Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/290

The total charge is $CV$, and the attraction towards the infinite plane is

${\begin{array}{l}-{\frac {1}{2}}V^{2}{\frac {dC}{db}}=V^{2}{\frac {ac}{4\pi b^{2}}}\left(1+{\frac {\frac {A}{a}}{1+{\frac {A}{a}}\log {\frac {a}{A}}}}+e^{-{\frac {a}{A}}}+\mathrm {etc.} \right)\\\\\qquad ={\frac {V^{2}c}{4\pi b^{2}}}\left\{a+{\frac {b}{\pi }}-{\frac {b^{2}}{\pi ^{2}a}}\log {\frac {a}{A}}+\mathrm {etc.} \right\}\end{array}}$

The equipotential lines and lines of force are given in Fig. XII.

Fig. XII.

EXAMPLE VIII. – Theory of a Grating of Parallel Wires. Fig. XIII.

203.] In many electrical instruments a wire grating is used to prevent certain parts of the apparatus from being electrified by induction. We know that if a conductor be entirely surrounded by a metallic vessel at the same potential with itself, no electricity can be induced on the surface of the conductor by any electrified body outside the vessel. The conductor, however, when completely surrounded by metal, cannot be seen, and therefore, in certain cases, an aperture is left which is covered with a grating of fine wire. Let us investigate the effect of this grating in diminishing the effect of electrical induction. We shall suppose the grating to consist of a series of parallel wires in one plane and at equal intervals, the diameter of the wires being small compared with the distance between them, while the nearest portions of the electrified bodies on the one side and of the protected conductor on the other are at distances from the plane of the screen, which are considerable compared with the distance between consecutive wires.

204.] The potential at a distance $r'$ from the axis of a straight wire of infinite length charged with a quantity of electricity $\lambda$ per unit of length is

$V=-2\lambda \log r'+C$

(1)

We may express this in terms of polar coordinates referred to an axis whose distance from the wire is unity, in which case we must make

$r'^{2}=1+2r\cos \theta +r^{2}$

(2)

and if we suppose that the axis of reference is also charged with the linear density $\lambda '$, we find

$V=-\lambda \log \left(1-2r\cos \theta +r^{2}\right)-2\lambda '\log r+C$

(3)

If we now make

$r=e^{2\pi {\frac {y}{a}}},\ \theta ={\frac {2\pi x}{a}}$

(4)