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 solve the problem of finding 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 Configuration**:
In a series LCR circuit, we have an inductor (L), a capacitor (C), and a resistor (R) connected in series. The same current flows through all components.

2. **Identify the Phase Relationships**:
- The voltage across the resistor (VR) is in phase with the current (I).
- The voltage across the inductor (VL) leads the current by 90 degrees (or π/2 radians).
- The voltage across the capacitor (VC) lags the current by 90 degrees (or π/2 radians).

3. **Express the Voltages in Terms of Current**:
- Let the current be represented as \( I = I_0 \sin(\omega t) \).
- The voltage across the inductor can be expressed as:
\[ V_L = I \cdot X_L = I_0 \cdot X_L \sin(\omega t + \frac{\pi}{2}) \]
- The voltage across the capacitor can be expressed as:
\[ V_C = I \cdot X_C = I_0 \cdot X_C \sin(\omega t - \frac{\pi}{2}) \]

4. **Determine the Phase Difference**:
- The phase of \( V_L \) is \( \frac{\pi}{2} \) (leads the current).
- The phase of \( V_C \) is \( -\frac{\pi}{2} \) (lags the current).
- To find the phase difference \( \Delta \phi \) between \( V_L \) and \( V_C \):
\[ \Delta \phi = \phi_L - \phi_C = \left(\frac{\pi}{2}\right) - \left(-\frac{\pi}{2}\right) = \frac{\pi}{2} + \frac{\pi}{2} = \pi \]

5. **Conclusion**:
The phase difference between the voltage across the inductor (L) and the voltage across the capacitor (C) is \( \pi \) radians or 180 degrees.

### Final Answer:
The phase difference between voltage across L and voltage across C is \( \pi \) radians (or 180 degrees).

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