Question

A parallel-plate capacitor whose plates have radius r is being charged through a resistor R by an EMF ε as shown in the Figure. If the switch is closed at t = 0, what is the maximum magnitude of the magnetic field between the plates of the capacitor a distance r/2 from the center?

Answer 1

We know from Ampere's Law (with J = 0 inside the capacitor) that 

If I choose a loop of radius ra within the capacitor then ΦE = Eπra² and

where I have used E = Q/(ε₀A), A = πr², and I = dQ/dt. The maximum magnetic field occurs when the current is maximum and solving for B_max at r_a = r/2 gives

where I used I_max = ε/R (which occurs at t = 0).