A distance d separates the infinitesimally thin and circular plates of a capacitor. The plates have radius

Question:

A distance d separates the infinitesimally thin and circular plates of a capacitor. The plates have radius R >> d and instantaneous charges ±Q(t) as they are slowly discharged by connection to a large resistor (not shown below). Use the non-uniform surface charge distribution of a perfectly conducting circular plate and model the actual current distribution on the right (left) plate using a single surface current density K_R(K_L).

(a) Integrate the continuity equation to find K_L(ρ, t) and K_R(ρ, t).
(b) Use the Amp`ere-Maxwell law to find the magnetic field on both sides of both plates whenρ < R. Ignore fringe fields.
(c) Show that the fields in (b) satisfy the matching conditions with the current densities in (a).