A Charged Isolated Conductor — Question 10

Question 10

A simple bulk-relaxation model explains the approach to zero interior charge density

For a homogeneous conducting medium, assume the local laws $\mathbf J=\gamma\mathbf E$, $\partial\rho/\partial t+\boldsymbol\nabla\cdot\mathbf J=0$, and $\boldsymbol\nabla\cdot\mathbf E=\rho/\epsilon_0$, with constant conductivity $\gamma>0$. Treat these as a quasistatic continuum model.

1. Derive the decay equation and relaxation time for an initial bulk charge density $\rho_0$.

2. For $\gamma/\epsilon_0=4.00$ s$^{-1}$, find the time for the density magnitude to fall to one percent of its initial value. Explain where the displaced charge can go in an isolated finite conductor.

Question 10 - Solution

A simple bulk-relaxation model explains the approach to zero interior charge density

(a) Method and calculation

$$\frac{\partial\rho}{\partial t}=-\gamma\boldsymbol\nabla\cdot\mathbf E=-\frac\gamma{\epsilon_0}\rho.$$$$\rho(t)=\rho_0e^{-t/\tau},\quad\tau=\epsilon_0/\gamma.$$

(b) Interpretation and further calculation

$$\tau=.250\,\mathrm s,\quad e^{-t/\tau}=.0100\Rightarrow t=\tau\ln100\approx1.15129\,\mathrm s.$$$$Q_{\mathrm{total}}=Q_{\mathrm{bulk}}+Q_{\mathrm{surfaces}}=\text{constant}.$$In an isolated finite sample, redistribution toward surfaces can remove bulk excess charge without destroying total charge.

Physical check

The exponential follows from the supplied homogeneous constitutive model; it is not a universal transient description at every microscopic time scale.