Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/278

 EXAMPLE IV. – Distribution of Electricity near an Edge of a Conductor formed by Two Plane Faces.

191.] In the case of an infinite plane face of a conductor charged with electricity to the surface-density $$\sigma_{0}$$, we find for the potential at a distance $$y$$ from the plane

$V=C-4\pi\sigma_{0}y$

where $$C$$ is the value of the potential of the conductor itself.

Assume a straight line in the plane as a polar axis, and transform into polar coordinates, and we find for the potential

$V=C-4\pi\sigma_{0}ae^{\rho}\sin\theta$

and for the quantity of electricity on a parallelogram of breadth unity, and length $$ae^{\rho}$$ measured from the axis

$E=\sigma_{0}ae^{\rho}$

Now let us make $$\rho=n\rho'$$ and $$\theta=n\theta'$$, then, since $$\rho'$$ and $$\theta'$$ are conjugate to $$\rho$$ and $$\theta$$, the equations

$V=C-4\pi\sigma_{0}ae^{n\rho'}\sin n\theta'$

and

$E=\sigma_{0}ae^{n\rho'}$

express a possible distribution of electricity and of potential.

If we write $$ae^{\rho'}$$, $$r$$ will be the distance from the axis, and $$\theta$$ the angle, and we shall have

$\begin{array}{l} V=C-4\pi\sigma_{0}\frac{r^{n}}{a^{n-1}}\sin n\theta\\ \\ E=\sigma_{0}\frac{r^{n}}{a^{n-1}} \end{array}$|undefined

$$V$$ will be equal to $$C$$ whenever $$n\theta=\pi$$ or a multiple of $$\pi$$.

Let the edge be a salient angle of the conductor, the inclination of the faces being $$\alpha$$, then the angle of the dielectric is $$2\pi-\alpha$$, so that when $$\theta=2\pi-\alpha$$ the point is in the other face of the conductor. We must therefore make

$n(2\pi-\alpha)=\pi$ or $n=\frac{\pi}{2\pi-\alpha}$.

Then

$\begin{array}{l} V=C-4\pi\sigma_{0}a\left(\frac{r}{a}\right)^{\frac{\pi}{2\pi-\alpha}}\sin\frac{\pi\theta}{2\pi-a}\\ \\ E=\sigma_{0}a\left(\frac{r}{a}\right)^{\frac{\pi}{2\pi-\alpha}} \end{array}$|undefined

The surface-density $$\sigma$$ at any distance $$r$$ from the edge is

$\sigma=\frac{dE}{dr}=\frac{\pi}{2\pi-\alpha}\sigma_{0}\left(\frac{r}{a}\right)^{\frac{\alpha-\pi}{2\pi-\alpha}}$|undefined