Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/147

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