Gauss' Law — Question 10

Question 10

A changing electric flux determines the net charge current through a boundary

Through a fixed closed surface the outward electric flux is $\Phi(t)=\Phi_0e^{-t/\tau}$ for $t\ge0$, with $\Phi_0,\tau>0$. Define outward conventional charge current $I_{\mathrm{out}}=-dQ_{\mathrm{enc}}/dt$. Assume Gauss law holds at each time.

1. Derive the enclosed charge and outward current as functions of time.

2. Find the total charge leaving between 0 and $\tau$, and the time when half the initial enclosed charge remains.

Question 10 - Solution

A changing electric flux determines the net charge current through a boundary

(a) Method and calculation

$$Q_{\mathrm{enc}}(t)=\epsilon_0\Phi_0e^{-t/\tau}.$$$$I_{\mathrm{out}}(t)=-\epsilon_0\frac{d\Phi}{dt}=\frac{\epsilon_0\Phi_0}{\tau}e^{-t/\tau}>0.$$

(b) Interpretation and further calculation

$$Q_{\mathrm{left}}(0,\tau)=\int_0^\tau I_{\mathrm{out}}\,dt=\epsilon_0\Phi_0(1-e^{-1}).$$$$Q(t_{1/2})=\tfrac12Q(0)\Rightarrow t_{1/2}=\tau\ln2.$$Electric flux is not itself a flow of charge; charge conservation relates its time derivative to the net current here.

Physical check

The decreasing positive enclosed charge corresponds to a positive net outward conventional current.