Question

A capacitor of capacitance C is given a charge Q. At t = 0, it is connected to an ideal battery of emf ε through a resistance R. Find the charge on the capacitor at time t.

Answer 1

Given:
Initial charge given to the capacitor = Q
When the capacitor is connected to a battery, it will charge through the battery. The initial charge will also decay.
The growth of charge across the capacitor due to the battery of emf E,
$q=CE\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)$
The decay of charge through the capacitor,
$q\text{'}=Q{\mathrm{e}}^{-\frac{t}{RC}}$
The net charge on the capacitor at any time t,
$$\begin{gathered} q_{\mathrm{net}}=q+q\text{'} \\ =CE\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)+Q{\mathrm{e}}^{-\frac{t}{RC}} \end{gathered}$$