Dielectrics and Gauss' Law — Question 5

Question 5

A homogeneous spherical dielectric has bound surface charge but no bound bulk charge in its gap

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 conducting sphere of radius a carries +Q and a concentric shell of radius b carries -Q. The gap is filled by a uniform linear dielectric of relative permittivity kappa.

1. Find the polarization field and bound volume density in the gap.

2. Find the bound surface densities at a and b and their total charges.

Question 5 - Solution

A homogeneous spherical dielectric has bound surface charge but no bound bulk charge in its gap

(a) Method and calculation

$$D_r=\frac{Q}{4\pi r^2},\quad P_r=(1-1/\kappa)\frac{Q}{4\pi r^2}.$$$$\rho_b=-\frac1{r^2}\frac d{dr}(r^2P_r)=0\quad(a<r<b).$$

(b) Interpretation and further calculation

$$\sigma_b(a)=-(1-1/\kappa)\frac{Q}{4\pi a^2},\quad\sigma_b(b)=+(1-1/\kappa)\frac{Q}{4\pi b^2}.$$$$Q_{b,a}=-(1-1/\kappa)Q,\quad Q_{b,b}=+(1-1/\kappa)Q,\quad Q_{b,\mathrm{net}}=0.$$

Physical check

The outward normal of the dielectric points inward at its inner boundary; missing that sign reverses the inner bound charge.