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

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88.]
RECIPROCAL PROPERTY OF THE COEFFICIENTS.
91

88.] Theorem I. The coefficients of relative to are equal to those of relative to .

If , the charge on , is increased by , the work spent in bringing from an infinite distance to the conductor whose potential is , is by the definition of potential in Art. 70,

,


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 , &c., the increment of the electric energy of the system will be

.


If, therefore, the electric energy is expressed as a function of the charges , , &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

.


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

for the term in the expansion of which involves the product , then, by differentiating with respect to , we find the term of the expansion of which involves to be .

Differentiating with respect to , we find the term in the expansion of which involves to be .

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

,


or, interpreting the symbols and  :—

The potential of due to a unit charge on is equal to the potential of due to a unit charge on .

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 and to be indefinitely small, we have the following reciprocal property of any two points :

The potential at any point , due to unit of electricity placed at in presence of any system of conductors, is a function of the positions of and in which the coordinates of and of enter in the same manner, so that the value of the function is unchanged if we exchange and .