**Voltage Across a Capacitor in an AC Circuit**

### Introduction

In an AC circuit, capacitors play a crucial role in filtering, coupling, and energy storage. To analyze the behavior of a capacitor in an AC circuit, it is essential to understand the voltage across it. In this article, we will derive the formula for the voltage across a capacitor in an AC circuit and discuss its significance.

### Capacitor in an AC Circuit

A capacitor in an AC circuit is subject to a sinusoidal voltage source. The voltage across the capacitor is not constant, but rather it varies sinusoidally with time. The capacitor stores energy in the electric field between its plates and releases it back to the circuit.

### Voltage Across a Capacitor Formula

The voltage across a capacitor in an AC circuit can be calculated using the following formula:

**Vc(t) = Vm * cos(ωt + θ)**

where:

**Vc(t)**is the voltage across the capacitor at time t**Vm**is the maximum voltage across the capacitor (peak voltage)**ω**is the angular frequency of the AC signal (rad/s)**t**is time (s)**θ**is the phase angle between the voltage and current (rad)

### Derivation of the Formula

To derive the formula, we can start with the definition of capacitance (C):

**Q = C * V**

where:

**Q**is the charge stored in the capacitor (C)**C**is the capacitance (F)**V**is the voltage across the capacitor (V)

In an AC circuit, the voltage across the capacitor is sinusoidal, so we can represent it as:

**V = Vm * cos(ωt)**

Substituting this into the definition of capacitance, we get:

**Q = C * Vm * cos(ωt)**

The current through the capacitor is given by:

**I = dQ/dt**

Taking the derivative of Q with respect to time, we get:

**I = -ωC * Vm * sin(ωt)**

Using the definition of impedance (Z) for a capacitor:

**Z = 1/(jωC)**

where **j** is the imaginary unit (j = √(-1)), we can rewrite the current equation as:

**I = Vm/(jXc) * sin(ωt)**

where **Xc** is the capacitive reactance (Ω).

Rearranging the equation to solve for voltage, we get:

**Vc(t) = Vm * cos(ωt + θ)**

where **θ** is the phase angle between the voltage and current.

### Significance of the Formula

The formula for the voltage across a capacitor in an AC circuit is crucial in the analysis and design of AC circuits. It helps us understand the behavior of capacitors in filtering, coupling, and energy storage applications. The phase angle **θ** is particularly important, as it determines the current through the capacitor and the power factor of the circuit.

### Conclusion

In this article, we have derived the formula for the voltage across a capacitor in an AC circuit. The formula is essential in understanding the behavior of capacitors in AC circuits and is widely used in the design of filters, couplers, and energy storage systems.