Page:A Dynamical Theory of the Electromagnetic Field.pdf/48

506 PART VII. — CALCULATION OF THE COEFFICIENTS OF ELECTROMAGNETIC INDUCTION.

General Methods.

(109) The electromagnetic relations between two conducting circuits, A and B, depend upon a function M of their form and relative position, as has been already shown.

M may be calculated in several different ways, which must of course all lead to the same result.

First Method. M is the electromagnetic momentum of the circuit B when A carries a unit current, or

$M=\int\left(F\frac{dx}{ds'}+G\frac{dy}{ds'}+H\frac{dz}{ds'}\right)ds'$

where F, G, H are the components of electromagnetic momentum due to a unit current in A, and $$ds'$$ is an element of length of B, and the integration is performed round the circuit of B.

To find F, G, H, we observe that by (B) and (C)

$\frac{d^{2}F}{dx^{2}}+\frac{d^{2}F}{dy^{2}}+\frac{d^{2}F}{dz^{2}}=-4\pi\mu p'$|undefined

with corresponding equations for G and H, $$p', q'$$, and $$r'$$ being the components of the current in A.

Now if we consider only a single element $$ds$$ of A, we shall have

$p'=\frac{dx}{ds}ds,\ q'=\frac{dy}{ds}ds,\ r'=\frac{dz}{ds}ds,$

and the solution of the equation gives

$F=\frac{\mu}{\rho}\frac{dx}{ds}ds,\ G=\frac{\mu}{\rho}\frac{dy}{ds}ds,\ H=\frac{\mu}{\rho}\frac{dz}{ds}ds,$

where $$\rho$$ is the distance of any point from $$ds$$. Hence

$\begin{array}{rl} M & =\iint\frac{\mu}{\rho}\left(\frac{dx}{ds}\frac{dx}{ds'}+\frac{dy}{ds}\frac{dy}{ds'}+\frac{dz}{ds}\frac{dz}{ds'}\right)dsds'\\ \\ & =\iint\frac{\mu}{\rho}\cos\theta dsds',\end{array}$

where $$\theta$$ is the angle between the directions of the two elements $$ds, ds'$$, and $$\rho$$ is the distance between them, and the integration is performed round both circuits.

In this method we confine our attention during integration to the two linear circuits alone.

(110) Second Method. M is the number of lines of magnetic force which pass through the circuit B when A carries a unit current, or

$M=\sum(\mu\alpha l+\mu\beta m+\mu\gamma n)dS'$

where $$\mu\alpha,\mu\beta,\mu\gamma$$, are the components of magnetic induction due to unit current in A,