Question

27.25 In a circuit the switch S has been open for a long time. It is suddenly closed. Take emf=10.0 V, R1=50.0 kOhms, R2=100 kOhms, and C=10.0 uF. Determine the time constant (a) before the switch is closed and (b) after the switch is closed. (c)Let the switch be closed at t=0. Determine the current in the switch as a function of time.

Answer 1

a)

$$\tau=(R_1+R_2)C\\=(50000+100000)10\cdot10^{-6}=1.50\ s$$

b)

$$\tau'=(R_2)C=(100000)10\cdot10^{-6}=1.00\ s$$

c)

$$\epsilon-I_1R_1=0$$

$$I_1=\frac{\epsilon}{R_1}=\frac{10}{50000}=2\cdot10^{-4}A$$

$$\frac{q}{C}-I_2R_2=0$$

$$I_2=\frac{q}{CR_2}=\frac{\epsilon}{R_2}e^{-\frac{t}{\tau'}}=\frac{10}{100000}e^{-\frac{t}{1}}$$

$$I=I_1+I_2=(2\cdot10^{-4}+(1\cdot10^{-4})e^{-t})A$$

$$I=(0.2+0.1e^{-t})\ mA$$