This exercise refers to the electric circuit in Example 5 on page 318 of Apostol. Assume the voltage in the circuit at time is given by

and

If prove the current at time is given by

and

and

*Proof.* By Kirchhoff’s law (page 317 of Apostol) we have

So, for and we have and . If then and so we have

Finally, if then and , so

A sketch of where we let , , and :

When t>b, shouldn’t the A(x) term be integrated on (b,t)?

The integration starts at the initial time, which is 0 here. The upper bound is t.

Just want to point out the graph looks suspicious. Doesn’t an inductor resist change in current, then attempts to maintain current when the source voltage is removed? So how can there be a negative current flow? Also, the term E/R(1-e^-(R(t-a)/L) can never be negative, being E is positive and e^-(R(t-a)/L is < 1.

Never mind :( If I looked carefully I would have seen the equations match the graph, but reading about the counter electromotive force set me straight.

How can it be negative? The book suggests that the L is positive, so the exp(…) is <= 1.