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

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

and let

Then if we make

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

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