Suppose J(r) is constant in time but ρ(r,t) is not—conditions that might prevail, for instance, during the charging of a capacitor.
(a) Show that the charge density at any particular point is a linear function of time:
ρ(r,t)=ρ(r,0)+ρ˙(r,0)t,
where ρ˙(r,0) is the time derivative of ρ at t=0. [Hint: Use the continuity equation.]
(b) Show that
B(r)=4πμ0∫r2J(r′)×r^dτ′
obeys Amp\`ere's law with Maxwell's displacement current term.