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

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

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

where F is got from the equation

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

$$r$$ 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

$\begin{array}{l} F=T+\frac{\mu\pi}{\rho}\left(-P+\frac{dT}{dt}\right)r^{2}+\left(\frac{\mu\pi}{\rho}\right)^{2}\frac{1}{1^{2}\cdot2^{2}}\frac{dT^{2}}{dt^{2}}r^{4}+etc.\\ \\p\rho=\left(P+\frac{dT}{dt}\right)-\frac{\mu\pi}{\rho}\frac{d^{2}T}{dt^{2}}r^{2}-\left(\frac{\mu\pi}{\rho}\right)^{2}\frac{1}{1^{2}\cdot2^{2}}\frac{d^{3}T}{dt^{3}}r^{4}-etc.\end{array}$|undefined

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

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

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

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

$\int\limits _{0}^{\infty}\int\limits _{0}^{r}2\pi\left(\frac{P}{\rho}-p\right)rdrdt=\frac{1}{\rho}T\pi r^{2}+\frac{1}{2}\frac{\mu\pi^{2}}{\rho^{2}}\frac{dT}{dt}r^{4}+\frac{\mu^{2}\pi^{3}}{\rho^{3}}\frac{1}{1^{2}\cdot2^{2}\cdot3}\frac{d^{2}T}{dt^{2}}r^{6}+etc.$|undefined

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

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

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

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

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).