Problem 1. This problem illustrates how the Poynting vector measures electromagnetic energy.r ow into a volume of space. Here that volume is the space between two capacitor plates. It synthesizes man}.r things we have studied separately, like RC circuits, parallel plate capacitors, and the AmpereMaxwell law. An RC circuit consists of an ideal battery of voltage V, a switch S, a resistor R and a parallelplate capacitor with circular plates of radius '0, lled with air, as shown in the gure below. The distance between the plates is d, and initially there is no charge on the capacitor's plates. After having been open for a long time, the switch is closed at t = 0. (a) What is the voltage across capacitor, V56}, as a function of time as it charges up? (b) What is the energy of the electlic eld, U5, stored in the (cylindrical) volume of the capacitor? Is this energy increasing or decreasing? What is the rate of change of energy, dUgfdt? (c) What is the magnitude of the electric eld between the plates, EU)? (Neglect edge effects.) (d) What is the magnetic eld BU", t) inside the capacitor, at time t and at the distance r from the center of the circular plates? Hint: use the AmpereMaxwell law. (e) How are E and E oriented? How is the Poynting vector 5" = $5,\" x E" oriented? What is the magnitude of the Poynting vector n], t} at the edge of the volume of the capacitor? (f) What is the integral I . dji', over the (cylindrical) surface that encloses the volume of the capacitor? This measures the ux of energy per unit time into this volume. Consider both the \"barrel\" and the \"endcaps\" of this volume at the capacitors plates. (Hint: howr is g oriented with respect to the area vector A? of each base?) (g) How does f 3' . dial compare to dUEfdt you computed in part {b}