Chapter 3.7, Problem 7P

Problems 7 through 10 deal with the RC circuit in Fig. 3.7.8, containing a resistor (R ohms), a capacitor (C farads), a switch, a source of emf, but no inductor Substitution of $L=0$ in Eq. (3) gives the linear first-order differential equation $R\frac{dQ}{dt}+\frac{1}{C}Q=E\left(t\right)$

for the charge $Q=Q\left(t\right)$ on the capacitor at time t. Note that $I\left(t\right)=Q\text{'}\left(t\right)$ .

(a) Find the charge Q (t ) and current I (t ) in the RC circuit if $E\left(t\right)\equiv E_0$ (a constant voltage supplied by a battery) and the switch is closed at time $t=0$, so that $Q\left(0\right)=0$. (b) Show that $\underset{t\to +\infty }{\lim }Q\left(t\right)=E_{0}C\text{and}\text{that}\underset{t\to +\infty }{\lim }I\left(t\right)=0$