Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/324

If $\rho$ denotes the specific resistance of the substance per unit of volume, the electromotive force at any point is $\rho w$, and this may be expressed in terms of the electric potential and the vector potential $H$ by equations (B), Art. 598,

Comparing the coefficients of like powers of $r$ in equations (3) and (5),

⁠		$T_{1}$	${}={\frac {\pi }{\rho }}\left({\frac {d\Psi }{dz}}+{\frac {dS}{dt}}+{\frac {dT_{0}}{dt}}\right)$,		(6)
		$T_{2}$	${}={\frac {\pi }{\rho }}{\frac {dT_{1}}{dt}}$,		(7)
		$T_{n}$	${}={\frac {\pi }{\rho }}{\frac {1}{n^{2}}}{\frac {dT_{n-1}}{dt}}$.		(8)

Hence we may write

690.] To find the total current $C$, we must integrate $w$ over the section of the wire whose radius is $a$,

Substituting the value of $\pi w$ from equation (3), we obtain

The value of $H$ at any point outside the wire depends only on the total current $C$, and not on the mode in which it is distributed within the wire. Hence we may assume that the value of $H$ at the surface of the wire is $AC$, where $A$ is a constant to be determined by calculation from the general form of the circuit. Putting $H=AC$ when $r=a$, we obtain

If we now write ${\frac {\pi a^{2}}{\rho }}=\alpha$, $\alpha$ is the value of the conductivity of unit of length of the wire, and we have

		$C$	${}=-\left(\alpha {\frac {dT}{dt}}+{\frac {2\alpha ^{2}}{1^{2}2^{2}}}{\frac {d^{2}T}{dt^{2}}}+\mathrm {\&c.} +{\frac {n\alpha ^{n}}{(|{\underline {n}})^{2}}}{\frac {d^{n}T}{dt^{n}}}+\mathrm {\&c.} \right)$,		(14)
		$AC-S$	${}=T+\alpha {\frac {dT}{dt}}+{\frac {\alpha ^{2}}{1^{2}2^{2}}}{\frac {d^{2}T}{dt^{2}}}+\mathrm {\&c.} +{\frac {\alpha ^{n}}{(|{\underline {n}})^{2}}}{\frac {d^{n}T}{dt^{n}}}+\mathrm {\&c.}$		(15)