Concepts/Formulas used:

Charge on Capacitor during Charging (RC Circuit):

A capacitor of capacitance with charge C is being charged with a battery of emf ϵ through a resistor of resistance R. A switch S is also connected in series with the capacitor. The switch is initially open. The capacitor is uncharged at first. At t=0, the switch is closed. The charge on the capacitor at any time t>0 is given by:

where

Note that the capacitor begins charging at t = 0.

Energy stored in a capacitor:

The energy stored in a capacitor with capacitance C , charge is given by:

where V is the potential difference across the capacitor.

Let a capacitor of capacitance C with no initial charge be attached to a battery of emf ϵ through a resistor of resistance R. Switch S in attached is series and in initially open. It is closed at t = 0.

Equilibrium is when no current flows and the potential across the battery is the same as the potential across the capacitor. The energy stored at equilibrium is:

Suppose the capacitor begins charging at t = 0.

Now, at any time t > 0, the energy stored is

Substituting the value for Q(t),

We want to find t when

Now, or

We reject the former as the capacitor begins charging at t = 0.

Hence, 1.23 time constants elapse .