Consider a very long solenoid/inductor of length D and radius a, with n coils per unit length. Recall that the magnetic field inside the solenoid is uniform and is given by B = μo n I, where I is the electrical current running through the wire. Note that D >> a. The current is a known function of time, /(t). If the time scale of current variations is large then the displacement current in Ampere's law can be neglected and the stored electric field energy in the solenoid is much less than the magnetic energy. For purposes of directions and signs in diagrams, assume that the time rate of change of /(t) is positive. Answer the following questions. Show all work, diagrams might help, as might a few words. (a) Find an expression for the electric field inside the solenoid as a function of radial distance, r, from the axis and time t. What is the direction of the E-field? (And the direction of the B-field.) (b) Find the Poynting vector, S, as a function of r in the interior of the solenoid? What is its direction? What is the divergence of the Poynting vector? (c) Find the time rate of change of the EM energy density (neglect the electric field part). Show that this agrees with part (b) and satisfies the Poynting theorem. (d) If you still have time, think beyond the interior of the solenoid and include the wire(s). Where might the power in the interior come from? How does this fit in with the Poynting theorem?

Fig: 1