Problem 1. This problem illustrates how the Poynting vector measures electromagnetic energy ow into a volume of space. Here that volume is the space between two capacitor plates. It synthesizes many things we have studied separately, like RC circuits, parallel plate capacitors, and the Ampere-Maxwell law. An RC circuit consists of an ideal battery of voltage V, a switch 3, a resistor R and a parallel-plate capacitor with circular plates of radius :r'u, 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 = U. (a) (b) S R 'What is the voltage across capacitor, VCR), as a function of time as it charges up? 1What is the energy of the electric eld, UE, stored in the (cylindrical) volume of the capacitor? Is this energy increasing or decreasing? "What is the rate of change of energy, dUE / dt? 1What is the magnitude of the electric eld between the plates, E[t)? (Neglect edge effects.) 1What 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 Ampere-Maxwell law. r p How are E and El oriented? How is the Poynting vector S = g x E oriented? What is the magnitude of the Poynting vector [rg,t) at the edge of the volume of the capacitor? (f) What is the integral f S . dA, over the (cylindrical) surface that encloses the volume of the capacitor? This measures the flux of energy per unit time into this volume. Consider both the "barrel" and the "endcaps" of this volume at the capacitor's plates. (Hint: how is S oriented with respect to the area vector A of each base?) (g) How does f S . dA compare to dUE/dt you computed in part (b)? I hope they come out to be the same