As a capacitor of capacitance C discharges through a resisto

As a capacitor of capacitance C discharges through a resistor R, what (a) is the time dependence of the charge that is initially on the positively charged plate, (b) is the time dependence of the potential between the two plates, and (c) what is the time dependence of the current through the resistor?

Solution

a)Relation between Current through capacitor and PD across it is as follow

    i = C dV/dt

if current through circuit is i then,

applying KVL:

V = - RC dV/dt

dt / (-RC) = dV/V

lnV = -t/RC

starting PD was V0 then

ln(V) - ln(V0) = =t/RC

ln(V/V0) = - t/RC

V/V0 = e^(-t/RC)

V = V0 e^(-t/RC)

and at any time charge on plate is

q = CV

q = CV0 e^(-t/RC)

and C V0 = q0

hence q = q0 e^(-t/RC)

where q0 is the initial charge stored.

b) V = V0 e^(-t/RC)

where V0 = q0/C is the initial PD across capacitor.

c) i = dq/dt

i = q0 e^(-t/RC) / (RC)   = (q0/RC) e^(-t/RC)