Answer Happy

to Problem 1: Energy in a charging capacitor R A long, straight cylindrical wire of radius R carries a con- stant current I uniformly distributed across it. The wire is cut neatly across its diameter, leaving a narrow gap of width d << R. As no current flows through the gap, charge builds up on either side of it over time. d (a) Assuming that there is no accumulated charge at time t = 0, find the electric field Ē(t) in the gap at all subsequent times. From this, determine the displacement current density in the gap. (b) Find the magnetic field in the gap as a function of distance s≤R from the wire's central axis. (c) Find both the electromagnetic energy density u and the Poynting vector 5 in the gap. (Take note of the direction of S; it may surprise you!) Verify that Poynting's theorem is satisfied in differential form, 3+5= 0, within the gap. (d) Calculate the total electromagnetic energy W(t) within the gap as a function of time. Likewise, compute the power flowing into the gap using the Poynting vector. Verify that the power input is equal to the rate of change of the energy (Poynting's theorem in integral form). dW