In LCR circuit, capacitor C is changed to 4C, then what should be the value of L to keep resonance frequency same

Options

A. 2 L
B. L/2
C. L/4
D. 4L

Answer

To solve the problem, we need to determine the new inductance \( L' \) required to keep the resonance frequency the same when the capacitance \( C \) is changed to \( 4C \).

### Step-by-Step Solution:

1. **Understand the Resonance Condition**:
In an LCR circuit, resonance occurs when the inductive reactance \( X_L \) equals the capacitive reactance \( X_C \). Mathematically, this is expressed as:
\[ X_L = X_C \]
where \( X_L = \omega L \) and \( X_C = \frac{1}{\omega C} \).

2. **Write the Resonance Frequency Formula**:
The resonance frequency \( f_r \) is given by:
\[ f_r = \frac{1}{2\pi\sqrt{LC}} \]
For the original circuit, we denote the inductance as \( L \) and capacitance as \( C \).

3. **Change in Capacitance**:
When the capacitance is changed to \( 4C \), we denote the new capacitance as \( C' = 4C \).

4. **Set Up the Equation for the New Resonance Frequency**:
The new resonance frequency \( f'_r \) with the new inductance \( L' \) and capacitance \( C' \) is:
\[ f'_r = \frac{1}{2\pi\sqrt{L'C'}} \]
Since we want to keep the resonance frequency the same, we set \( f_r = f'_r \):
\[ \frac{1}{2\pi\sqrt{LC}} = \frac{1}{2\pi\sqrt{L' \cdot 4C}} \]

5. **Cancel Common Terms**:
We can cancel \( \frac{1}{2\pi} \) from both sides:
\[ \sqrt{LC} = \sqrt{L' \cdot 4C} \]

6. **Square Both Sides**:
Squaring both sides gives:
\[ LC = L' \cdot 4C \]

7. **Solve for \( L' \)**:
Dividing both sides by \( C \) (assuming \( C \neq 0 \)):
\[ L = 4L' \]
Rearranging gives:
\[ L' = \frac{L}{4} \]

### Final Answer:
To keep the resonance frequency the same when the capacitance is changed to \( 4C \), the new inductance \( L' \) should be:
\[ L' = \frac{L}{4} \]