4.4 Charging of a Capacitor through a resistor:

Let us consider a capacitor of capacitance $(C)$ and a resistor of resistance $(R)$ are connected in a series with a potential $(V)$, as shown in the figure.

Figure 1: Circuit diagram for a charging Capacitor

Initially, there is no charge in the capacitor. i.e. at time $t = 0$, the charge in the capacitor $C$ is also zero.

After a time $(t)$ sec, the charge on a capacitor is $q$. Let the current in the circuit is $I$, with a potential difference across the capacitor is $V_{C}$, and the potential difference across the resistor is $V_{R}$. Then,

$V{_C}$ = $\frac{q}{c}$ $\;\;\;$ and $\;\;\;$ $V_{R}$ = $IR$

If $q_{0}$ is the maximum charge stored in the capacitor, then

$q_{0}$ = $CV$

From Figure, $V$ = $V_{C}$ + $V_{R}$ .......... (i)

or, $\frac{q_{0}}{C}$ = $\frac{q}{C}$ + $IR$

or, $\frac{q_{0}- q}{C}$ = $R$ $\frac{dq}{dt}$

or, $\frac{dq}{q_{0}- q}$ = $\frac{1}{RC}$$dt$ .......... (ii)

Integrating equation (ii), we get

or, $\int_{0}^{q}\frac{dq}{q_{0}- q}$ = $\int_{0}^{t}\frac{1}{RC}$$dt$

or, $[$-$\;\ln(q_{0}-q)$]$_{0}^{q}$ = $\frac{1}{RC}[t]_{0}^{t}$

or, - $\ln(q_{0} - q)$ + $\ln q_{0}$ = $\frac{t}{RC}$

or, $\ln(q_{0} - q) - \ln q_{0}$ = $- \; \frac{t}{RC}$

or, $\ln$$\frac{q_{0} - q}{q_{0}}$ = $- \frac{t}{RC}$

or, $\frac{q_{0} - q}{q_{0}}$ = $e^{-\frac{t}{RC}}$

or, $q_{0} - q$ = $q_{0}\;$$e^{-\frac{t}{RC}}$

or, $q$ = $q_{0}( 1$ - $e^{-\frac{t}{RC}})$ .......... (iii)

A graph between charge $(q)$ and time $(t)$ during charging of a capacitor as shown in figure.

Figure 2: Variation of charge with time in a charging capacitor 

If $t = RC$, then from equation (iii) we get,

or, $q$  =  $q_{0}( 1 -$ $e^{-\frac{RC}{RC}}$$)$  =  $q_{0}(1 -$ $e^{-1}$$)$

or, $q$ = $q_{0}(1 - 0.37)$

or, $q$ = $q_{0} * 0.63$

or, $q$ = $63 \; \%$ of $q_{0}$ .......... (iv)

This is the equation for growth of charge.

∴ The charging time constant (or $R-C$ time constant) of a capacitor is defined as the time interval in which the capacitor charges by about $63 \%$ of its maximum charge.

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