Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/291

then, by the theory of conjugate functions,

$V=-\lambda \log \left(1-2e^{\frac {2\pi y}{a}}\cos {\frac {2\pi x}{a}}+e^{\frac {4\pi y}{a}}\right)-2\lambda '\log e^{\frac {2\pi y}{a}}+C$

(5)

where $x$ and $y$ are rectangular coordinates, will be the value of the potential due to an infinite series of fine wires parallel to $z$ in the plane of $yz$, and passing through points in the axis of $x$ for which $x$ is a multiple of $a$.

Each of these wires is charged with a linear density $\lambda$.

The term involving $\lambda '$ indicates an electrification, producing a constant force $-{\frac {4\pi \lambda '}{a}}$ in the direction of $y$.

The forms of the equipotential surfaces and lines of force when $\lambda '=0$ are given in Fig. XIII. The equipotential surfaces near the wires are nearly cylinders, so that we may consider the solution approximately true, even when the wires are cylinders of a diameter which is finite but small compared with the distance between them.

Fig. XIII.

The equipotential surfaces at a distance from the wires become more and more nearly planes parallel to that of the grating.

If in the equation we make $y=b_{1}$, a quantity large compared with $a$, we find approximately,

$V_{1}=-{\frac {4\pi b_{1}}{a}}(\lambda +\lambda ')+C$

(6)

If we next make $y=-b_{2}$ where $b_{2}$ is a negative quantity large compared with $a$, we find approximately,

$V_{2}=-{\frac {4\pi b_{2}}{a}}(\lambda -\lambda ')+C\ \mathrm {nearly} .$

If $c$ is the radius of the wires of the grating, $c$ being small compared with $a$, we may find the potential of the grating itself by supposing that the surface of the wire coincides with the equipotential surface which cuts the plane of $yz$ at a distance $c$ from the axis of $z$. To find the potential of the grating we therefore put $x=c$, and $y=0$, whence

$V=-2\lambda \log 2\sin {\frac {\pi c}{a}}+C$

(8)

205.] We have now obtained expressions representing the electrical state of a system consisting of a grating of wires whose diameter is small compared with the distance between them, and two plane conducting surfaces, one on each side of the grating, and at distances which are great compared with the distance between the wires.