20.4.3 AC Through a Capacitor | Alternating Current | Physics 12

When an alternating voltage is applied across a pure capacitor, the circuit behaves very differently from a resistive circuit. Two key phenomena occur: a phase difference between current and voltage, and a frequency-dependent opposition called capacitive reactance.

1. Circuit Analysis

Let the applied alternating voltage be:

$V = V_{0} sin (ω t)$

The charge stored on the capacitor at any instant is:

$q = C V = C V_{0} sin (ω t)$

The instantaneous current is the rate of change of charge:

$I = \frac{d q}{d t} = ω C V_{0} cos (ω t)$

This can be rewritten as:

$I = I_{0} sin (ω t + \frac{π}{2})$

where $I_{0} = ω C V_{0}$ is the peak current.

2. Phase Relationship

Comparing the expressions for $V$ and $I$ :

Quantity	Expression
Voltage	$V_{0} sin (ω t)$
Current	$I_{0} sin (ω t + π /2)$

The current leads the voltage by $9 0^{\circ}$ ( $π /2$ radians) in a purely capacitive AC circuit.

Equivalently, the voltage lags the current by $9 0^{\circ}$ .

Phasor Diagram

In a phasor diagram, the current phasor $I$ is drawn $9 0^{\circ}$ ahead (counter-clockwise) of the voltage phasor $V$ .

3. Capacitive Reactance ( $X_{C}$ )

The capacitive reactance is the opposition offered by a capacitor to the flow of alternating current. It is defined as:

$X_{C} = \frac{V _{0}}{I _{0}} = \frac{V _{rms}}{I _{rms}}$

Substituting $I_{0} = ω C V_{0}$ :

$X_{C} = \frac{1}{ω C} = \frac{1}{2 π f C}$

SI Unit: Ohm ( $Ω$ ) — same as resistance.

4. Effect of Frequency on $X_{C}$

Since $X_{C} = \frac{1}{2 π f C}$ , capacitive reactance is inversely proportional to frequency:

$X_{C} \propto \frac{1}{f}$

Frequency	$X_{C}$	Effect
$f = 0$ (D.C.)	$X_{C} \to \infty$	Capacitor blocks D.C. completely
Low $f$	Large $X_{C}$	Little current flows
High $f$	Small $X_{C}$	Current flows easily

A capacitor blocks D.C. but passes A.C. — the higher the frequency, the more easily it passes.

5. Power in a Purely Capacitive Circuit

The instantaneous power is:

$P = V I = V_{0} sin (ω t) \cdot I_{0} cos (ω t) = \frac{V _{0} I _{0}}{2} sin (2 ω t)$

The average value of $sin (2 ω t)$ over a complete cycle is zero, therefore:

$P_{avg} = 0$

This can also be seen from the power factor:

$P = V_{rms} I_{rms} cos θ = V_{rms} I_{rms} cos (9 0^{\circ}) = 0$

During one quarter-cycle the capacitor stores energy (charges up), and during the next quarter-cycle it returns that energy to the source. No net energy is dissipated.

Summary Table

Property	Purely Capacitive Circuit
Phase of $I$ relative to $V$	$I$ leads $V$ by $9 0^{\circ}$
Reactance formula	$X_{C} = 1/ (2 π f C)$
Effect of increasing $f$	$X_{C}$ decreases
Average power	Zero
Power factor	$cos 9 0^{\circ} = 0$

1. Circuit Analysis

Let the applied alternating voltage be:

$V = V_{0} sin (ω t)$

The charge stored on the capacitor at any instant is:

$q = C V = C V_{0} sin (ω t)$

The instantaneous current is the rate of change of charge:

$I = \frac{d q}{d t} = ω C V_{0} cos (ω t)$

This can be rewritten as:

$I = I_{0} sin (ω t + \frac{π}{2})$

where $I_{0} = ω C V_{0}$ is the peak current.

2. Phase Relationship

Comparing the expressions for $V$ and $I$ :

Quantity	Expression
Voltage	$V_{0} sin (ω t)$
Current	$I_{0} sin (ω t + π /2)$

The current leads the voltage by $9 0^{\circ}$ ( $π /2$ radians) in a purely capacitive AC circuit.

Equivalently, the voltage lags the current by $9 0^{\circ}$ .

Phasor Diagram

In a phasor diagram, the current phasor $I$ is drawn $9 0^{\circ}$ ahead (counter-clockwise) of the voltage phasor $V$ .

3. Capacitive Reactance ( $X_{C}$ )

The capacitive reactance is the opposition offered by a capacitor to the flow of alternating current. It is defined as:

$X_{C} = \frac{V _{0}}{I _{0}} = \frac{V _{rms}}{I _{rms}}$

Substituting $I_{0} = ω C V_{0}$ :

$X_{C} = \frac{1}{ω C} = \frac{1}{2 π f C}$

SI Unit: Ohm ( $Ω$ ) — same as resistance.

4. Effect of Frequency on $X_{C}$

Since $X_{C} = \frac{1}{2 π f C}$ , capacitive reactance is inversely proportional to frequency:

$X_{C} \propto \frac{1}{f}$

Frequency	$X_{C}$	Effect
$f = 0$ (D.C.)	$X_{C} \to \infty$	Capacitor blocks D.C. completely
Low $f$	Large $X_{C}$	Little current flows
High $f$	Small $X_{C}$	Current flows easily

A capacitor blocks D.C. but passes A.C. — the higher the frequency, the more easily it passes.

5. Power in a Purely Capacitive Circuit

The instantaneous power is:

$P = V I = V_{0} sin (ω t) \cdot I_{0} cos (ω t) = \frac{V _{0} I _{0}}{2} sin (2 ω t)$

The average value of $sin (2 ω t)$ over a complete cycle is zero, therefore:

$P_{avg} = 0$

This can also be seen from the power factor:

$P = V_{rms} I_{rms} cos θ = V_{rms} I_{rms} cos (9 0^{\circ}) = 0$

During one quarter-cycle the capacitor stores energy (charges up), and during the next quarter-cycle it returns that energy to the source. No net energy is dissipated.

Summary Table

Property	Purely Capacitive Circuit
Phase of $I$ relative to $V$	$I$ leads $V$ by $9 0^{\circ}$
Reactance formula	$X_{C} = 1/ (2 π f C)$
Effect of increasing $f$	$X_{C}$ decreases
Average power	Zero
Power factor	$cos 9 0^{\circ} = 0$