Exercises

1. Prove the mean value theorem: for charge-free space the value of the electrostatic potential at any point is equal to the average of the potential over the surface of any sphere centered on that point.

2. Prove Green's reciprocation theorem: if $\phi$ is the potential due to a volume charge density $\rho$ within a volume $V$ and a surface charge density $\sigma$ on the conducting surface $S$ bounding the volume $V$ , while $\phi'$ is the potential due to another charge distribution $\rho'$ and $\sigma'$ (non-simultaneously occupying the same volume and the same surface, respectively), then
   $$\int_V \rho\,\phi'\,dV + \int_S \sigma\,\phi'\,dS = \int_V \rho'\,\phi\,dV+\int_S\,\sigma'\,\phi\,dS.$$

3. Two infinite grounded parallel conducting planes are separated by a distance $d$ . A point charge $q$ is placed between the planes. Use Green's reciprocation theorem to prove that the total charge induced on one of the planes is equal to $(-q)$ times the fractional perpendicular distance of the point charge from the other plane. [Hint: Choose as your comparison electrostatic problem with the same surfaces one whose charge densities and potential are known and simple.]

4. Consider two insulated conductors, labeled 1 and 2. Let $\phi_1$ be the potential of the first conductor when it is uncharged and the second conductor holds a charge $Q$ . Likewise, let $\phi_2$ be the potential of the second conductor when it is uncharged and the first conductor holds a charge $Q$ . Use Green's reciprocation theorem to demonstrate that
   $$\phi_1 = \phi_2.$$

5. Consider two insulated spherical conductors. Let the first have radius $a$ . Let the second be sufficiently small that it can effectively be treated as a point charge, and let it also be located a distance $b>a$ from the center of the first. Suppose that the first conductor is uncharged, and that the second carries a charge $q$ . What is the potential of the first conductor? [Hint: Consider the result proved in Exercise 1.]

6. Consider a set of $N$ conductors distributed in a vacuum. Suppose that the $i$ th conductor carries the charge $Q_i$ and is at the scalar potential $\phi_i$ . It follows from the linearity of Maxwell's equations and Ohm's law that a linear relationship exists between the potentials and the charges: that is,
   $$\phi_i = \sum_{j=1,N} p_{ij}\,Q_j.$$
   Here, the $p_{ij}$ are termed the coefficients of potential. Demonstrate that $p_{ij} = p_{ji}$ for all $i,j$ . [Hint: Consider the result proved in Exercise 1.] Show that the total electrostatic potential energy of the charged conductors is
   $$W = \frac{1}{2}\sum_{i,j=1,N}p_{ij}\,Q_i\,Q_j.$$

7. Demonstrate that the Green's function for Poisson's equation in two dimensions (i.e., $\partial/\partial z\equiv 0$ ) is
   $$G({\bf r}, {\bf r}') = \frac{\ln\vert{\bf r}-{\bf r}'\vert}{2\pi},$$
   where ${\bf r}= (x,\,y)$ , et cetera. Hence, deduce that the scalar potential field generated by the two-dimensional charge distribution $\rho({\bf r})$ is
   $$\phi({\bf r}) = - \frac{1}{2\pi\,\epsilon_0}\int\rho({\bf r}')\,\ln\vert{\bf r}-{\bf r'}\vert\,dV'.$$

8. A electric dipole of fixed moment ${\bf p}$ is situated at position ${\bf r}$ in a non-uniform external electric field ${\bf E}({\bf r})$ . Demonstrate that the net force on the dipole can be written ${\bf f} = - \nabla W$ , where
   $$W =- {\bf p}\cdot{\bf E}.$$

9. Demonstrate that the electric field generated by an electric dipole of dipole moment ${\bf p}$ is
   $${\bf E}({\bf r}) = \frac{3\,({\bf p}\cdot{\bf r})\,{\bf r}-r^{\,2}\,{\bf p}}{4\pi\,\epsilon_0\,r^{\,5}},$$
   where ${\bf r}$ represents vector displacement relative to the dipole. Show that the potential energy of an electric dipole of moment ${\bf p}_1$ in the electric field generated by a second dipole of moment ${\bf p}_2$ is
   $$W = \frac{r^{\,2}\,({\bf p}_1\cdot{\bf p}_2) - 3\,({\bf p}_1\cdot{\bf r})\,({\bf p}_2\cdot{\bf r})}{4\pi\,\epsilon_0\,r^{\,5}},$$
   where ${\bf r}$ is the displacement of one dipole from another.

10. Show that the torque on an electric dipole of moment ${\bf p}$ in a uniform external electric field ${\bf E}$ is
       $\tau$ $$= {\bf p}\times {\bf E}.$$
    Hence, deduce that the potential energy of the dipole is
    $$W =- {\bf p}\cdot{\bf E}.$$

11. Consider two coplanar electric dipoles with their centers a fixed distance apart. Show that if the angles the dipoles make with the line joining their centers are $\theta$ and $\theta'$ , respectively, and $\theta$ is held fixed, then
    $$\tan\theta=-\frac{1}{2}\,\tan\theta'$$
    in equilibrium.