Consider a parallel plate capacitor that has circular plates

Consider a parallel plate capacitor that has circular plates. Let the radius of the plates be R, and the distance between the plates to be d. Suppose that the capacitor is discharging through a resistor of resistance R. a) Find an expression for the Poynting vector S~ on the side of the capacitor. This can be done by determining E~ and B~ at the side surface of the capacitor. b) Integrate S over the cylindrical surface to find the rate at which energy is leaving the capacitor.

Solution

Magnetic Field and Poynting Flux in a Charging Capacitor- When a circular capacitor with radius and plate separation is charged up, the electric field , and hence the electric flux , between the plates changes. According to Ampère\'s law as extended by Maxwell, this change in flux induces a magnetic field that can be found from integral of B * dl = Mu 0 (i + Epsilon 0 (delta flux/delta t)) = ampere maxwell law We can solve this equation to obtain the field inside a capacitor: B(r) = Mu 0 (ir/2piR^2) theta where r is the radial distance from the axis of the capacitor.

S = 1/Mu 0 (E X B) = S(t) = (i^2/(2pi^2R^3epsilon 0))t