Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/293

This investigation is approximate only when $b_{1}$ and $b_{2}$ are large compared with $a$, and when $a$ is large compared with $c$. The quantity $a$ is a line which may be of any magnitude. It becomes infinite when $c$ is indefinitely diminished.

If we suppose $c={\tfrac {1}{2}}a$ there will be no apertures between the wires of the grating, and therefore there will be no induction through it. We ought therefore to have for this case $\alpha =0$. The formula (11), however, gives in this case

$\alpha =-{\frac {a}{2\pi }}\log _{e}2,\ =-0.11a$,

which is evidently erroneous, as the induction can never be altered in sign by means of the grating. It is easy, however, to proceed to a higher degree of approximation in the case of a grating of cylindrical wires. I shall merely indicate the steps of this process.

Method of Approximation.

206.] Since the wires are cylindrical, and since the distribution of electricity on each is symmetrical with respect to the diameter parallel to $y$, the proper expansion of the potential is of the form

$V=C_{0}\log r+\Sigma C_{i}r^{i}\cos i\theta$

(14)

where $r$ is the distance from the axis of one of the wires, and $\theta$ the angle between $r$ and $y$, and, since the wire is a conductor, when $r$ is made equal to the radius $V$ must be constant, and therefore the coefficient of each of the multiple cosines of $\theta$ must vanish.

For the sake of conciseness let us assume new coordinates $\xi ,\eta$, &c. such that

$a\xi =2\pi x,\ a\eta =2\pi y,\ a\rho =2\pi r,\ a\beta =2\pi b,\ \mathrm {etc.}$

(15)

and let

$F_{\beta }=\log \left(\epsilon ^{\eta +\beta }+\epsilon ^{-(\eta +\beta )}-2\cos \xi \right)$

(16)

Then if we make

$V=A_{0}F+A_{1}{\frac {dV}{d\eta }}+A_{e}{\frac {d^{2}F}{d\eta ^{2}}}+\mathrm {etc.}$

(17)

by giving proper values to the coefficients $A$ we may express any potential which is a function of $\eta$ and $\cos \xi$, and does not become infinite except when $\eta +\beta =0$ and $\cos \xi =1$.

When $\beta =0$ the expansion of $F$ in terms of $\rho$ and $\theta$ is

$F_{0}=2\log \rho +{\frac {1}{12}}\rho ^{2}\cos 2\theta -{\frac {1}{1440}}\rho ^{4}\cos 4\theta +\mathrm {etc.}$

(18)

For finite values of $\beta$ the expansion of $F$ is

$F_{\beta }=\beta +2\log \left(1-e^{-\beta }\right)+{\frac {1+e^{-\beta }}{1-e^{-\beta }}}\rho \cos \theta -{\frac {e^{-\beta }}{\left(1-e^{-\beta }\right)^{2}}}\rho ^{2}\cos 2\theta +\mathrm {etc} .$

(19)