A circuit consisting of a capacitor \( C \), a resistor of resistance \( R \), and an ideal battery of emf \( V \), as shown in the figure, is known as an RC series circuit.
As soon as the circuit is completed by closing key \( S_1 \) (keeping \( S_2 \) open), charges begin to flow between the capacitor plates and the battery terminals. The charge on the capacitor increases and consequently the potential difference \( V_C = \frac{q}{C} \) across the capacitor also increases with time. When this potential difference equals the potential difference across the battery, the capacitor is fully charged (\( Q = VC \)). During this process of charging, the charge \( q \) on the capacitor changes with time as:
\[
q = Q\left( 1 - e^{-\frac{t}{RC}} \right)
\]
The charging current can be obtained by differentiating it and using:
\[
I = \frac{dq}{dt} = \frac{V}{R} e^{-\frac{t}{RC}}
\]
Consider the case when \( R = 20 \, \text{k}\Omega \), \( C = 500 \, \mu\text{F} \), and \( V = 10 \, \text{V} \).
