Phase difference between voltage and current in a capacitor in an ac circuit is

Options

A. `pi`
B. `pi//2`
C. 0
D. `pi//3`

Explanation

To find the phase difference between voltage and current in a capacitor in an AC circuit, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the AC Circuit with a Capacitor**:
- In an AC circuit containing a capacitor, the voltage across the capacitor and the current through it are not in phase.

2. **Write the Voltage Equation**:
- The voltage across the capacitor can be expressed as:
\[ V(t) = V_0 \sin(\omega t) \]
- Here, \(V_0\) is the peak voltage, and \(\omega\) is the angular frequency.

3. **Relate Charge and Voltage**:
- The charge \(Q\) on the capacitor is related to the voltage by the equation:
\[ Q = C \cdot V \]
- Therefore, substituting the voltage expression:
\[ Q(t) = C \cdot V_0 \sin(\omega t) \]

4. **Differentiate Charge to Find Current**:
- The current \(I\) is the rate of change of charge with respect to time:
\[ I(t) = \frac{dQ}{dt} = \frac{d}{dt}(C \cdot V_0 \sin(\omega t)) \]
- Using the chain rule:
\[ I(t) = C \cdot V_0 \cdot \omega \cos(\omega t) \]

5. **Express Current in Sine Form**:
- We can express the cosine function in terms of sine:
\[ I(t) = C \cdot V_0 \cdot \omega \cdot \cos(\omega t) = C \cdot V_0 \cdot \omega \cdot \sin\left(\omega t + \frac{\pi}{2}\right) \]
- This shows that the current leads the voltage by \(90^\circ\) or \(\frac{\pi}{2}\) radians.

6. **Determine the Phase Difference**:
- Since the current leads the voltage, we can conclude that the phase difference \(\phi\) between the voltage and current in a capacitor is:
\[ \phi = -\frac{\pi}{2} \]
- This indicates that the voltage lags the current by \(90^\circ\) or \(\frac{\pi}{2}\) radians.

### Final Answer:
The phase difference between voltage and current in a capacitor in an AC circuit is \(-\frac{\pi}{2}\) radians (or \(90^\circ\)), meaning the voltage lags the current.

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