Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/131

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 $A_{s}$.