In a series LCR circuit the rms voltage across the inductance , capacitance and resistance are respectively 4 V, 8 V and 5 V . The RMS voltage of the AC source in the circuit is

Options

A. 17 V
B. 13 V
C. 5 V
D. 6.4 V

Answer

To find the RMS voltage of the AC source in a series LCR circuit where the RMS voltages across the inductance (VL), capacitance (VC), and resistance (VR) are given, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- RMS voltage across the inductor, \( V_L = 4 \, V \)
- RMS voltage across the capacitor, \( V_C = 8 \, V \)
- RMS voltage across the resistor, \( V_R = 5 \, V \)

2. **Understand the Phase Relationships**:
- In a series LCR circuit, the voltage across the resistor (VR) is in phase with the current.
- The voltage across the inductor (VL) leads the current by \( 90^\circ \).
- The voltage across the capacitor (VC) lags behind the current by \( 90^\circ \).

3. **Use the Phasor Diagram**:
- The voltage across the inductor and capacitor can be considered as vectors at right angles to each other.
- The effective voltage across the inductor and capacitor can be calculated as:
\[ V_{LC} = V_C - V_L \]

4. **Calculate the Resultant Voltage**:
- The total RMS voltage \( V \) of the AC source can be calculated using the formula:
\[ V = \sqrt{V_R^2 + (V_C - V_L)^2} \]

5. **Substitute the Values**:
- First, calculate \( V_C - V_L \):
\[ V_C - V_L = 8 \, V - 4 \, V = 4 \, V \]
- Now substitute into the formula:
\[ V = \sqrt{(5 \, V)^2 + (4 \, V)^2} \]
- Calculate:
\[ V = \sqrt{25 + 16} = \sqrt{41} \]

6. **Calculate the Final Result**:
- The numerical value of \( \sqrt{41} \) is approximately \( 6.4 \, V \).
- Therefore, the RMS voltage of the AC source is:
\[ V \approx 6.4 \, V \]

### Final Answer:
The RMS voltage of the AC source in the circuit is approximately **6.4 V**.

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