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4.4 Discharging of a Capacitor through Resistor:


To discharge a capacitor through a resistor, let us consider a capacitor of capacitance (C) is initially charged to a potential difference V0 and charge q0, then
q0=CV0 .......... (i)
Now, the charged capacitor is joined to a resistor of resistance (R) in series as shown in figure.
Figure 1: Circuit diagram for discharging Capacitor

At a time (t) after the discharging through (C) has begun, let the VC is the potential across the capacitor and q be the charge. If current (I) flows through R, then we have
VC=VR .......... (ii)
or, qc = IR
or, qc = dqdtR    [∵ I = dqdt;  '-ve' sign shows that q decreases with increasing t]
or, dqq = 1RC
Integrating both sides, we get
or, q0qdqq=t01RC
or, [lnq]qq0 = 1RC[t]t0
or, lnqlnq0 = tRC
or, ln (qq0) = tRC
or, qq0 = etRC
q=q0etRC .......... (iii)

This is called the decay of charge equation. Clearly, q decreases exponentially with time (t) as shown in figure.
Figure 2: Variation of charge with time in a discharging Capacitor

If t = RC in above equation, then from the equation (iii), we get
or, q=q0eRCRC = q0e1 q0e =0.37q0=37% of q0
Thus the discharging time constant may be defined as the time at which the charge on the capacitor during discharging becomes about 37 % of the initial charge.

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