A very long solenoid with n turns per unit length and radius a carries a current i that is increasing at a constant rate of di/dt. (a) Calculate the magnetic field and induced electric field at a point inside the solenoid at a distance r from the solenoid’s axis. (Hint: for the induced electric field, use the integral form of Faraday’s law and consider a circular loop of integration of radius r. What’s the rate of change of magnetic flux through this loop?) 3 (b) Compute the magnitude and direction of the Poynting vector ⃗Sat this point. Show that ⃗Sis directed inward toward the solenoid axis. (c) Consider a cylindrical surface of radius a and length ℓ that coincides with the coils of the solenoid. Integrate ⃗Sover this surface to find the total rate at which electromag- netic energy is flowing into the solenoid through the solenoid walls. (d) Find the magnetic energy stored in a length ℓ of the solenoid, and the rate at which that energy is increasing due to the increase of the current. (e) Compare the rate of change of magnetic-field energy from part (c) with the result of part (d). Discuss why the energy stored in a current-carrying solenoid can be thought of as entering through the cylindrical walls of the solenoid.