Chapter 7.13, Problem 4MP

The diagram shows a "relaxation" oscillator. The charge $q$ on the capacitor builds up until the neon tube fires and discharges the capacitor (we assume instantaneously). Then the cycle repeats itself over and over.

(a) The charge $q$ on the capacitor satisfies the differential equation $R\frac{dq}{dt}+\frac{q}{c}=V$, where $R$ is the resistance, $C$ is the capacitance, and $V$ is the constant d-c voltage, as shown in the diagram. Show that if $q=0$ when $t=0,$ then at any later time $t$ (during one cycle, that is, before the neon tube fires) $q=CV\left(1-e^{-t/RC}\right)$.

(b) Suppose the neon tube fires at $t=\frac{1}{2}RC.$ Sketch $q$ as a function of $t$ for several cycles.

(c) Expand the periodic $q$ in part (b) in an appropriate Fourier series.