Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/428

 

$\left.{\begin{aligned}\nabla ^{2}F+4\pi \mu C{\frac {dF}{dt}}&=0{\mbox{,}}\\\nabla ^{2}G+4\pi \mu C{\frac {dG}{dt}}&=0{\mbox{,}}\\\nabla ^{2}H+4\pi \mu C{\frac {dH}{dt}}&=0{\mbox{.}}\end{aligned}}\right\}$

(1)

Each of these equations is of the same form as the equation of the diffusion of heat given in Fourier's Traité de Chaleur.

802.] Taking the first as an example, the component $F$ of the vector-potential will vary according to time and position in the same way as the temperature of a homogeneous solid varies according to time and position, the initial and the surface-conditions heing made to correspond in the two cases, and the quantity $4\pi \mu C$ being numerically equal to the reciprocal of the thermometric conductivity of the substance, that is to say, the number of units of volume of the substance which would be heated one degree by the heat which passes through a unit cube of the substance, two opposite faces of which differ by one degree of temperature, while the other faces are impermeable to heat^[1].

The different problems in thermal conduction, of which Fourier has given the solution, may be transformed into problems in the diffusion of electromagnetic quantities, remembering that $F$, $G$, $H$ are the components of a vector, whereas the temperature, in Fourier's problem, is a scalar quantity.

Let us take one of the cases of which Fourier has given a complete solution^[2], that of an infinite medium, the initial state of which is given.

The state of any point of the medium at the time $t$ is found by taking the average of the state of every part of the medium, the weight assigned to each part in taking the average being

$e^{-{\frac {\pi \mu Cr^{2}}{t}}}$,

where $r$ is the distance of that part from the point considered. This average, in the case of vector-quantities, is most conveniently taken by considering each component of the vector separately.

1. See Maxwell's Theory of Heat, p. 235.
2. Traité de la Chaleur, Art. 384. The equation which determines the temperature, $v$, at a point $(x,\;y,\;z)$ after a time $t$, in terms of $f(\alpha ,\;\beta ,\;\gamma )$, the initial temperature at the point $(\alpha ,\;\beta ,\;\gamma )$, is

   $v=\iiint {\frac {d\alpha \,d\beta \,d\gamma }{2^{3}{\sqrt {k^{3}\pi ^{3}t^{3}}}}}e^{-\left({\frac {(\alpha -x)^{2}+(\beta -y)^{2}+(\gamma -z)^{2}}{4kt}}\right)}f(\alpha ,\;\beta ,\;\gamma )$,

   where $k$ is the thermometric conductivity.