Problem 60P

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.

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