Introduction:

When an alternating current (AC) voltage is applied to a capacitor, the capacitor exhibits unique behavior compared to its response in direct current (DC) circuits. Understanding this behavior is crucial for analyzing AC circuits, especially in Class 12 Physics.



Behavior of a Capacitor in an AC Circuit:

In a DC circuit, a capacitor charges up to the applied voltage and then blocks any further current flow, acting as an open circuit. However, in an AC circuit, the voltage across the capacitor is continuously changing, causing the capacitor to charge and discharge periodically. This results in a continuous current flow through the circuit.

The voltage (v) across the capacitor and the current (i) through the capacitor are related by the equation:

v = q / C

where q is the charge on the capacitor and C is the capacitance.

Applying Kirchhoff's loop rule to an AC circuit with a capacitor, we get:

v_m sin(ωt) = q / C

Differentiating both sides with respect to time (t), we obtain the current (i):

i = dq/dt = ω C v_m cos(ωt)

This can be rewritten as:

i = i_m sin(ωt + π/2)

where i_m = ω C v_m is the maximum current.

Capacitive Reactance:

The opposition offered by a capacitor to the flow of AC is called capacitive reactance, denoted by X_C. It is given by:

X_C = 1 / (ω C) = 1 / (2π f C)

where ω = 2π f is the angular frequency and f is the frequency of the AC source.

Capacitive reactance decreases with increasing frequency and capacitance, implying that a capacitor allows higher frequency signals to pass through more easily.

Phase Relationship Between Current and Voltage:

In a purely capacitive AC circuit, the current leads the voltage by 90 degrees. This phase difference is a key characteristic of capacitors in AC circuits and is crucial in applications like tuning circuits and filters.

Read Also: Scalars and Vectors-Class 11 Physics Study Notes

Power in a Capacitive AC Circuit:

The instantaneous power (p(t)) supplied to the capacitor is:

p(t) = v(t) * i(t) = v_m sin(ωt) * i_m sin(ωt + π/2)

Since sin(ωt + π/2) = cos(ωt), we have:

p(t) = v_m i_m sin(ωt) cos(ωt) = (v_m i_m / 2) sin(2ωt)

The average power over a complete cycle is zero, indicating that, on average, no net energy is transferred over one complete cycle in a purely capacitive AC circuit.

Conclusion:

Understanding the behavior of capacitors in AC circuits is essential for analyzing and designing various electronic systems.

The key points to remember are:

- In AC circuits, capacitors allow current to flow continuously due to the periodic charging and discharging.

- The current leads the voltage by 90 degrees in a purely capacitive AC circuit.

- Capacitive reactance inversely depends on the frequency and capacitance.

- The average power consumed in a purely capacitive AC circuit over a complete cycle is zero.

These principles are foundational for further studies in AC circuit analysis and applications.