An ideal circular parallel plate capacitor of radius a and plate separation d

(a) Calculate the electric and magnetic fields between the plates to second order in powers of the frequency (or wave number), neglecting the effects of fringing fields.

(b) Calculate the volume integrals of we and wm that enter the definition of the reactance X, (6.140), to second order in ?. Show that in terms of the input current I_i defined by I_i = -i?Q, where Q is the total charge on one plate, these energies are

(c) Show that the equivalent series circuit has ? ? ??₀?²/d, L ? ?₀d/8?, and that an estimate for the resonant frequency of the system is ?_res ? 2?2 c/a. Compare with the first root of J₀(x).

I(t) a I(t)