Current Density — Question 6

Question 6

A spatially increasing current density depletes local charge

Within a slab $0<x<L$, $\vec J=(J_0+ax)\hat x$, where a is constant in time and has units A/m$^3$. Initially $\rho_q=\rho_0$. There is no lateral flux.

1. Use $\partial\rho_q/\partial t+\nabla\cdot\vec J=0$ to find the charge density.

2. For slab area A, verify the total charge change directly from endpoint currents.

Question 6 - Solution

A spatially increasing current density depletes local charge

(a) Method and calculation

$$\nabla\cdot\vec J=\partial J_x/\partial x=a.$$$$\rho_q(x,t)=\rho_0-at.$$

(b) Interpretation and further calculation

$$I_{\mathrm{in}}=AJ_0,\quad I_{\mathrm{out}}=A(J_0+aL).$$$$dQ/dt=I_{\mathrm{in}}-I_{\mathrm{out}}=-aAL=AL\,\partial\rho_q/\partial t.$$

Physical check

Positive divergence removes positive charge from a volume; it does not create charge.