Consider the circuit shown in Fig. The circuit elements

QUESTION:

Consider the circuit shown in Fig. P30.76. The circuit elements are as follows: \(\mathcal{E}=32.0 \mathrm{~V}, L=0.640 \mathrm {~H}, C=2.00 ~\mu \mathrm{F} \text {, and } R=400 ~\Omega\). At time t = 0, switch S is closed. The current through the inductor is \(i_{1}\), the current through the capacitor branch is \(i_{2}\), and the charge on the capacitor is \(q_{2}\).

(a) Using Kirchhoff’s rules, verify the circuit equations

\(R\left(i_{1}+i_{2}\right)+L\left(\frac{d i_{1}}{d t}\right)=\mathcal{E}\)

\(R\left(i_{1}+i_{2}\right)+\frac{q_{2}}{C}=\mathcal{E}\)

(b) What are the initial values of \(i_{1}, i_{2} \text {, and } q_{2} ?\)

(c) Show by direct substitution that the following solutions for \(i_{1} \text { and } q_{2}\) satisfy the circuit equations from part (a). Also, show that they satisfy the initial conditions

\(i_{1}=\left(\frac{\mathcal{E}}{R}\right)\left[1-e^{-\beta t}\left\{(2 \omega R C)^{-1} \sin (\omega t)+\cos (\omega t)\right\}\right]\)

\(q_{2}=\left(\frac{\mathcal{E}}{\omega R}\right) e^{-\beta t} \sin (\omega t)\)

Where \(\beta=(2 R C)^{-1} \text { and } \omega=\left[(L C)^{-1}-(2 R C)^{-2}\right]^{1 / 2}\).

(d) Determine the time \(t_{1}\) at which \(i_{2}\) first becomes zero.