An LCR series ac circuit is at resonance with 10 V each across L, C and R. If the resistance is halved, the respective voltage across L, C and R are

Options

A. 10 V, 10 V and 5 V
B. 10 V, 10 V and 10 V
C. 20 V, 20 V and 5 V
D. 20 V, 20 V and 10 V

Answer

To solve the problem step by step, let's analyze the given LCR series AC circuit at resonance and how the voltages across the components change when the resistance is halved.

### Step-by-Step Solution:

1. **Understanding Resonance in LCR Circuit:**
At resonance, the voltages across the inductor (L), capacitor (C), and resistor (R) are equal. Given that the voltage across each component is 10 V, we have:
\[ V_L = V_C = V_R = 10 \text{ V} \]

2. **Effect of Halving the Resistance:**
When the resistance (R) is halved, the new resistance becomes \( R' = \frac{R}{2} \). According to Ohm's law, the current in the circuit is inversely proportional to the resistance. Therefore, if the resistance is halved, the current doubles:
\[ I' = 2I \]

3. **Voltage Across the Resistor (R):**
The voltage across the resistor can be calculated using Ohm's law:
\[ V_R' = I' \cdot R' = (2I) \cdot \left(\frac{R}{2}\right) = I \cdot R = V_R = 10 \text{ V} \]

4. **Voltage Across the Capacitor (C):**
The voltage across the capacitor can be expressed in terms of the current and the capacitive reactance (X_C). Since the circuit is at resonance, the total impedance is purely resistive, and we can say:
\[ V_C' = I' \cdot X_C = (2I) \cdot X_C \]
At resonance, the voltage across the capacitor is equal to the voltage across the inductor. Therefore, since the current doubles:
\[ V_C' = 2 \cdot 10 \text{ V} = 20 \text{ V} \]

5. **Voltage Across the Inductor (L):**
Similarly, for the inductor:
\[ V_L' = I' \cdot X_L = (2I) \cdot X_L \]
Again, at resonance, the voltage across the inductor is equal to the voltage across the capacitor:
\[ V_L' = 20 \text{ V} \]

### Final Result:
Thus, after halving the resistance, the respective voltages across the resistor, capacitor, and inductor are:
- Voltage across Resistor (R): \( V_R' = 10 \text{ V} \)
- Voltage across Capacitor (C): \( V_C' = 20 \text{ V} \)
- Voltage across Inductor (L): \( V_L' = 20 \text{ V} \)

### Summary of Results:
- \( V_R' = 10 \text{ V} \)
- \( V_C' = 20 \text{ V} \)
- \( V_L' = 20 \text{ V} \)