Exercises

1. Two concentric spheres have radii $a$ , $b$ ($b>a$ ) and are each divided into two hemispheres by the same horizontal plane. The upper hemisphere of the inner sphere and the lower hemisphere of the outer sphere are maintained at potential $V$ . The other hemispheres are at zero potential. Demonstrate that the potential in the region $a\leq r\leq b$ can be written
   $$\phi(r,\theta) = \sum_{l=0,\infty} \left(\alpha_l\,r^{\,l} +\beta_l\,r^{-l-1}\right)P_l(\cos\theta),$$
   where

   $$\alpha_l = \frac{V}{2}\left[\frac{a^{\,l+1}-(-1)^l\,b^{\,l+1}}{a^{\,2\,l+1}-b^{\,2\,l+1}}\right]\left[P_{l-1}(0)-P_{l+1}(0)\right],$$

   $$\beta_l = \frac{V}{2}\left[\frac{a^{-l}-(-1)^l\,b^{-l}}{a^{-2\,l-1}-b^{-2\,l-1}}\right]\left[P_{l-1}(0)-P_{l+1}(0)\right].$$

   
   
   Here, $r$ , $\theta$ , $\varphi$ are conventional spherical coordinates whose origin coincides with the common center of the spheres, and are such that the dividing plane corresponds to $\theta=\pi/2$ .

2. A spherical surface of radius $R$ has charge uniformly distributed over its surface with density $Q/4\pi\,R^{\,2}$ , except for a spherical cap at the north pole, defined by the cone $\theta=\alpha$ . Here, $r$ , $\theta$ , $\varphi$ are conventional spherical coordinates whose origin coincides with the center of the surface.
   1. Show that the potential inside the spherical surface can be expressed as
      $$\phi(r,\theta) = \frac{Q}{8\pi\,\epsilon_0}\sum_{l=0,\infty}\frac... ...\alpha)-P_{l-1}(\cos\alpha)\right]\frac{r^{\,l}}{R^{\,l+1}}\, P_l(\cos\theta),$$
      where $P_{-1}(\cos\alpha)=-1$ .
   2. Show that the electric field at the origin is
      $${\bf E}({\bf0}) = \frac{Q}{16\,\pi\,\epsilon_0\,R^{\,2}}\,\sin^2\alpha\,{\bf e}_z.$$

   3. Show that in the limit $\alpha\rightarrow 0$ ,
      $$\phi(r,\theta) \rightarrow \frac{Q}{4\pi\,\epsilon_0\,R}-\frac{Q\... ...6\pi\,\epsilon_0\,R}\sum_{l=0,\infty}\frac{r^{\,l}}{R^{\,l}}\,P_l(\cos\theta).$$

   4. Show that in the limit $\alpha\rightarrow\pi$ ,
      $$\phi(r,\theta) \rightarrow\frac{Q\,(\pi-\alpha)^{\,2}}{16\pi\,\epsilon_0\,R}\sum_{l=0,\infty}(-1)^l\,\frac{r^{\,l}}{R^{\,l}}\,P_l(\cos\theta).$$

3. The Dirichlet Green's function for the unbounded space between planes at $z=0$ and $z=L$ allows a discussion of a point charge, or a distribution of charge, between parallel conducting planes held at zero potential.
   1. Using cylindrical coordinates, show that one form of the Green's function is

      $$G({\bf r},{\bf r}') =-\frac{1}{\pi\,L}\sum_{n=1,\infty}\sum_{m=-\infty,\infty}\sin\le... ...\sin\left(\frac{n\,\pi}{L}\,z'\right)\,{\rm e}^{\,{\rm i}\,m\,(\theta-\theta')}$$

      $$\phantom{==}I_m\left(\frac{n\,\pi}{L}\,r_<\right)K_m\left(\frac{n\,\pi}{L}\,r_>\right).$$

      
      

   2. Show that an alternative form of the Green's function is
      $$G({\bf r}, {\bf r}')=-\frac{1}{2\pi}\sum_{m=-\infty,\infty}\int_0... ...h(k\,L)}\,J_m(k\,r)\,J_m(k\,r')\,{\rm e}^{\,{\rm i}\,m\,(\theta-\theta')}\,dk.$$

4. From the results of the previous exercise, show that the potential due to a point charge $q$ placed between two infinite parallel conducting planes held at zero potential can be written as
   $$\phi(z,r)= \frac{q}{\pi\,\epsilon_0\,L}\sum_{n=1,\infty}\sin\left... ...ght)\sin\left(\frac{n\,\pi}{L}\,z\right)\,K_0\left(\frac{n\,\pi}{L}\,r\right),$$
   where the planes are at $z=0$ and $z=L$ , and the charge is on the $z$ -axis at $z=z_0$ . Show that induced surface charge densities on the lower and upper planes are

   $$\sigma_-(r) = - \frac{q}{\pi\,L}\sum_{n=1,\infty}\left(\frac{n\,\pi}{L}\right)\sin\left(\frac{n\,\pi}{L}\,z_0\right)\,K_0\left(\frac{n\,\pi}{L}\,r\right),$$

   $$\sigma_+(r) = \frac{q}{\pi\,L}\sum_{n=1,\infty}\cos(n\,\pi)\left(\frac{n\,\pi... ...t) \sin\left(\frac{n\,\pi}{L}\,z_0\right)\,K_0\left(\frac{n\,\pi}{L}\,r\right),$$

   
   
   respectively.

5. Show that the potential due to a conducting disk of radius $a$ carrying a charge $q$ is
   $$\phi(r,z) =\frac{q}{4\pi\,\epsilon_0\,a}\int_0^\infty e^{-k\,\vert z\vert}\,J_0(k\,r)\,\frac{\sin(k\,a)}{k}\,dk$$
   in cylindrical coordinates (whose origin coincides with the center of the disk, and whose symmetry axis coincides with that of the disk.)

6. A conducting spherical shell of radius $a$ is placed in a uniform electric field ${\bf E}$ . Show that the force tending to separate two halves of the sphere across a diametral plane perpendicular to ${\bf E}$ is given by
   $$F = \frac{9}{4}\,\pi\,\epsilon_0\,a^{\,2}\,E^{\,2}.$$