Current Density — Question 9

Question 9

An interface mismatch grows a surface charge sheet

At a flat interface with normal $+\hat x$ from medium 1 to 2, uniform normal currents are $J_{1n}=J_a$ and $J_{2n}=J_a(1-e^{-t/\tau})$. Initially the surface charge density is zero.

1. Find the surface charge density by applying conservation to a thin pillbox.

2. Find the charge accumulated over area A and its long-time limit.

Question 9 - Solution

An interface mismatch grows a surface charge sheet

(a) Method and calculation

$$A\,d\sigma_s/dt=A(J_{1n}-J_{2n})=AJ_ae^{-t/\tau}.$$$$\sigma_s(t)=J_a\tau(1-e^{-t/\tau}).$$

(b) Interpretation and further calculation

$$Q_s=A\sigma_s=AJ_a\tau(1-e^{-t/\tau}).$$$$Q_s(\infty)=AJ_a\tau,\quad d\sigma_s/dt\to0.$$

Physical check

The pillbox turns volume conservation into a surface-charge boundary condition.