Dielectrics and Gauss' Law — Question 6

Question 6

A uniformly distributed free charge induces compensating bulk and surface bound charges

Use $\mathbf D=\epsilon_0\mathbf E+\mathbf P$, $\nabla\cdot\mathbf D=\rho_f$, $\rho_b=-\nabla\cdot\mathbf P$ and $\sigma_b=\mathbf P\cdot\hat{\mathbf n}$, where n points out of the dielectric. A nonconducting sphere of radius R and uniform relative permittivity kappa contains fixed uniform free volume density rho0. Outside is vacuum; no free surface charge is present.

1. Find the interior D, E and bound volume density.

2. Find the bound surface density and show that the net bound charge is zero.

Question 6 - Solution

A uniformly distributed free charge induces compensating bulk and surface bound charges

(a) Method and calculation

$$4\pi r^2D_r=\frac{4\pi r^3}{3}\rho_0\Rightarrow D_r=\rho_0r/3,\quad E_r=\rho_0r/(3\kappa\epsilon_0).$$$$P_r=(1-1/\kappa)\rho_0r/3\Rightarrow\rho_b=-(1-1/\kappa)\rho_0.$$

(b) Interpretation and further calculation

$$\sigma_b(R)=P_r(R)=(1-1/\kappa)\rho_0R/3.$$$$Q_{b,\mathrm{volume}}=-\frac{4\pi R^3}{3}(1-1/\kappa)\rho_0=-Q_{b,\mathrm{surface}}.$$

Physical check

The bound charges redistribute a neutral dielectric; they do not change its total charge.