Consider a nested shell capacitor whose inner and outer conductors are very close together compared to their radii.
a) Argue that the energy stored in the electric field of such a capacitor is Ufield=1/2CVo2, if Vo is the potential difference. (Hint: Argue that between the plates, the field strength is EVo/d, where dRo -RI is the distance between the conductors. Use the equation for the capacitance of a nested sell capacitor and the fact that Ro Ri, to eliminate d in the equation for E in favor of C. Roughly how much volume does this field occupy?)
b) Imagine we discharge this capacitor by connecting its plates with a wire of resistance R. The rate at which thermal energy is produced in the wire at any given instant of time is
dUth =P =VR2=VC2
dt R R
Where Vc is the time dependent potential difference across the capacitor's plates. If we integrate both sides with respect to t, we find that the total energy converted to thermal energy in the wire as the capacitor discharges is
Uth =1/R∫Vc2dt =1/R∫ (Voe-t/RC)2dt
Where the upper bound is infinity and the lower bound is 0
Do this integral and show that Uth =1/2CVo2, which is consistent with energy initially stored in the field according to part (a)