Chapter 3.7, Problem 10P

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 $I\left(t\right)=Q\text{'}\left(t\right)$ .

An emf of voltage $E\left(t\right)=E_0\cos \omega t$ is applied to the RC circuit of Fig. 3.7.8 at time $t=0$ (with the switch closed), and $Q\left(0\right)=0$. Substitute $Q_{\text{sp}}\left(t\right)=A\cos \omega t+B\sin \omega t$ in the differential equation to show that the steady periodic charge on the capacitor is $Q_{\text{sp}}\left(t\right)=\frac{E_{0}C}{\sqrt{1+{\omega }^2{R}^2{C}^2}}\cos \left(\omega t-\beta \right)$ where $\beta ={\tan }^{-1}\left(\omega RC\right)$ .