Page:A Dynamical Theory of the Electromagnetic Field.pdf/49

S' is a surface bounded by the current B, and $l,m,n$ are the direction-cosines of the normal to the surface, the integration being extended over the surface.

We may express this in the form

$M=\mu \sum {\frac {1}{\rho ^{2}}}\sin \theta \sin \theta '\sin \varphi dS'ds$

where $d$S' is an element of the surface bounded by B, $ds$ is an element of the circuit A, $\rho$ is the distance between them, $\theta$ and $\theta '$ are the angles between $\rho$ and $ds$ and between $\rho$ and the normal to $d$S' respectively, and $\varphi$ is the angle between the planes in which $\theta$ and $\theta '$ are measured. The integration is performed round the circuit A and over the surface bounded by B.

This method is most convenient in the case of circuits lying in one plane, in which case sin $\sin \theta =1$, and $\sin \varphi =1$.

111. Third Method. M is that part of the intrinsic magnetic energy of the whole field which depends on the product of the currents in the two circuits, each current being unity.

Let $\alpha ,\beta ,\gamma$ be the components of magnetic intensity at any point due to the first circuit, $\alpha ',\beta ',\gamma '$ the same for the second circuit; then the intrinsic energy of the element of volume $d$V of the field is

${\frac {\mu }{8\pi }}\left((\alpha +\alpha ')^{2}+(\beta +\beta ')^{2}+(\gamma +\gamma ')^{2}\right)dV$

The part which depends on the product of the currents is

${\frac {\mu }{4\pi }}(\alpha \alpha '+\beta \beta '+\gamma \gamma ')dV$

Hence if we know the magnetic intensities I and I' due to unit current in each circuit, we may obtain M by integrating

${\frac {\mu }{4\pi }}\sum \mu II'\cos \theta dV$

over all space, where $\theta$ is the angle between the directions of I and I'.

Application to a Coil.

(112) To find the coefficient (M) of mutual induction between two circular linear conductors in parallel planes, the distance between the curves being everywhere the same, and small compared with the radius of either.

If $r$ be the distance between the curves, and $a$ the radius of either, then when $r$ is very small compared with $a$, we find by the second method, as a first approximation,

$M=4\pi \left(\log _{e}{\frac {8a}{r}}-2\right)$

To approximate more closely to the value of M, let $a$ and $a_{1}$ be the radii of the circles, and $b$ the distance between their planes; then

$r^{2}=\left(a-a_{1}\right)^{2}+b^{2}$