A Treatise on Electricity and Magnetism/Volume 1/Part 1/Chapter 3

by James Clerk Maxwell

CHAPTER III.

SYSTEMS OF CONDUCTORS.

On the Superposition of Electrical Systems.

84.] Let $E_{l}$ be a given electrified system of which the potential at a point $P$ is $V_{1}$, and let $E_{2}$ be another electrified system of which the potential at the same point would be $V_{2}$ if $E_{l}$ did not exist. Then, if $E_{1}$ and $E_{2}$ exist together, the potential of the combined system will be $V_{1}+V_{2}$.

Hence, if $V$ be the potential of an electrified system $E$, if the electrification of every part of $E$ be increased in the ratio of $n$ to 1 , the potential of the new system $nE$ will be $nV$.

Energy of an Electrified System.

85.] Let the system be divided into parts, $A_{1}$, $A_{2}$ , &c. so small that the potential in each part may be considered constant through out its extent. Let $e_{l}$ ,$e_{2}$ , &c. be the quantities of electricity in each of these parts, and let $V_{1}$, $V_{2}$ &c. be their potentials.

If now $e_{1}$ is altered to $ne_{1}$, $e_{2}$ to $ne_{2}$, &c., then the potentials will become $nV_{1}$, $nV_{2}$, &c.

Let us consider the effect of changing $n$ into $n+dn$ in all these expressions. It will be equivalent to charging $A_{1}$ with a quantity of electricity $e_{l}dn$, $A_{2}$ with $e_{2}dn$, &c. These charges must be supposed to be brought from a distance at which the electrical action of the system is insensible. The work done in bringing $e_{1}dn$ of electricity to $A_{1}$, whose potential before the charge is $nV_{1}$, and after the charge $(n+dn)V_{1}$, lf must lie between

$nV_{1}e_{1}\,dn\,\!$ and $(n+dn)V_{1}e_{1}\,dn\,\!$.

In the limit we may neglect the square of $dn$, and write the expression

$V_{1}e_{1}n\,dn\,\!$

Similarly the work required to increase the charge of $A_{2}$ is $V_{2}e_{2}ndn$, so that the whole work done in increasing the charge of the system is

$(v_{1}e_{1}+V_{2}e_{2}+etc.)\,n\,dn\,\!$.

If we suppose this process repeated an indefinitely great number of times, each charge being indefinitely small, till the total effect becomes sensible, the work done will be

$\sum (Ve)\int n\,dn={1 \over 2}\sum (Ve)(n_{1}^{2}-n_{0}^{2})$;

where $\sum (Ve)$ means the sum of all the products of the potential of each element into the quantity of electricity in that element when $n=1$, and $n_{0}$ is the initial and $n_{1}$ the final value of $n$.

If we make $n_{0}=0$ and $n_{1}=1$, we find for the work required to charge an unelectrified system so that the electricity is $e$ and the potential $V$ in each element,

$Q={1 \over 2}\sum (Ve)$.

General Theory of a System of Conductors.

86.] Let $A_{1},A_{2},\ldots A_{N}$ be any number of conductors of any form. Let the charge or total quantity of electricity on each of these be $E_{1},E_{2},\ldots E_{N}$ and let their potentials be $V_{1},V_{2},\ldots V_{N}$ respectively.

Let us suppose the conductors to be all insulated and originally free of charge, and at potential zero.

Now let $A_{1}$ be charged with unit of electricity, the other bodies being without charge. The effect of this charge on $A_{1}$ will be to raise the potential of $A_{1}$ to $p_{11}$, that of $A_{2}$ to $p_{12}$, and that of $A-n$ to $p_{1n}$ , where $p_{11}$, &c. are quantities depending on the form and relative position of the conductors. The quantity $p_{11}$ may be called the Potential Coefficient of $A_{l}$ on itself, and $p_{12}$ may be called the Potential Coefficient of $A_{1}$ on $A_{2}$ , and so on.

If the charge upon $A_{1}$ is now made $E_{l}$ , then, by the principle of superposition, we shall have

$V_{1}=p_{11}E_{1}\ldots \ldots v_{n}=p_{1n}E_{1}\,\!$.

Now let $A_{1}$ be discharged, and $A_{2}$ charged with unit of electricity, and let the potentials of $A_{1},A2,$ ... $A_{n}$ be $p_{21}$,$p_{22}$,...$p_{2n}$ potentials due to $E_{2}$ on $A_{2}$ will be

$V_{1}=p_{21}E_{2}\ldots \ldots V_{n}=p_{2n}E_{2}$.

Similarly let us denote the potential of $A_{s}$ due to a unit charge on $A_{r}$ by $p_{rs}$ , and let us call $p_{rs}$ the Potential Coefficient of $A_{r}$ on $A_{s}$, then we shall have the following equations determining the potentials in terms of the charges:

${\begin{matrix}V_{1}=p_{11}E_{1}\ldots +p_{r1}E_{r}\ldots +p_{n1}E_{n},\\-\qquad -\qquad -\qquad -\qquad -\qquad -\\V_{s}=p_{1s}E_{1}\ldots +p_{rs}E_{r}\ldots +p_{ns}E_{n},\\-\qquad -\qquad -\qquad -\qquad -\qquad -\\V_{n}=p_{1n}E_{1}\ldots +p_{rn}E_{r}\ldots +p_{nn}E_{n},\end{matrix}}$

(1)

We have here $n$ linear equations containing $n^{2}$ coefficients of potential.

87.] By solving these equations for $E_{1}$, $E_{2}$ , &c. we should obtain $n$ equations of the form

${\begin{matrix}E_{1}=q_{11}V_{1}\ldots +q_{1s}V_{s}\ldots +q_{1n}V_{n},\\-\qquad -\qquad -\qquad -\qquad -\qquad -\\E_{r}=q_{r1}V_{1}\ldots +q_{rs}V_{s}\ldots +q_{rn}V_{n},\\-\qquad -\qquad -\qquad -\qquad -\qquad -\\E_{n}=q_{n1}V_{1}\ldots +q_{ns}V_{s}\ldots +q_{nn}V_{n},\end{matrix}}$

(2)

The coefficients in these equations may be obtained directly from those in the former equations. They may be called Coefficients of Induction.

Of these $q_{11}$ is numerically equal to the quantity of electricity on $A_{l}$ when $A_{l}$ is at potential unity and all the other bodies are at potential zero. This is called the Capacity of $A_{1}$. It depends on the form and position of all the conductors in the system.

Of the rest $q_{rs}$ is the charge induced on $A_{r}$ when $A_{s}$ is maintained at potential unity and all the other conductors at potential zero. This is called the Coefficient of Induction of $A_{s}$ on $A_{r}$.

The mathematical determination of the coefficients of potential and of capacity from the known forms and positions of the conductors is in general difficult. We shall afterwards prove that they have always determinate values, and we shall determine their values in certain special cases. For the present, however, we may suppose them to be determined by actual experiment.

Dimensions of these Coefficients.

Since the potential of an electrified point at a distance $r$ is the charge of electricity divided by the distance, the ratio of a quantity of electricity to a potential may be represented by a line. Hence all the coefficients of capacity and induction $(q)$ are of the nature of lines, and the coefficients of potential $(p)$ are of the nature of the reciprocals of lines.

88.] Theorem I. The coefficients of $A_{r}$ relative to $A_{8}$ are equal to those of $A_{8}$ relative to $A_{r}$ .

If $E_{r}$, the charge on $A_{r}$, is increased by $\delta E_{r}$ , the work spent in bringing $\delta E_{r}$ from an infinite distance to the conductor $A_{r}$ whose potential is $V_{r}$, is by the definition of potential in Art. 70,

$V_{r}\delta E_{r}\,\!$,

and this expresses the increment of the electric energy caused by this increment of charge.

If the charges of the different conductors are increased by $\delta E_{1}$, &c., the increment of the electric energy of the system will be

$\delta Q=V_{1}\delta E_{1}+etc.+V_{r}\delta E_{r}+etc.\,\!$.

If, therefore, the electric energy $Q$ is expressed as a function of the charges $E_{1}$, $E_{2}$, &c., the potential of any conductor may be expressed as the partial differential coefficient of this function with respect to the charge on that conductor, or

$V_{r}=({\frac {dQ}{dE_{r}}})\ldots \ldots V_{s}=({\frac {dQ}{dE_{s}}})$.

Since the potentials are linear functions of the charges, the energy must be a quadratic function of the charges. If we put

$CE_{r}E_{s}\,\!$

for the term in the expansion of $Q$ which involves the product $E_{r}E_{s}$, then, by differentiating with respect to $E_{s}$, we find the term of the expansion of $V_{s}$ which involves $E_{r}$ to be $CE_{r}$.

Differentiating with respect to $E_{r}$, we find the term in the expansion of $V_{r}$ which involves $E_{s}$ to be $CE_{s}$.

Comparing these results with equations (1), Art. 86, we find

$p_{rs}\;=\;C\;=p_{sr}\,\!$,

or, interpreting the symbols $p_{rs}$ and $p_{sr}$ :—

The potential of $A_{s}$ due to a unit charge on $A_{r}$ is equal to the potential of $A_{r}$ due to a unit charge on $A_{s}$.

This reciprocal property of the electrical action of one conductor on another was established by Helmholtz and Sir W. Thomson.

If we suppose the conductors $A_{r}$ and $A_{s}$ to be indefinitely small, we have the following reciprocal property of any two points :

The potential at any point $A_{s}$, due to unit of electricity placed at $A_{r}$ in presence of any system of conductors, is a function of the positions of $A_{r}$ and $A_{s}$ in which the coordinates of $A_{r}$ and of $A_{s}$ enter in the same manner, so that the value of the function is unchanged if we exchange $A_{r}$ and $A_{s}$. This function is known by the name of Green's Function.

The coefficients of induction $q_{rs}$ and $q_{sr}$ are also equal. This is easily seen from the process by which these coefficients are obtained from the coefficients of potential. For, in the expression for $q_{rs}$, $p_{rs}$ and $p_{sr}$ enter in the same way as $p_{sr}$ and $p_{rs}$ do in the expression for $q_{sr}$ . Hence if all pairs of coefficients $p_{rs}$ and $p_{sr}$ are equal, the pairs $q_{rs}$ and $q_{sr}$ are also equal.

89.] Theorem II. Let a charge $E_{r}$ be placed on $A_{r}$, and let all the other conductors he at potential zero, and let the charge induced on $A_{s}$ be $-n_{rs}E_{r}$, then if $A_{r}$ is discharged and insulated, and $A_{s}$ brought to potential $V_{s}$, the other conductors being at potential zero, then the potential of $A_{r}$ will be $+n_{rs}V_{s}$.

For, in the first case, if $V_{r}$ is the potential of $A_{r}$, we find by equations (2),

$E_{s}\,=\,q_{rs}\,V_{r}{\color {White}xxxx}$, and ${\color {White}xxx}E_{r}=q_{rr}V_{r}$.

Hence ${\color {White}xxxx}E_{s}={\frac {q_{rs}}{q_{rr}}}E_{r}{\color {White}xxxx}$, and ${\color {White}xxxx}n_{rs}=-{\frac {q_{rs}}{q_{rr}}}$

In the second case, we have

$E_{r}=0=q_{rr}V_{r}+q_{rs}V_{s}\,\!$.

Hence ${\color {White}xxxxx}V_{r}=-{\frac {q_{rs}}{q_{rr}}}V_{s}=n_{rs}V_{s}$.

From this follows the important theorem, due to Green: If a charge unity, placed on the conductor $A_{0}$ in presence of conductors $A_{1}$, $A_{2}$, &c. at potential zero induces charges $-n_{1}$, $-n_{2}$, &c. in these conductors, then, if $A_{0}$ is discharged and insulated, and these conductors are maintained at potentials $V_{1}$, $V_{2}$, &c., the potential of $A_{0}$ will be

${n_{1}}{V_{1}}+{n_{2}}{V_{2}}+$&c.

The quantities $(n)$ are evidently numerical quantities, or ratios.

The conductor $A_{0}$ may be supposed reduced to a point, and $A_{1}$, $A_{2}$, &c. need not be insulated from each other, but may be different elementary portions of the surface of the same conductor. We shall see the application of this principle when we investigate Green's Functions.

90.] Theorem III. The coefficients of potential are all positive,but none of the coefficients $p_{rs}$ is greater than $p_{rr}$ or $p_{ss}$.

For let a charge unity be communicated to $A_{r}$, the other conductors being uncharged. A system of equipotential surfaces will be formed. Of these one will be the surface of $A_{r}$ and its potential will be $p_{rr}$. If $A_{s}$ is placed in a hollow excavated in $A_{r}$ so as to be completely enclosed by it, then the potential of $A_{s}$ will also be $p_{rr}$.

If, however, $A_{s}$ is outside of $A_{r}$ its potential $p_{rs}$ will lie between $p_{rr}$ and zero.

For consider the lines of force issuing from the charged conductor $A_{r}$. The charge is measured by the excess of the number of lines which issue from it over those which terminate in it. Hence, if the conductor has no charge, the number of lines which enter the conductor must be equal to the number which issue from it. The lines which enter the conductor come from places of greater potential, and those which issue from it go to places of less potential. Hence the potential of an uncharged conductor must be intermediate between the highest and lowest potentials in the field, and therefore the highest and lowest potentials cannot belong to any of the uncharged bodies.

The highest potential must therefore be $p_{rr}$, that of the charged body $A_{r}$ , and the lowest must be that of space at an infinite distance, which is zero, and all the other potentials such as $p_{rs}$ must lie between $p_{rr}$ and zero.

If $A_{s}$ completely surrounds $A_{t}$ then $p_{rs}$ = $p_{rt}$.

91.] Theorem IV. None of the coefficients of induction are positive, and the sum of all those belonging to a single conductor is not numerically greater than the coefficient of capacity of that conductor, which is always positive.

For let $A_{r}$ be maintained at potential unity while all the other conductors are kept at potential zero, then the charge on $A_{r}$ is $q_{rr}$, and that on any other conductor $A_{s}$ is $q_{rs}$.

The number of lines of force which issue from $A_{r}$ is $p_{rr}$. Of these some terminate in the other conductors, and some may proceed to infinity, but no lines of force can pass between any of the other conductors or from them to infinity, because they are all at potential zero.

No line of force can issue from any of the other conductors such as $A_{s}$ , because no part of the field has a lower potential than $A_{s}$. If $A_{s}$ is completely cut off from $A_{r}$ by the closed surface of one of the conductors, then $q_{rs}$ is zero. If $A_{s}$ is not thus cut off, $q_{rs}$ is a negative quantity.

If one of the conductors $A_{t}$ completely surrounds $A_{r}$, then all the lines of force from $A_{r}$ fall on $A_{t}$ and the conductors within it, and the sum of the coefficients of induction of these conductors with respect to $A_{r}$ will be equal to $q_{rr}$ with its sign changed. But if $A_{r}$ is not completely surrounded by a conductor the arithmetical sum of the coefficients of induction $q_{rs}$, &c. will be less than $q_{rr}$.

We have deduced these two theorems independently by means of electrical considerations. We may leave it to the mathematical student to determine whether one is a mathematical consequence of the other.

Resultant Mechanical Force on any Conductor in terms of the Charges.

92.] Let $\delta \phi$ be any mechanical displacement of the conductor, and let $\Phi$ be the the component of the force tending to produce that displacement, then $\Phi \delta \phi$ is the work done by the force during the displacement. If this work is derived from the electrification of the system, then if $Q$ is the electric energy of the system,

$\Phi \,\delta \phi +\delta Q=0\,$,

(3)

or ${\color {White}xxxxxx}\Phi =-{\frac {\delta Q}{\delta \Phi }}$.

(4)

Here

$Q={\tfrac {1}{2}}(E_{1}V_{1}+E_{2}V_{2}+\And \!\!{\mbox{c.}})$

(5)

If the bodies are insulated, the variation of $Q$ must be such that $E_{1},E_{2}$, &c. remain constant. Substituting therefore for the values of the potentials, we have

$Q={\tfrac {1}{2}}\Sigma _{r}\Sigma _{s}(E_{r}E_{s}p_{rs})$,

(6)

where the symbol of summation $\Sigma$ includes all terms of the form within the brackets, and $r$ and $s$ may each have any values from 1 to $n$. From this we find

$\Phi =-{\frac {dQ}{d\phi }}=-{\tfrac {1}{2}}\Sigma _{r}\Sigma _{s}(E_{r}E_{s}{\frac {dp_{rs}}{d\phi }})$

(7)

as the expression for the component of the force which produces variation of the generalized coordinate $\phi$.

Resultant Mechanical Force in terms of the Potentials.

93.] The expression for $\Phi$ in terms of the charges is

$\Phi =-{\tfrac {1}{2}}\Sigma _{r}\Sigma _{s}(E_{r}E_{s}{\frac {dp_{rs}}{d\phi }})$

(8)

where in the summation $r$ and $s$ have each every value in succession from 1 to $n$.

Now $E_{r}=\Sigma _{1}^{t}(V_{t}q_{rt})$ where $t$ may have any value from 1 to $n$, so that

$\Phi =-{\tfrac {1}{2}}\Sigma _{r}\Sigma _{s}\Sigma _{t}(E_{s}V_{t}q_{rt}{\frac {dp_{rs}}{d\phi }})$.

(9)

Now the coefficients of potential are connected with those of induction by n equations of the form

$\Sigma _{r}(p_{ar}q_{ar})=1\,\!$,

(10)

and ${\tfrac {1}{2}}n(n-1)$ of the form

$\Sigma _{r}(p_{ar}q_{ar})=0\,\!$.

(11)

Differentiating with respect to $\phi$ we get ${\tfrac {1}{2}}n(n+1)$ equations of the form

$\Sigma _{r}(p_{ar}{\frac {dq_{br}}{d\phi }})+\Sigma _{r}(q_{br}{\frac {dp_{ar}}{d\phi }})=0$,

(12)

where $a$ and $b$ may be the same or different.

Hence, putting $a$ and $b$ equal to $r$ and $s$,

$\Phi ={\tfrac {1}{2}}\Sigma _{r}\Sigma _{s}\Sigma _{t}(E_{s}V_{t}p_{rs}{\frac {dq_{rt}}{d\phi }})$,

(13)

but \Sigma_s(E_s p_{rs})=V_r, so that we may write

$\Phi ={\tfrac {1}{2}}\Sigma _{r}\Sigma _{t}(V_{r}V_{t}{\frac {dq_{rt}}{d\phi }})$,

(14)

where $r$ and $t$ may have each every value in succession from 1 to $n$. This expression gives the resultant force in terms of the potentials.

If each conductor is connected with a battery or other contrivance by which its potential is maintained constant during the displacement, then this expression is simply

$\Phi ={\frac {dQ}{d\phi }}\,\!$,

(15)

under the condition that all the potentials are constant.

The work done in this case during the displacement $\delta \phi$ is $\Phi \delta \phi$, and the electrical energy of the system of conductors is increased by $\delta Q$; hence the energy spent by the batteries during the displacement is

$\Phi \delta \phi +\delta Q=2\Phi \delta \phi =2\delta Q\,\!$.

(16)

It appears from Art. 92, that the resultant force $\Phi$ is equal to $-{\tfrac {dQ}{d\phi }}$, under the condition that the charges of the conductors are constant. It is also, by Art. 93, equal to ${\tfrac {dQ}{d\phi }}$, under the condition that the potentials of the conductors are constant. If the conductors are insulated, they tend to move so that their energy is diminished, and the work done by the electrical forces during the displacement is equal to the diminution of energy.

If the conductors are connected with batteries, so that their potentials are maintained constant, they tend to move so that the energy of the system is increased, and the work done by the electrical forces during the displacement is equal to the increment of the energy of the system. The energy spent by the batteries is equal to double of either of these quantities, and is spent half in mechanical, and half in electrical work.

On the Comparison of Similar Electrified Systems.

94.] If two electrified systems are similar in a geometrical sense., so that the lengths of corresponding lines in the two systems are as $L$ to $L'$, then if the dielectric which separates the conducting bodies is the same in both systems, the coefficients of induction and of capacity will be in the proportion of $L$ to $L'$. For if we consider corresponding portions, $A$ and $A'$, of the two systems, and suppose the quantity of electricity on $A$ to be $E$, and that on $A'$ to be $E'$, then the potentials $V$ and $V'$ at corresponding points $B$ and $B'$, due to this electrification, will be

$V={\frac {E}{AB}}$,${\color {White}xxx}$and ${\color {White}xxx}$$V={\frac {E'}{A'B'}}$

But $AB$ is to $A'B'$ as $L$ to $L'$, so that we must have

$E\ :E'\ ::LV\ :\ L'V'\,\!$

But if the inductive capacity of the dielectric is different in the two systems, being $K$ in the first and $K'$ in the second, then if the potential at any point of the first system is to that at the corresponding point of the second as $V$ to $V'$ and if the quantities of electricity on corresponding parts are as $E$ to $E'$, we shall have

$E\ :E'\ ::LVK\ :\ L'V'K'\,\!$

By this proportion we may find the relation between the total electrification of corresponding parts of two systems, which are in the first place geometrically similar, in the second place composed of dielectric media of which the dielectric inductive capacity at corresponding points is in the proportion of $K$ to $K'$ and in the third place so electrified that the potentials of corresponding points are as $V$ to $V'$.

From this it appears that if $q$ be any coefficient of capacity or induction in the first system, and $q'$ the corresponding one in the second,

$q\ :q'\ ::LK\ :\ L'K'\,\!$;

and if $p$ and $p'$ denote corresponding coefficients of potential in the two systems,

$p\ :p'\ ::{\frac {1}{LK}}\ :\ {\frac {1}{L'K'}}\,\!$.

If one of the bodies be displaced in the first system, and the corresponding body in the second system receive a similar displacement, then these displacements are in the proportion of $L$ to $L'$, and if the forces acting on the two bodies are as $F$ to $F'$, then the work done in the two systems will be as $FL$ to $F'L'$.

But the total electrical energy is half the sum of the quantities of electricity multiplied each by the potential of the electrified body, so that in the similar systems, if $Q$ and $Q'$ be the total electrical energy,

$Q\ :\ Q'\ ::\ EV\ :\ E'V'$,

and the difference of energy after similar displacements in the two systems will be in the same proportion. Hence, since $FL$ is proportional to the electrical work done during the displacement,

$FL\ :\ F'L'\ ::\ EV\ :\ E'V'$.

Combining these proportions, we find that the ratio of the resultant force on any body of the first system to that on the corresponding body of the second system is

$F\ :\ F'\ ::\ V^{2}K\ :\ V'^{2}K1$,

or${\color {White}xxxxxxx}$$F\ :\ F'\ ::\ {\frac {E^{2}}{L^{2}K}}:\ {\frac {E'^{2}}{L'^{2}K'}}$.

The first of these proportions shews that in similar systems the force is proportional to the square of the electromotive force and to the inductive capacity of the dielectric, but is independent of the actual dimensions of the system.

Hence two conductors placed in a liquid whose inductive capacity is greater than that of air, and electrified to given potentials, will attract each other more than if they had been electrified to the same potentials in air.

The second proportion shews that if the quantity of electricity on each body is given, the forces are proportional to the squares of the electrifications and inversely to the squares of the distances, and also inversely to the inductive capacities of the media.

Hence, if two conductors with given charges are placed in a liquid whose inductive capacity is greater than that of air, they will attract each other less than if they had been surrounded with air and electrified with the same charges of electricity.