### Spring 2007

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Spring 2007 20 Math

A thin wire carries constant current *I* into one plate of a charging capacitor, and another thin wire carries constant current *l* out of the other plate. The capacitor plates are disks of radius *a* and separation *w* « *a* (so edge effects can safely be neglected). The region between the plates has $\epsilon ~ \epsilon_0$, $\mu ~ \mu_0$, but does have a non-negligible, constant conductivity $\sigma$.

*Note:* The capacitor is not an ideal capacitor, since the material between the plates is not a perfect insulator.

(a) Supposing that the charges are uniformly distributed on the plates, find a differential equation for the charge Q(t) on the plates, and solve it for Q(t), taking Q(O) = O.

(b) Find the electric and magnetic fields in the gap. Approximate the electric field as just that due to the charged plates. When computing the magnetic field,

include all sourcing contributions.

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Answer

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