Consider a capacitor with circular parallel plates that have radius r and are separated by a distance d. The capacitor is being charged linearly in time with a constant current I so that q(t) = It.

(a) Determine the electric field within the capacitor as a function of time

(b) Determine the magnetic field within the capacitor as a function of time based on Ampere's law in the following form: ∫B^→.ds^→ = μ₀ε₀dΦE/dt which holds where there is no actual charge current as between the capacitor plates. (Hint: The magnetic field lines are circular within the capacitor)

(c) Determine the Poynting vector S^→ = 1/μ₀ E^→ x B^→ on the surface of the cylindrical volume between the capacitor plates (curved part and on the bases of the cylinder). Make a sketch to illustrate the pattern formed by S^→.

(d) Determine the rate at which energy is flowing into the capacitor by calculating ∫S^→.dA^→ over the surface of the cylindrical volume between the capacitor plates (hint: Notice and describe how the integral simply turns into a product of the value |S| on the curved surface time the area of that surface.)

(e) compare the result of problem (d) to the rate at which the energy stored in the capacitor is increasing.