Suppose J(r) is constant in time but ρ(r, t) is not—conditions that might prevail, for i...

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:

where is the time derivative of ρ at t = 0. [Hint: Use the continuity equation.]

This is not an electrostatic or magnetostatic configuration;34 nevertheless, rather surprisingly, both Coulomb’s law (Eq. 2.8) and the Biot-Savart law (Eq. 5.42) hold, as you can confirm by showing that they satisfy Maxwell’s equations. In particular:

(b) Show that

obeys Ampère’s law with Maxwell’s displacement current term.