In Section 20.9, in order to find the energy stored in an inductor, we assumed that the current was increased from zero at a constant rate. In this problem, you will prove that the energy stored in an inductor is UL = 1/2LI2 - that is, it only depends on the current I and not on the previous time dependence of the current.
(a) If the current in the inductor increases from i to i + Δi in a very short time Δt, show that the energy added to the inductor is ΔU = LiΔi.
(b) Show that, on a graph of Li versus i, for any small current interval Δi, the energy added to the inductor can be interpreted as the area under the graph for that interval.
(c) Now show that the energy stored in the inductor when a current I flows is U = 1/2LI2.