Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/281

The negative values of $x'$ on the same lines will correspond to values of $x$ less than unity, for which, as we have seen,

$\phi =0,\ \psi =\cos ^{-1}x=\cos ^{-1}e^{\frac {x'}{b}}$

(8)

The properties of the axis of $y$ in the first figure will belong to a series of lines in the second figure parallel to $x'$, for which

$y'=b\pi \left(n'+{\frac {1}{2}}\right)$

(9)

The value of $\psi$ along these lines is $\psi =\pi \left(n'+{\frac {1}{2}}\right)$ for all points both positive and negative, and

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

(10)

194.] If we consider $\phi$ as the potential function, and $\psi$ as the function of flow, we may consider the case to be that of an in definitely long strip of metal of breadth $\pi b$ with a non-conducting division extending from the origin indefinitely in the positive direction, and thus dividing the positive part of the strip into two separate channels. We may suppose this division to be a narrow slit in the sheet of metal.

If a current of electricity is made to flow along one of these divisions and back again along the other, the entrance and exit of the current being at an indefinite distance on the positive side of the origin, the distribution of potential and of current will be given by the functions $\phi$ and $\psi$ respectively.

If, on the other hand, we make $\psi$ the potential, and $\phi$ the function of flow, then the case will be that of a current in the general direction of $y$, flowing through a sheet in which a number of non-conducting divisions are placed parallel to $x$, extending from the axis of $y$ to an indefinite distance in the negative direction.

195.] We may also apply the results to two important cases in statical electricity.

(1) Let a conductor in the form of a plane sheet, bounded by a straight edge but otherwise unlimited, be placed in the plane of $xz$ on the positive side of the origin, and let two infinite conducting planes be placed parallel to it and at distances ${\tfrac {1}{2}}\pi b$ on either side. Then, if $\psi$ is the potential function, its value is for the middle conductor and ${\tfrac {1}{2}}\pi$ for the two planes.

Let us consider the quantity of electricity on a part of the middle conductor, extending to a distance 1 in the direction of $z$, and from the origin to $x=a$.

The electricity on the part of this strip extending from $x_{1}$ to $x_{2}$ is ${\tfrac {1}{4\pi }}\left(\phi _{2}-\phi _{1}\right)$.