In series LCR circuit, the phase difference between voltage across L and voltage across C is

Options

A. Zero
B. `pi`
C. `(pi)/(2)`
D. `2pi`

Answer

To find the phase difference between the voltage across the inductor (L) and the voltage across the capacitor (C) in a series LCR circuit, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Circuit Components**: In a series LCR circuit, we have an inductor (L), a capacitor (C), and a resistor (R) connected in series. The alternating current (AC) voltage is supplied across this combination.

2. **Phase Relationships**:
- The voltage across the resistor (VR) is in phase with the current. If we denote the supplied voltage as \( V = V_0 \sin(\omega t) \), then \( V_R = V_0 \sin(\omega t) \).
- The voltage across the inductor (VL) leads the current by \( \frac{\pi}{2} \) (90 degrees). Therefore, \( V_L = V_0 \sin(\omega t + \frac{\pi}{2}) \).
- The voltage across the capacitor (VC) lags the current by \( \frac{\pi}{2} \) (90 degrees). Thus, \( V_C = V_0 \sin(\omega t - \frac{\pi}{2}) \).

3. **Determine the Phase Difference**:
- To find the phase difference between VL and VC, we need to analyze their respective phase angles:
- The phase angle of VL is \( \frac{\pi}{2} \).
- The phase angle of VC is \( -\frac{\pi}{2} \).
- The phase difference \( \Delta \phi \) between VL and VC can be calculated as:
\[ \Delta \phi = \phi_L - \phi_C = \left(\frac{\pi}{2}\right) - \left(-\frac{\pi}{2}\right) = \frac{\pi}{2} + \frac{\pi}{2} = \pi \]

4. **Conclusion**: The phase difference between the voltage across the inductor and the voltage across the capacitor in a series LCR circuit is \( \pi \) radians.

### Final Answer:
The phase difference between the voltage across L and voltage across C is \( \pi \).

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