How many time constants will elapse before the energy stored in the capacitor reaches half of its equilibrium value in a charging RC circuit?
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:
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
We reject the former as the capacitor begins charging at t = 0.
Hence, 1.23 time constants elapse .
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