whole surface $S$.

But if unit of electricity had been placed at $Q$, we should have had by equation (1),

${\frac {1}{r_{qa'}}}=-G_{qa'}$

(3)

$=-\iint {\frac {\sigma _{qa}}{r_{aa'}}}dS;$

(4)

where $\sigma _{qa}$ is the density induced by $Q$ on an element $dS$ at $A$, and $r_{aa'}$ is the distance between $A$ and $A'$. Substituting this value of ${\frac {1}{r_{qa'}}}$ in the expression for $G_{pq}$, we find

$G_{pq}=-\iiiint {\frac {\sigma _{qa}\sigma _{pa'}}{r_{aa'}}}dS\ dS'.$

(5)

Since this expression is not altered by changing $_{p}$ into $_{q}$ and $_{q}$ into $_{p}$,we find that

$G_{pq}=G_{qp};$

(6)

a result which we have already shewn to be necessary in Art. 88, but which we now see to be deducible from the mathematical process by which Green’s function may be calculated.

If we assume any distribution of electricity whatever, and place in the field a point charged with unit of electricity, and if the surface of potential zero completely separates the point from the assumed distribution, then if we take this surface for the surface $S$, and the point for $P$, Green’s function, for any point on the same side of the surface as $P$, will be the potential of the assumed distribution on the other side of the surface. In this way we may construct any number of cases in which Green’s function can be found for a particular position of $P$. To find the form of the function when the form of the surface is given and the position of $P$ is arbitrary, is a problem of far greater difficulty, though, as we have proved, it is mathematically possible.

Let us suppose the problem solved, and that the point $P$ is taken within the surface. Then for all external points the potential of the superficial distribution is equal and opposite to that of $P$. The superficial distribution is therefore centrobaric^[4], and its action on all external points is the same as that of a unit of negative electricity placed at $P$.

Method of Approximating to the Values of Coefficients of Capacity, &c.

102.] Let a region be completely bounded by a number of surfaces $S_{0},S_{1},S_{2}$, &c., and let $K$ be a quantity, positive or zero but not negative, given at every point of this region. Let $V$ be a function subject to the conditions that its values at the surfaces $S_{1},S_{2}$, &c. are the constant quantities $C_{1},C_{2}$, &c., and that at the surface $S_{0}$

${\frac {dV}{d\nu }}=0,$

(1)

where $\nu$ is a normal to the surface $S_{0}$. Then the integral

$Q={\frac {1}{8\pi }}\iiint K\left(\left({\frac {dV}{dx}}\right)^{2}+\left({\frac {dV}{dy}}\right)^{2}+\left({\frac {dV}{dz}}\right)^{2}\right)dx\ dy\ dz,$

(2)

taken over the whole region, has a unique minimum when $V$ satisfies the equation

${\frac {d}{dx}}K{\frac {dV}{dx}}+{\frac {d}{dy}}K{\frac {dV}{dy}}+{\frac {d}{dz}}K{\frac {dV}{dz}}=0$

(3)

throughout the region, as well as the original conditions.

We have already shewn that a function $V$ exists which fulfils the conditions (1) and (3), and that it is determinate in value. We have next to shew that of all functions fulfilling the surface-conditions it makes $Q$ a minimum.

Let $V_{0}$ be the function which satisfies (1) and (3), and let

$V=V_{0}+U\,$

(4)

be a function which satisfies (1).

It follows from this that at the surfaces $S_{1},S_{2}$, &c. $U=0$.

The value of $Q$ becomes

$Q={\frac {1}{8\pi }}\iiint \left\{K\left(\left({\frac {dV_{0}}{dx}}\right)^{2}+\mathrm {\&c.} \right)+K\left(\left({\frac {dU}{dx}}\right)^{2}+\mathrm {\&c.} \right)+2K\left({\frac {dV_{0}}{dx}}{\frac {dU}{dx}}+\mathrm {\&c.} \right)\right\}dx\ dy\ dz.$

(5)

Let us confine our attention to the last of these three groups of terms, merely observing that the other groups are essentially positive. By Green’s theorem

${\begin{array}{c}\iiint K\left({\frac {dV_{0}}{dx}}{\frac {dU}{dx}}+{\frac {dV_{0}}{dy}}{\frac {dU}{dy}}+{\frac {dV_{0}}{dz}}{\frac {dU}{dz}}\right)dx\ dy\ dz=\iint KU{\frac {dV_{0}}{d\nu }}dS\\\\-\iiint U\left({\frac {d}{dx}}K{\frac {dV_{0}}{dx}}+{\frac {d}{dy}}K{\frac {dV_{0}}{dy}}+{\frac {d}{dz}}K{\frac {dV_{0}}{dz}}\right)dx\ dy\ dz;\end{array}}$

(6)

the first integral of the second member being extended over the surface of the region and the second throughout the enclosed space. But on the surfaces $S_{1},S_{2}$ &c. $U=0$, so that these contribute nothing to the surface-integral.

Again, on the surface $S_{0}$, ${\tfrac {dV_{0}}{d\nu }}=0$, so that this surface contributes nothing to the integral. Hence the surface-integral is zero.

The quantity within brackets in the volume-integral also disappears by equation (3), so that the volume-integral is also zero. Hence $Q$ is reduced to

$Q={\frac {1}{8\pi }}\iiint K\left({\overline {\frac {dV_{0}}{dx}}}{\Bigg \vert }^{2}+\mathrm {\&c.} \right)dx\ dy\ dz+{\frac {1}{8\pi }}\iiint K\left({\overline {\frac {dU}{dx}}}{\Bigg \vert }^{2}+\mathrm {\&c.} \right)dx\ dy\ dz.$

(7)

Both these quantities are essentially positive, and therefore the minimum value of $Q$ is when

${\frac {dU}{dx}}={\frac {dU}{dy}}={\frac {dU}{dz}}=0$

(8)

or when $U$ is a constant. But at the surfaces $S$, &c. $U=0$. Hence $U=0$ everywhere, and $V_{0}$ gives the unique minimum value of $Q$.

Calculation of a Superior Limit of the Coefficients of Capacity.

The quantity $Q$ in its minimum form can be expressed by means of Green’s theorem in terms of $V_{1},V_{2}$, &c., the potentials of $S_{1},S_{2}$, and $E_{1},E_{2}$, &c., the charges of these surfaces,

$Q={\frac {1}{2}}\left(V_{1}E_{1}+V_{2}E_{2}+\mathrm {etc.} \right);$

(9)

or, making use of the coefficients of capacity and induction as defined in Article 87,

$Q={\frac {1}{2}}\left(V_{1}^{2}q_{11}+V_{2}^{2}q_{22}+\mathrm {etc.} \right)+V_{1}V_{2}q_{12}+\mathrm {etc.} ;$

(10)

The accurate determination of the coefficients $q$ is in general difficult, involving the solution of the general equation of statical electricity, but we make use of the theorem we have proved to determine a superior limit to the value of any of these coefficients.

To determine a superior limit to the coefficient of capacity $q_{11}$, make $V_{1}=1$, and $V_{2},V_{3}$, &c. each equal to zero, and then take any function $V$ which shall have the value 1 at $S_{1}$, and the value 0 at the other surfaces.

From this trial value of $V$ calculate $Q$ by direct integration, and let the value thus found be $Q'$. We know that $Q'$ is not less than the absolute minimum value $Q$, which in this case is ${\tfrac {1}{2}}q_{11}$.

Hence

$q_{11}\ \mathrm {is\ not\ greater\ than} \ 2Q'$

(11)

If we happen to have chosen the right value of the function $V$, then $q_{11}=2Q'$, but if the function we have chosen differs slightly from the true form, then, since $Q$ is a minimum, $Q'$ will still be a close approximation to the true value.

Superior Limit of the Coefficients of Potential.

We may also determine a superior limit to the coefficients of potential defined in Article 86 by means of the minimum value of the quantity $Q$ in Article 98, expressed in terms of $a,b,c$.

By Thomson’s theorem, if within a certain region bounded by the surfaces $S_{0},S_{1}$ &c. the quantities $a,b,c$ are subject to the condition

${\frac {da}{dx}}+{\frac {db}{dy}}+{\frac {dc}{dz}}=0;$

(12)

and if

$la+mb+nc=q\,$

(13)

be given all over the surface, where $l,m,n$ are the direction-cosines of the normal, then the integral

$Q={\frac {1}{8\pi }}\iiint {\frac {1}{K}}\left(a^{2}+b^{2}+c^{2}\right)dx\ dy\ dz$

(14)

is an absolute and unique minimum when

$a=K{\frac {dV}{dx}},\ b=K{\frac {dV}{dy}},\ c=K{\frac {dV}{dz}}.$

(15)

When the minimum is attained $Q$ is evidently the same quantity which we had before.

If therefore we can find any form for $a,b,c$ which satisfies the condition (12) and at the same time makes

$\iint q\ dS_{1}=E_{1},\ \iint q\ dS_{2}=E_{2}$ &c.;

(16)

and if $Q''$ be the value of $Q$ calculated by (14) from these values of $a,b,c$, then $Q''$ is not less than

${\frac {1}{2}}\left(E_{1}^{2}p_{11}+E_{2}^{2}p_{22}\right)+E_{1}E_{2}p_{12}.$

(17)

If we take the case in which one of the surfaces, say $S_{2}$, surrounds the rest at an infinite distance, we have the ordinary case of conductors in an infinite region; and if we make $E_{2}=-E_{1}$, and $E=0$ for all the other surfaces, we have $V_{2}=0$ at infinity, and $p_{11}$ is not greater than ${\tfrac {2Q''}{E_{1}}}$.

In the very important case in which the electrical action is entirely between two conducting surfaces $S_{1}$ and $S_{2}$, of which $S_{2}$ completely surrounds $S_{1}$ and is kept at potential zero, we have $E_{1}=-E_{2}$ and $q_{11}p_{11}=1$.

Hence in this case we have

$q_{11}\ \mathrm {not\ less\ than} \ {\frac {E_{1}}{2Q''}};$

(18)

and we had before

$q_{11}\ \mathrm {not\ greater\ than} \ 2Q';$

(19)

so that we conclude that the true value of $q_{11}$, the capacity of the internal conductor, lies between these values.

This method of finding superior and inferior limits to the values of these coefficients was suggested by a memoir 'On the Theory of Resonance,' by the Hon. J. W. Strutt, Phil. Trans., 1871. See Art. 308.

---

1. Cambridge and Dublin Mathematical Journal, February, 1848.
2. ‚Ueber Integrate der Hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen‘, Crelle, 1858. Translated by Tait in Phil. Mag., 1867.
3. ‚On Vortex Motion‘, Trans. R. S. Edin., xxv. part i. p. 241 (1868).
4. Thomson and Tait’s Natural Philosophy, § 526.