Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/284

At $n$ times the breadth of the strip on the positive side, the density is less than the normal density by about ${\tfrac {1}{2^{2n+1}}}$.

At $n$ times the breadth of the strip on the negative side, the density is about ${\tfrac {1}{2^{n}}}$ of the normal density.

These results indicate the degree of accuracy to be expected in applying this method to plates of limited extent, or in which irregularities may exist not very far from the boundary. The same distribution would exist in the case of an infinite series of similar plates at equal distances, the potentials of these plates being alternately $+V$ and $-V$. In this case we must take the distance between the plates equal to $B$.

197.] (2) The second case we shall consider is that of an infinite series of planes parallel to $xz$ at distances $B=\pi b$, and all cut off by the plane of $yz$, so that they extend only on the negative side of this plane. If we make $\phi$ the potential function, we may regard these planes as conductors at potential zero.

Let us consider the curves for which $\phi$ is constant.

When $y'=n\pi b$, that is, in the prolongation of each of the planes, we have

$x'=b\log {\frac {1}{2}}\left(e^{\phi }+e^{-\phi }\right)$

(21)

when $y'=\left(n+{\frac {1}{2}}\right)b\pi$, that is, in the intermediate positions

$x'=b\log {\frac {1}{2}}\left(e^{\phi }-e^{-\phi }\right)$

(22)

Hence, when $\phi$ is large, the curve for which is constant is an undulating line whose mean distance from the axis of $y'$ is approximately

$a=b\left(\phi -\log _{e}2\right)$

(23)

and the amplitude of the undulations on either side of this line is

${\frac {1}{2}}b\log {\frac {e^{\phi }+e^{-\phi }}{e^{\phi }-e^{-\phi }}}$

(24)

When $\phi$ is large this becomes $be^{-2\phi }$, so that the curve approaches to the form of a straight line parallel to the axis of $y'$ at a distance $a$ from $ab$ on the positive side.

If we suppose a plane for which $x'=a$, kept at a constant potential while the system of parallel planes is kept at a different potential, then, since $b\phi =a+b\log _{e}2$, the surface-density of the electricity induced on the plane is equal to that which would have been induced on it by a plane parallel to itself at a potential equal to that of the series of planes, but at a distance greater than that of the edges of the planes by $b\log _{e}2$.