Dielectrics and Gauss' Law — Question 4

Question 4

A radial permittivity gradient creates bound volume charge in a coaxial 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 long coaxial capacitor has radii a,b and inner free line charge lambda. Its gap has $\kappa(r)=\kappa_0r/a$ with positive kappa0. Ignore end effects.

1. Find D, E and capacitance per length.

2. Find the polarization and bound volume charge density in a<r<b using cylindrical divergence.

Question 4 - Solution

A radial permittivity gradient creates bound volume charge in a coaxial gap

(a) Method and calculation

$$D_r=\lambda/(2\pi r),\quad E_r=\frac{\lambda a}{2\pi\epsilon_0\kappa_0r^2}.$$$$V=\frac{\lambda}{2\pi\epsilon_0\kappa_0}(1-a/b),\quad\frac CL=\frac{2\pi\epsilon_0\kappa_0}{1-a/b}.$$

(b) Interpretation and further calculation

$$P_r=D_r-\epsilon_0E_r=\frac{\lambda}{2\pi}\left(\frac1r-\frac{a}{\kappa_0r^2}\right).$$$$\rho_b=-\frac1r\frac d{dr}(rP_r)=-\frac{\lambda a}{2\pi\kappa_0r^3}.$$

Physical check

Free charge is absent from the dielectric volume, but a spatially varying polarization can still produce bound volume charge.