19.6 Charging and Discharging of a Capacitor | Electric Potential And Capacitor | Physics 12

When a capacitor is connected to a DC source through a resistor, it does not charge instantaneously. Similarly, when it is disconnected and allowed to discharge through a resistor, it does not lose its charge instantly. Both processes follow exponential behaviour governed by the product $R C$ .

The Time Constant ( $τ$ )

The time constant of an RC circuit is defined as:

$τ = R C$

where $R$ is resistance in ohms ( $Ω$ ) and $C$ is capacitance in farads (F).

Unit of $R C$ :

$[R C] = [Ω] [F] = [\frac{V}{A}] [\frac{C}{V}] = [\frac{C}{A}] = [\frac{C}{C / s}] = [s]$

So $R C$ has the SI unit of seconds.

Charging of a Capacitor

When a capacitor $C$ is connected in series with a resistor $R$ and a battery of EMF $E$ , charge builds up on the capacitor. The charge $q$ at any time $t$ is:

$q = q_{0} (1 - e^{- t / R C})$

where $q_{0} = C E$ is the maximum (equilibrium) charge.

The voltage across the capacitor during charging:

$V = V_{0} (1 - e^{- t / R C})$

Key values:

At $t = 0$ : $q = 0$ (uncharged)
At $t = R C$ : $q = q_{0} (1 - e^{- 1}) \approx 0.632 q_{0}$ (63.2% of maximum)
At $t = 5 R C$ : $q \approx 0.993 q_{0}$ (practically fully charged)

Discharging of a Capacitor

When a fully charged capacitor (initial charge $q_{0}$ , initial voltage $V_{0}$ ) is discharged through a resistor $R$ , the charge, voltage, and current all decay exponentially:

$q = q_{0} e^{- t / R C}$

$V = V_{0} e^{- t / R C}$

$I = I_{0} e^{- t / R C} where I_{0} = \frac{V _{0}}{R}$

Key values during discharge:

At $t = 0$ : $q = q_{0}$ , $V = V_{0}$ , $I = I_{0}$
At $t = R C$ : $q = q_{0} e^{- 1} \approx 0.368 q_{0}$ (36.8% of initial)
At $t = 2 R C$ : $q \approx 0.135 q_{0}$ (13.5% of initial)
At $t = 5 R C$ : $q \approx 0.007 q_{0}$ (practically fully discharged)

Graphs of V, Q, and I vs Time

During Charging:

$Q$ – $t$ and $V$ – $t$ : exponential rise, starting at zero and asymptotically approaching $q_{0}$ or $V_{0}$ .
$I$ – $t$ : exponential decay from $I_{0} = V_{0} / R$ toward zero (as the capacitor opposes further charging).

During Discharging:

$Q$ – $t$ , $V$ – $t$ , and $I$ – $t$ : all show exponential decay from their initial values toward zero.

All curves have the same time constant $τ = R C$ .

Effect of $R C$ on graph shape:

Small $R C$ : steep curve — rapid charging/discharging.
Large $R C$ : flat curve — slow charging/discharging.

Unit Proof for $R C$

The Time Constant ( $τ$ )

The time constant of an RC circuit is defined as:

$τ = R C$

where $R$ is resistance in ohms ( $Ω$ ) and $C$ is capacitance in farads (F).

Unit of $R C$ :

$[R C] = [Ω] [F] = [\frac{V}{A}] [\frac{C}{V}] = [\frac{C}{A}] = [\frac{C}{C / s}] = [s]$

So $R C$ has the SI unit of seconds.

Charging of a Capacitor

When a capacitor $C$ is connected in series with a resistor $R$ and a battery of EMF $E$ , charge builds up on the capacitor. The charge $q$ at any time $t$ is:

$q = q_{0} (1 - e^{- t / R C})$

where $q_{0} = C E$ is the maximum (equilibrium) charge.

The voltage across the capacitor during charging:

$V = V_{0} (1 - e^{- t / R C})$

Key values:

At $t = 0$ : $q = 0$ (uncharged)
At $t = R C$ : $q = q_{0} (1 - e^{- 1}) \approx 0.632 q_{0}$ (63.2% of maximum)
At $t = 5 R C$ : $q \approx 0.993 q_{0}$ (practically fully charged)

Discharging of a Capacitor

When a fully charged capacitor (initial charge $q_{0}$ , initial voltage $V_{0}$ ) is discharged through a resistor $R$ , the charge, voltage, and current all decay exponentially:

$q = q_{0} e^{- t / R C}$

$V = V_{0} e^{- t / R C}$

$I = I_{0} e^{- t / R C} where I_{0} = \frac{V _{0}}{R}$

Key values during discharge:

At $t = 0$ : $q = q_{0}$ , $V = V_{0}$ , $I = I_{0}$
At $t = R C$ : $q = q_{0} e^{- 1} \approx 0.368 q_{0}$ (36.8% of initial)
At $t = 2 R C$ : $q \approx 0.135 q_{0}$ (13.5% of initial)
At $t = 5 R C$ : $q \approx 0.007 q_{0}$ (practically fully discharged)

Graphs of V, Q, and I vs Time

During Charging:

$Q$ – $t$ and $V$ – $t$ : exponential rise, starting at zero and asymptotically approaching $q_{0}$ or $V_{0}$ .
$I$ – $t$ : exponential decay from $I_{0} = V_{0} / R$ toward zero (as the capacitor opposes further charging).

During Discharging:

$Q$ – $t$ , $V$ – $t$ , and $I$ – $t$ : all show exponential decay from their initial values toward zero.

All curves have the same time constant $τ = R C$ .

Effect of $R C$ on graph shape:

Small $R C$ : steep curve — rapid charging/discharging.
Large $R C$ : flat curve — slow charging/discharging.