The current through an inductor of inductance \(L\) is given by \(I(t)=I_{\max } \sin (\omega t)\).

(a) Derive an expression for the induced emf in the inductor as a function of time.

(b) At \(t=0\), is the current through the inductor increasing or decreasing?

(c) At \(t=0\), is the induced emf opposing or aiding the flow of the charge carriers? (Remember that the direction of a positive induced \(\mathrm{emf}\) is the same as the current direction and the direction of a negative induced emf is opposite the current direction.)

(d) How are the answers to parts \(b\) and \(c\) consistent with the behavior of inductors discussed in the text?