Equation for voltage across capacitor of initial charge Q0 in RC circuit after switch is closed.

Question

We have a circuit with a capacitor C and resistor R with a switch which is open at time t=0. The capacitor has an initial charge of Q₀. What is the voltage across the capacitor as a function of time after the switch is closed?

Answer 1

With the switch closed, after time t, the charge on the capacitor is q and the current through the resistor is -dq/dt. Applying Kirchoff's loop rule to this circuit gives q/C - IR = q/C + dq/dt R = 0 ⇒ -∫dt/(RC) = ∫dq/q ⇒ ln(q/Q₀) = -t/(RC) ⇒ q = Q₀e^-t/(RC) ⇒ V = V₀e^-t/(RC).