JEE Main & Advanced Physics Electrostatics & Capacitance Charging and Discharging of Capacitor in Series RC Circuit

As shown in the following figure (A) when switch S is closed, capacitor start charging. In this transient state potential difference appears across capacitor as well as resistor. When capacitor gets fully charged the entire potential difference appeared across the capacitor and nothing is left for the resistor. [Shown in figure (B)]

(i) Charging : In transient state of charging charge on the capacitor at any instant \[Q={{Q}_{0}}\,\left( 1-{{e}^{\frac{-t}{RC}}} \right)\] and potential difference across the capacitor at any instant \[V={{V}_{0}}\,\left( 1-{{e}^{\frac{-t}{RC}}} \right)\]

(Here Q and V are the instantaneous values of charge and potential difference while maximum charge on capacitor is  \[{{Q}_{0}}=C{{V}_{0}}\])

(ii) Discharging : After the completion of charging, if battery is removed capacitor starts discharging. In transient state charge on the capacitor at any instant \[Q={{Q}_{0}}{{e}^{-t/RC}}\] and potential difference cross the capacitor at any instant \[V={{V}_{0}}{{e}^{-t/CR}}\].

(iii) Time constant \[(\tau )\]: The quantity RC is called the time constant as it has the dimension of time during charging if \[t=\tau =RC\], \[Q={{Q}_{0}}(1-{{e}^{-1}})=0.63\,{{Q}_{0}}=63%\] of \[{{Q}_{0}}\,(\frac{1}{e}=0.37)\]  or  during discharging it is defined as the time during which charge on a capacitor falls to 0.37 times (37%) of the initial charge on the capacitor.