# A Treatise on Electricity and Magnetism/Part I/Chapter III

A Treatise on Electricity and Magnetism by James Clerk Maxwell
Part I, Chapter III: Systems of Conductors

## CHAPTER III. SYSTEMS OF CONDUCTORS.

### On the Superposition of Electrical Systems.

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

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

### Energy of an Electrified System.

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

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

Let us consider the effect of changing ${\displaystyle n}$ into ${\displaystyle n+dn}$ in all these expressions. It will be equivalent to charging ${\displaystyle A_{1}}$ with a quantity of electricity ${\displaystyle e_{l}dn}$, ${\displaystyle A_{2}}$ with ${\displaystyle e_{2}dn}$, &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 ${\displaystyle e_{1}dn}$ of electricity to ${\displaystyle A_{1}}$, whose potential before the charge is ${\displaystyle nV_{1}}$, and after the charge ${\displaystyle (n+dn)V_{1}}$, lf must lie between

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

In the limit we may neglect the square of ${\displaystyle dn}$, and write the expression

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

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

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

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

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

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

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

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

### General Theory of a System of Conductors.

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

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

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

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

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

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

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

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

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

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

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

 ${\displaystyle {\begin{matrix}E_{1}=q_{11}V_{1}\ldots +q_{1s}V_{s}\ldots +q_{1n}V_{n},\\-\qquad -\qquad -\qquad -\qquad -\qquad -\\E_{r}=q_{r1}V_{1}\ldots +q_{rs}V_{s}\ldots +q_{rn}V_{n},\\-\qquad -\qquad -\qquad -\qquad -\qquad -\\E_{n}=q_{n1}V_{1}\ldots +q_{ns}V_{s}\ldots +q_{nn}V_{n},\end{matrix}}}$ (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 ${\displaystyle q_{11}}$ is numerically equal to the quantity of electricity on ${\displaystyle A_{l}}$ when ${\displaystyle A_{l}}$ is at potential unity and all the other bodies are at potential zero. This is called the Capacity of ${\displaystyle A_{1}}$. It depends on the form and position of all the conductors in the system.

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

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 ${\displaystyle r}$ 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 ${\displaystyle (q)}$ are of the nature of lines, and the coefficients of potential ${\displaystyle (p)}$ are of the nature of the reciprocals of lines.

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

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

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

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 ${\displaystyle \delta E_{1}}$, &c., the increment of the electric energy of the system will be

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

If, therefore, the electric energy ${\displaystyle Q}$ is expressed as a function of the charges ${\displaystyle E_{1}}$, ${\displaystyle E_{2}}$, &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

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

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

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

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

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

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

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

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

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

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

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

The coefficients of induction ${\displaystyle q_{rs}}$ and ${\displaystyle q_{sr}}$ 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 ${\displaystyle q_{rs}}$, ${\displaystyle p_{rs}}$ and ${\displaystyle p_{sr}}$ enter in the same way as ${\displaystyle p_{sr}}$ and ${\displaystyle p_{rs}}$ do in the expression for ${\displaystyle q_{sr}}$ . Hence if all pairs of coefficients ${\displaystyle p_{rs}}$ and ${\displaystyle p_{sr}}$ are equal, the pairs ${\displaystyle q_{rs}}$ and ${\displaystyle q_{sr}}$ are also equal.

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

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

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

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

In the second case, we have

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

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

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

The quantities ${\displaystyle (n)}$ are evidently numerical quantities, or ratios.

The conductor ${\displaystyle A_{0}}$ may be supposed reduced to a point, and ${\displaystyle A_{1}}$, ${\displaystyle A_{2}}$, &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 ${\displaystyle p_{rs}}$ is greater than ${\displaystyle p_{rr}}$ or ${\displaystyle p_{ss}}$.

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

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

For consider the lines of force issuing from the charged conductor ${\displaystyle A_{r}}$. 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 ${\displaystyle p_{rr}}$, that of the charged body ${\displaystyle A_{r}}$ , and the lowest must be that of space at an infinite distance, which is zero, and all the other potentials such as ${\displaystyle p_{rs}}$ must lie between ${\displaystyle p_{rr}}$ and zero.

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

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

The number of lines of force which issue from ${\displaystyle A_{r}}$ is ${\displaystyle p_{rr}}$. 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 ${\displaystyle A_{s}}$ , because no part of the field has a lower potential than ${\displaystyle A_{s}}$. If ${\displaystyle A_{s}}$ is completely cut off from ${\displaystyle A_{r}}$ by the closed surface of one of the conductors, then ${\displaystyle q_{rs}}$ is zero. If ${\displaystyle A_{s}}$ is not thus cut off, ${\displaystyle q_{rs}}$ is a negative quantity.

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

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

 ${\displaystyle \Phi \,\delta \phi +\delta Q=0\,}$, (3)

 or ${\displaystyle {\color {White}xxxxxx}\Phi =-{\frac {\delta Q}{\delta \Phi }}}$. (4)

 Here ${\displaystyle Q={\tfrac {1}{2}}(E_{1}V_{1}+E_{2}V_{2}+\And \!\!{\mbox{c.}})}$ (5)

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

 ${\displaystyle Q={\tfrac {1}{2}}\Sigma _{r}\Sigma _{s}(E_{r}E_{s}p_{rs})}$, (6)

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

 ${\displaystyle \Phi =-{\frac {dQ}{d\phi }}=-{\tfrac {1}{2}}\Sigma _{r}\Sigma _{s}(E_{r}E_{s}{\frac {dp_{rs}}{d\phi }})}$ (7)

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

### Resultant Mechanical Force in terms of the Potentials.

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

 ${\displaystyle \Phi =-{\tfrac {1}{2}}\Sigma _{r}\Sigma _{s}(E_{r}E_{s}{\frac {dp_{rs}}{d\phi }})}$ (8)

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

Now ${\displaystyle E_{r}=\Sigma _{1}^{t}(V_{t}q_{rt})}$ where ${\displaystyle t}$ may have any value from 1 to ${\displaystyle n}$, so that
 ${\displaystyle \Phi =-{\tfrac {1}{2}}\Sigma _{r}\Sigma _{s}\Sigma _{t}(E_{s}V_{t}q_{rt}{\frac {dp_{rs}}{d\phi }})}$. (9)

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

 ${\displaystyle \Sigma _{r}(p_{ar}q_{ar})=1\,\!}$, (10)

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

 ${\displaystyle \Sigma _{r}(p_{ar}q_{ar})=0\,\!}$. (11)

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

 ${\displaystyle \Sigma _{r}(p_{ar}{\frac {dq_{br}}{d\phi }})+\Sigma _{r}(q_{br}{\frac {dp_{ar}}{d\phi }})=0}$, (12)

where ${\displaystyle a}$ and ${\displaystyle b}$ may be the same or different.

Hence, putting ${\displaystyle a}$ and ${\displaystyle b}$ equal to ${\displaystyle r}$ and ${\displaystyle s}$,

 ${\displaystyle \Phi ={\tfrac {1}{2}}\Sigma _{r}\Sigma _{s}\Sigma _{t}(E_{s}V_{t}p_{rs}{\frac {dq_{rt}}{d\phi }})}$, (13)

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

 ${\displaystyle \Phi ={\tfrac {1}{2}}\Sigma _{r}\Sigma _{t}(V_{r}V_{t}{\frac {dq_{rt}}{d\phi }})}$, (14)

where ${\displaystyle r}$ and ${\displaystyle t}$ may have each every value in succession from 1 to ${\displaystyle n}$. 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

 ${\displaystyle \Phi ={\frac {dQ}{d\phi }}\,\!}$, (15)

under the condition that all the potentials are constant.

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

 ${\displaystyle \Phi \delta \phi +\delta Q=2\Phi \delta \phi =2\delta Q\,\!}$. (16)

It appears from Art. 92, that the resultant force ${\displaystyle \Phi }$ is equal to ${\displaystyle -{\tfrac {dQ}{d\phi }}}$, under the condition that the charges of the conductors are constant. It is also, by Art. 93, equal to ${\displaystyle {\tfrac {dQ}{d\phi }}}$, 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 ${\displaystyle L}$ to ${\displaystyle L'}$, 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 ${\displaystyle L}$ to ${\displaystyle L'}$. For if we consider corresponding portions, ${\displaystyle A}$ and ${\displaystyle A'}$, of the two systems, and suppose the quantity of electricity on ${\displaystyle A}$ to be ${\displaystyle E}$, and that on ${\displaystyle A'}$ to be ${\displaystyle E'}$, then the potentials ${\displaystyle V}$ and ${\displaystyle V'}$ at corresponding points ${\displaystyle B}$ and ${\displaystyle B'}$, due to this electrification, will be

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

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

 ${\displaystyle E\ :E'\ ::LV\ :\ L'V'\,\!}$

But if the inductive capacity of the dielectric is different in the two systems, being ${\displaystyle K}$ in the first and ${\displaystyle K'}$ 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 ${\displaystyle V}$ to ${\displaystyle V'}$ and if the quantities of electricity on corresponding parts are as ${\displaystyle E}$ to ${\displaystyle E'}$, we shall have

 ${\displaystyle E\ :E'\ ::LVK\ :\ L'V'K'\,\!}$

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 ${\displaystyle K}$ to ${\displaystyle K'}$ and in the third place so electrified that the potentials of corresponding points are as ${\displaystyle V}$ to ${\displaystyle V'}$.

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

 ${\displaystyle q\ :q'\ ::LK\ :\ L'K'\,\!}$;

and if ${\displaystyle p}$ and ${\displaystyle p'}$ denote corresponding coefficients of potential in the two systems,

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

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 ${\displaystyle L}$ to ${\displaystyle L'}$, and if the forces acting on the two bodies are as ${\displaystyle F}$ to ${\displaystyle F'}$, then the work done in the two systems will be as ${\displaystyle FL}$ to ${\displaystyle F'L'}$.

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 ${\displaystyle Q}$ and ${\displaystyle Q'}$ be the total electrical energy,

 ${\displaystyle Q\ :\ Q'\ ::\ EV\ :\ E'V'}$,

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

 ${\displaystyle FL\ :\ F'L'\ ::\ EV\ :\ E'V'}$.

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

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

 or${\displaystyle {\color {White}xxxxxxx}}$${\displaystyle 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.