Q. 72

How many time constants will elapse before the energy stored in the capacitor reaches half of its equilibrium value in a charging RC circuit?

Answer :

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 .


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