Charging and discharging of a capacitor:
Charging: Let us consider a capacitor C is connected to a battery of emf E through a resistance R. During charging of the capacitor the potential difference across the plates becomes equal to the emf of the battery. Let at any time t, I is the current through the resistance R and q is the charge of capacitor. Using KVL, VC + VR = E
Or, + IR = E
Or, + R = E
Or, R = E –
Or, CRdq = (EC – q)dt
Or, – loge =
Or, 1 – = e– t/RC
Or, q = EC(1 – e– t/RC)
Or, q = q0(1 – e– t/RC) at t = , q = q0 = EC.
= CR is called capacitive time constant, i.e. the time in which charge of the capacitor is 0.632 times of its maximum value during charging.
The charging current is I = = = I0e– t/RC where I0 = = .
At t = 0, I = I0 i.e. initially it acts as short circuit or as a simple conducting wire.
At t = , I = 0 it acts as an open circuit.
Discharging: If a capacitor C with charge q0 is discharged through a resistance R then at any time t, the PD of the resistance is V = IR.
Using KVL, VC + VR = 0
Or, R + = 0
Or, = –
Or, q = q0e– t/RC [This is the charge of the capacitor during discharging.]
The capacitive time constant = CR is the time in which charge becomes i.e. 0.368 times the initial value of charge q0.
The discharging current is I = – = – = I0e– t/RC where I0 = = .
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