What is an example of an RC circuit practice problem?

Sample practice problem -A08 A resistor $R = 10 \Omega$ is connected with $20 \mu F \mathmr{and} 10 \mu F$ capacitors and a $10 V$ DC battery as shown below. Analyze the circuit to find 1. Current in the loop after three time constants 2. Steady state energy stored in each capacitor. Aug 25, 2017

Equivalent capacitance $C$ of both capacitors connected in series is

${C}_{T} = \frac{{C}_{1} \times {C}_{2}}{{C}_{1} + {C}_{2}} = \frac{20 \times 10}{20 + 10} = \frac{200}{30} = 6. \overline{6} \mu F$

Time constant $\tau$ of the circuit $R C = \left(10\right) \left(6. \overline{6} \times {10}^{-} 6\right) = 6. \overline{6} \times {10}^{-} 5 s$
Current $I$ in the circuit is given by the expression

$I = {I}_{0} {e}^{- \frac{t}{\tau}}$
where ${I}_{0}$ is the initial current in the loop as the switch is closed.

${I}_{0} = \frac{\varepsilon}{R} = \frac{10}{10} = 1 A$

1. Current after three time constants is

$I = {I}_{0} {e}^{- \frac{3 \tau}{\tau}}$
$\implies I = 1 \times {e}^{- 3}$
$\implies I = 0.05 A$

2. We know that
${C}_{T} = \frac{Q}{\varepsilon}$
$\implies Q = 6. \overline{6} \times {10}^{-} 6 \times 10 = 6. \overline{6} \times {10}^{-} 5 C$

In the steady state there is no current flowing in the loop. Hence, voltage drop across resistor $= 0$. Voltage of battery is divided across capacitors in the inverse fraction of respective capacitances.

${V}_{10} = \varepsilon \times {C}_{T} / {C}_{10} = 10 \times \frac{6. \overline{6}}{10} = 6. \overline{6} V$

The balance voltage appears across other capacitor.

$\therefore {V}_{20} = 10 - 6. \overline{6} = 3. \overline{3} V$

Now energy stored in a capacitor is given by the expression

$E = \frac{1}{2} C {V}^{2}$
$\therefore {E}_{10} = \frac{1}{2} \times \left(10 \times {10}^{-} 6\right) {\left(6. \overline{6}\right)}^{2}$
$\implies {E}_{10} = 2. \overline{2} \times {10}^{-} 4 J$

Similarly

${E}_{20} = \frac{1}{2} \times \left(20 \times {10}^{-} 6\right) {\left(3. \overline{3}\right)}^{2} = 1. \overline{1} \times {10}^{-} 4 J$