Question

1. An 18 µF capacitor has been charged to 100 V. A 15 kΩ resistor and a 5 kΩ resistor, are connected in series with the capacitor.

(a) What is the time constant of this circuit?

(b) Approximately how long (in terms of τ) will it take for the charge stored on the resistor to drop to 0.1% of its original charge?

Once the capacitor is fully discharged the 5 kΩ resistor is removed and replaced with a 12 V battery.

(c) What is the time constant of the circuit now?

(d) Approximately how long will it take for the charge stored on the resistor to rise to 95% of its maximum charge (in terms of τ)?

Answer 1

a) $\tau=(R_1+R_2)C=(15+5)\cdot18=360$ s

b) $q=q_0e^{-t/\tau}$

$0.001q_0=q_0e^{-t/\tau}$

$ln0.001=-t/\tau$

$t=-\tau ln0.001=3\tau ln10$

c) $\tau=15\cdot18=270$ s

d) $q=q_{max}(1-e^{-t/\tau})$

$0.95q_{max}=q_{max}(1-e^{-t/\tau})$

$e^{-t/\tau}=0.05$

$t=-\tau ln0.05$