The Electrical Properties of Solids — Question 10

Question 10

Charge imbalance relaxes on a conductivity-dependent time scale

In a uniform ohmic material, $\boldsymbol J=\sigma\boldsymbol E$, $\nabla\cdot\boldsymbol E=\rho_q/\varepsilon$, and charge continuity is $\partial_t\rho_q+\nabla\cdot\boldsymbol J=0$. Treat $\sigma,\varepsilon$ as constants.

1. Derive the decay law for a small bulk free-charge imbalance.

2. Find the relaxation time for $\varepsilon=4\varepsilon_0$, $\sigma=2.00\times10^{-8}\,\mathrm{S/m}$ and $\varepsilon_0=8.854\times10^{-12}\,\mathrm{F/m}$.

Question 10 - Solution

Charge imbalance relaxes on a conductivity-dependent time scale

(a) Method and calculation

$$\partial_t\rho_q=-\sigma\nabla\cdot\boldsymbol E=-(\sigma/\varepsilon)\rho_q.$$$$\boxed{\rho_q(t)=\rho_q(0)e^{-t/\tau},\quad\tau=\varepsilon/\sigma}.$$

(b) Interpretation and further calculation

$$\tau=4(8.854\times10^{-12})/(2.00\times10^{-8}).$$$$\boxed{\tau=1.7708\,\mathrm{ms}}.$$

Physical check

The derivation concerns bulk free-charge relaxation under a local linear constitutive law. It does not imply that all surface charge on an isolated object disappears.