Page:A Dynamical Theory of the Electromagnetic Field.pdf/52

acts on a cylindrical wire of specific resistance $\rho$, we have

$p\rho =P-{\frac {dF}{dt}}$

where F is got from the equation

${\frac {d^{2}F}{dr^{2}}}+{\frac {1}{r}}{\frac {dF}{dr}}=-4\pi \mu p,$

$r$ being the distance from the axis of the cylinder.

Let one term of the value of F be of the form $Tr^{n}$, where T is a function of the time, then the term of $p$ which produced it is of the form

$-{\frac {1}{4\pi \mu }}n^{2}Tr^{n-2}$

Hence if we write

${\begin{array}{l}F=T+{\frac {\mu \pi }{\rho }}\left(-P+{\frac {dT}{dt}}\right)r^{2}+\left({\frac {\mu \pi }{\rho }}\right)^{2}{\frac {1}{1^{2}\cdot 2^{2}}}{\frac {dT^{2}}{dt^{2}}}r^{4}+etc.\\\\p\rho =\left(P+{\frac {dT}{dt}}\right)-{\frac {\mu \pi }{\rho }}{\frac {d^{2}T}{dt^{2}}}r^{2}-\left({\frac {\mu \pi }{\rho }}\right)^{2}{\frac {1}{1^{2}\cdot 2^{2}}}{\frac {d^{3}T}{dt^{3}}}r^{4}-etc.\end{array}}$

The total counter current of self-induction at any point is

$\int \left({\frac {P}{\rho }}-p\right)dt={\frac {1}{\rho }}T+{\frac {\mu \pi }{\rho ^{2}}}{\frac {dT}{dt}}r^{2}+{\frac {\mu ^{2}\pi ^{2}}{\rho ^{3}}}{\frac {1}{1^{2}2^{2}}}{\frac {d^{2}T}{dt^{2}}}r^{4}+etc.$

from $t=0$ to $t=\infty$.

${\begin{array}{llc}\mathrm {When} \ t=0,\ p=0,&\therefore \left({\frac {dT}{dt}}\right)_{0}=P,&\left({\frac {d^{2}T}{dt^{2}}}\right)_{0}=0,\ \mathrm {etc.} \\\\\mathrm {When} \ t=\infty ,\ p={\frac {P}{\rho }},&\therefore \left({\frac {dT}{dt}}\right)_{\infty }=0,&\left({\frac {d^{2}T}{dt^{2}}}\right)_{\infty }=0,\ \mathrm {etc.} \end{array}}$

$\int \limits _{0}^{\infty }\int \limits _{0}^{r}2\pi \left({\frac {P}{\rho }}-p\right)rdrdt={\frac {1}{\rho }}T\pi r^{2}+{\frac {1}{2}}{\frac {\mu \pi ^{2}}{\rho ^{2}}}{\frac {dT}{dt}}r^{4}+{\frac {\mu ^{2}\pi ^{3}}{\rho ^{3}}}{\frac {1}{1^{2}\cdot 2^{2}\cdot 3}}{\frac {d^{2}T}{dt^{2}}}r^{6}+etc.$

from $t=0$ to $=\infty$.

${\begin{array}{llc}\mathrm {When} \ t=0,\ p=0\ \mathrm {throughout\ the\ section} ,&\therefore \left({\frac {dT}{dt}}\right)_{0}=P,&\left({\frac {d^{2}T}{dt^{2}}}\right)_{0}=0,\ \mathrm {etc.} \\\\\mathrm {When} \ t=\infty ,\ p=0\ \mathrm {throughout} ,&\therefore \left({\frac {dT}{dt}}\right)_{\infty }=0,&\left({\frac {d^{2}T}{dt^{2}}}\right)_{\infty }=0,\ \mathrm {etc.} \end{array}}$

Also if $l$ be the length of the wire, and R its resistance,

$R={\frac {\rho l}{\pi r^{2}}}$

and if C be the current when established in the wire, $C={\frac {Pl}{R}}$.

The total counter current may be written

${\frac {l}{R}}\left(T_{\infty }-T_{0}\right)-{\frac {1}{2}}\mu {\frac {l}{R}}C=-{\frac {LC}{R}}$ by § (35).