Dielectrics and Gauss' Law — Question 9

Question 9

A constant radial field in a graded spherical dielectric requires free bulk charge

In a spherical annulus a<r<b, relative permittivity is $\kappa(r)=\kappa_0r/a$. A desired field is $\mathbf E=E_0\hat{\mathbf r}$, with E0>0. An inner conductor of radius a supplies charge Qa. Use Gauss law for D.

1. Find the free volume density required in the annulus and the inner conductor charge.

2. Find the voltage drop and total enclosed free charge just inside r=b; verify the latter by adding volume and conductor charges.

Question 9 - Solution

A constant radial field in a graded spherical dielectric requires free bulk charge

(a) Method and calculation

$$D_r=\epsilon_0\kappa_0E_0r/a\Rightarrow\rho_f=\frac1{r^2}\frac d{dr}(r^2D_r)=\frac{3\epsilon_0\kappa_0E_0}{a}.$$$$Q_a=4\pi a^2D_r(a)=4\pi\epsilon_0\kappa_0E_0a^2.$$

(b) Interpretation and further calculation

$$V_a-V_b=E_0(b-a),\quad Q_f(<b)=4\pi b^2D_r(b)=4\pi\epsilon_0\kappa_0E_0b^3/a.$$$$Q_a+\frac{4\pi\rho_f}{3}(b^3-a^3)=4\pi\epsilon_0\kappa_0E_0b^3/a.$$

Physical check

A radial field of constant magnitude has nonzero divergence; it is not equivalent to a spatially uniform Cartesian vector field.