Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/156

Let us confine our attention to the last of these three groups of terms, merely observing that the other groups are essentially positive. By Green’s theorem

${\begin{array}{c}\iiint K\left({\frac {dV_{0}}{dx}}{\frac {dU}{dx}}+{\frac {dV_{0}}{dy}}{\frac {dU}{dy}}+{\frac {dV_{0}}{dz}}{\frac {dU}{dz}}\right)dx\ dy\ dz=\iint KU{\frac {dV_{0}}{d\nu }}dS\\\\-\iiint U\left({\frac {d}{dx}}K{\frac {dV_{0}}{dx}}+{\frac {d}{dy}}K{\frac {dV_{0}}{dy}}+{\frac {d}{dz}}K{\frac {dV_{0}}{dz}}\right)dx\ dy\ dz;\end{array}}$

(6)

the first integral of the second member being extended over the surface of the region and the second throughout the enclosed space. But on the surfaces $S_{1},S_{2}$ &c. $U=0$, so that these contribute nothing to the surface-integral.

Again, on the surface $S_{0}$, ${\tfrac {dV_{0}}{d\nu }}=0$, so that this surface contributes nothing to the integral. Hence the surface-integral is zero.

The quantity within brackets in the volume-integral also disappears by equation (3), so that the volume-integral is also zero. Hence $Q$ is reduced to

$Q={\frac {1}{8\pi }}\iiint K\left({\overline {\frac {dV_{0}}{dx}}}{\Bigg \vert }^{2}+\mathrm {\&c.} \right)dx\ dy\ dz+{\frac {1}{8\pi }}\iiint K\left({\overline {\frac {dU}{dx}}}{\Bigg \vert }^{2}+\mathrm {\&c.} \right)dx\ dy\ dz.$

(7)

Both these quantities are essentially positive, and therefore the minimum value of $Q$ is when

${\frac {dU}{dx}}={\frac {dU}{dy}}={\frac {dU}{dz}}=0$

(8)

or when $U$ is a constant. But at the surfaces $S$, &c. $U=0$. Hence $U=0$ everywhere, and $V_{0}$ gives the unique minimum value of $Q$.

Calculation of a Superior Limit of the Coefficients of Capacity.

The quantity $Q$ in its minimum form can be expressed by means of Green’s theorem in terms of $V_{1},V_{2}$, &c., the potentials of $S_{1},S_{2}$, and $E_{1},E_{2}$, &c., the charges of these surfaces,

$Q={\frac {1}{2}}\left(V_{1}E_{1}+V_{2}E_{2}+\mathrm {etc.} \right);$

(9)

or, making use of the coefficients of capacity and induction as defined in Article 87,

$Q={\frac {1}{2}}\left(V_{1}^{2}q_{11}+V_{2}^{2}q_{22}+\mathrm {etc.} \right)+V_{1}V_{2}q_{12}+\mathrm {etc.} ;$

(10)

The accurate determination of the coefficients $q$ is in general difficult, involving the solution of the general equation of statical electricity, but we make use of the theorem we have proved to determine a superior limit to the value of any of these coefficients.