Displacement Current — Question 7

Question 7

Charge conservation makes the generalized current divergence-free

In vacuum, local charge conservation is $\nabla\cdot\mathbf J=-\partial_t\rho$ and Gauss law is $\nabla\cdot\mathbf E=\rho/\epsilon_0$.

1. Show that $\mathbf J+\epsilon_0\partial_t\mathbf E$ has zero divergence.

2. Integrate over a fixed volume to relate net outward conduction and displacement currents.

Question 7 - Solution

Charge conservation makes the generalized current divergence-free

(a) Method and calculation

$$\nabla\cdot(\mathbf J+\epsilon_0\partial_t\mathbf E)=-\dot\rho+\epsilon_0\partial_t(\rho/\epsilon_0)=0.$$$$\boxed{\nabla\cdot\mathbf J_{\mathrm{generalized}}=0}.$$

(b) Interpretation and further calculation

$$\oint\mathbf J\cdot d\mathbf A=-dQ_{\mathrm{enc}}/dt,\quad\oint\epsilon_0\partial_t\mathbf E\cdot d\mathbf A=dQ_{\mathrm{enc}}/dt.$$$$\boxed{I_{c,\mathrm{out}}+I_{d,\mathrm{out}}=0}.$$

Physical check

Conduction current alone can converge where charge accumulates; the generalized current has no such divergence.