Displacement Current — Question 8

Question 8

An isolated leaky capacitor can have cancelling internal generalized currents

A homogeneous dielectric with permittivity epsilon and conductivity sigma fills an isolated charged parallel-plate capacitor. Neglect edge effects, magnetic transients, and external leads. Plate charge relaxes through the material.

1. Derive E(t) from conservation of free surface charge.

2. Find conduction and displacement current densities and their sum.

Question 8 - Solution

An isolated leaky capacitor can have cancelling internal generalized currents

(a) Method and calculation

$$Q=\epsilon AE,\quad\dot Q=-\sigma AE\Rightarrow\dot E=-(\sigma/\epsilon)E.$$$$\boxed{E(t)=E_0e^{-t/\tau},\quad\tau=\epsilon/\sigma}.$$

(b) Interpretation and further calculation

$$J_c=\sigma E,\quad J_d=\epsilon\dot E=-\sigma E.$$$$\boxed{J_c+J_d=0\quad\text{in this uniform relaxation model}}.$$

Physical check

Finite leakage transport can coexist with zero total local source term in the Ampere-Maxwell equation.