A cylindrical solenoid of radius \(R\) and height \(b\) consists of \(N\) windings. There is a current through the windings, and this current increases with time as \(I=\alpha t\), where \(\alpha\) is a constant.

(a) Describe the directions of the electric field, the magnetic field, and the Poynting vector. 

\((b)\) Determine the magnitude of the Poynting vector at a radial distance \(r=R\) from the center of the long center axis of the solenoid.

(c) Use the result of part \(b\) to derive an expression for the power input to the solenoid as a function of time.

(d) Using the formula for the inductance of an ideal solenoid, calculate the power from the rate of change of the stored magnetic energy. Does this answer agree with your answer in part \(c\) ?