Page:On Faraday's Lines of Force.pdf/43

If $\alpha ,\beta ,\gamma$ be the electro-motive forces, $p$ the electric tension, and $k$ the coefficient of resistance, then the above equation is identical with the equation of continuity

${\frac {da_{2}}{dx}}+{\frac {db_{2}}{dy}}+{\frac {dc_{2}}{dz}}+4\pi \rho =0;$

and the theorem shows that when the electro-motive forces and the rate of production of electricity at every part of space are given, the value of the electric tension is determinate.

Since the mathematical laws of magnetism are identical with those of electricity, as far as we now consider them, we may regard $\alpha ,\beta ,\gamma$ as magnetizing forces, $p$ as magnetic tension, and $\rho$ as real magnetic density, $k$ being the coefficient of resistance to magnetic induction.

The proof of this theorem rests on the determination of the minimum value of

$\scriptstyle Q=\iiint \left\{{\frac {1}{k}}\left(\alpha -{\frac {dp}{dx}}-k{\frac {dV}{dx}}\right)^{2}+{\frac {1}{k}}\left(\beta -{\frac {dp}{dy}}-k{\frac {dV}{dy}}\right)^{2}+{\frac {1}{k}}\left(\gamma -{\frac {dp}{dz}}-k{\frac {dV}{dz}}\right)^{2}\right\}dxdydz$;

where $V$ is got from the equation

${\frac {d^{2}V}{dx^{2}}}+{\frac {d^{2}V}{dy^{2}}}+{\frac {d^{2}V}{dz^{2}}}+4\pi \rho =0$,

and $p$ has to be determined.

The meaning of this integral in electrical language may be thus brought out. If the presence of the media in which $k$ has various values did not affect the distribution of forces, then the “quantity” resolved in $x$ would be simply ${\frac {dV}{dx}}$ and the intensity $k{\frac {dV}{dx}}$. But the actual quantity and intensity are ${\frac {1}{k}}\left(\alpha -{\frac {dp}{dx}}\right)$ and $\alpha ={\frac {dp}{dx}}$, and the parts due to the distribution of media alone are therefore

${\frac {1}{k}}\left(\alpha -{\frac {dp}{dx}}\right)-{\frac {dV}{dx}}$ and $\alpha ={\frac {dp}{dx}}-k{\frac {dV}{dx}}$.

Now the product of these represents the work done on account of this distribution of media, the distribution of sources being determined, and taking in the terms in $y$ and $z$ we get the expression Q for the total work done