Exercises

1. Starting from Equation (201), derive the result $\rho_b = - \nabla \cdot {\bf P}$ .

2. Consider an electron of charge $-e$ moving in a circular orbit of radius $a_0$ about a charge $+e$ in a field directed at right angles to the plane of the orbit. Show that the polarizability $\alpha$ is approximately $4\pi\,a_0^{\,3}$ .

3. A point charge $q$ is located in free space a distance $d$ from the center of a dielectric sphere of radius $a$ ($a<d$ ) and dielectric constant $\epsilon$ . Find the potential at all points in space as an expansion in spherical harmonics. Calculate the rectangular components of the electric field in the vicinity of the center of the sphere.

4. A dielectric sphere of radius $a$ and dielectric constant $\epsilon_1$ is imbedded in an infinite dielectric block of dielectric constant $\epsilon_2$ . The block is placed in a uniform electric field ${\bf E} = E_0\,{\bf e}_z$ . In other words, if $\epsilon_1=\epsilon_2$ then the electric field would be ${\bf E} = E_0\,{\bf e}_z$ . Find the potential both inside and outside the sphere (assuming that $\epsilon_1\neq \epsilon_2$ ), and the distribution of bound charges on the surface of the sphere.

5. An electric dipole of moment ${\bf p} = p\,{\bf e}_z$ lies at the center of a spherical cavity of radius $a$ in a uniform dielectric material of relative dielectric constant $\epsilon$ . Find the electrostatic potential throughout space. Find the bound charge sheet density on the surface of the cavity.

6. A cylindrical coaxial cable consists of a thin inner conductor of radius $a$ , surrounded by a dielectric sheath of dielectric constant $\epsilon_1$ and outer radius $b$ , surrounded by a second dielectric sheath of dielectric constant $\epsilon_2$ and outer radius $c$ , surrounded by a thin outer conductor. All components of the cable are touching. What is the capacitance per unit length of the cable?

7. A very long, right circular, cylindrical shell of dielectric constant $\epsilon$ and inner and outer radii $a$ and $b$ , respectively, is placed in a previously uniform electric field $E_0$ with its axis perpendicular to the field. The medium inside and outside the cylinder has a dielectric constant of unity. Determine the potential in the three regions, neglecting end effects. Discuss the limiting forms of your solutions for a solid dielectric cylinder in a uniform field, and a cylindrical cavity in a uniform dielectric.

8. Suppose that
   $${\bf D} = \epsilon_0 \,$$$\epsilon$ $$\cdot{\bf E},$$
   where the dielectric tensor, $\epsilon$ , is constant (i.e., it is indepedent of ${\bf E}$ ). Demonstrate that
   $$U = \int_{\bf0}^{\bf D }\int_V {\bf E} \cdot \delta{\bf D}\,dV$$
   can only be integrated to give
   $$U = \frac{1}{2} \int_V {\bf E}\cdot{\bf D}\,dV$$
   if $\epsilon$ is symmetric. (Incidentally, because we generally expect a dielectric system to be conservative, this proves that $\epsilon$ must be a symmetric tensor, otherwise the final energy of a dielectric system would not be independent of its past history.)

9. Show that for an electret (i.e., a material of fixed ${\bf P}$ ) the integral $\int {\bf E}\cdot{\bf D}\,dV$ over all space vanishes.

10. Two long, coaxial, cylindrical conducting surfaces of radii $a$ and $b$ ($b>a$ ) are lowered vertically into a liquid dielectric. If the liquid rises a mean height $h$ between the electrodes when a potential difference $V$ is established between them, show that the susceptibility of the liquid is
    $$\chi_e = \frac{(b^{\,2}-a^{\,2})\,\rho_m\,g\,h\,\ln(b/a)}{\epsilon_0\,V^{\,2}}$$
    where $\rho_m$ is the mass density of the liquid, $g$ the acceleration due to gravity, and the susceptibility of air is neglected.