In a RLC circuit capacitance is changed from C to 2 C. For the resonant frequency to remain unchanged, the inductance should be changed from L to

Options

A. 4 L
B. 2 L
C. `(L)/(2)`
D. `(L)/(4)`

Answer

To solve the problem, we need to analyze the relationship between the inductance (L), capacitance (C), and the resonant frequency in an RLC circuit.

### Step-by-Step Solution:

1. **Understand the Resonant Frequency Formula**:
The resonant frequency \( \omega \) for an RLC circuit is given by the formula:
\[ \omega = \frac{1}{\sqrt{LC}} \]
where \( L \) is the inductance and \( C \) is the capacitance.

2. **Set Up the Initial Condition**:
Let the initial inductance be \( L \) and the initial capacitance be \( C \). The initial resonant frequency can be expressed as:
\[ \omega_1 = \frac{1}{\sqrt{LC}} \]

3. **Change the Capacitance**:
According to the problem, the capacitance is changed from \( C \) to \( 2C \). We denote the new capacitance as \( C_2 = 2C \).

4. **Maintain the Same Resonant Frequency**:
We want the resonant frequency to remain unchanged. Therefore, the new resonant frequency \( \omega_2 \) must equal \( \omega_1 \):
\[ \omega_2 = \frac{1}{\sqrt{L_2 C_2}} = \frac{1}{\sqrt{L_2 \cdot 2C}} \]

5. **Set the Frequencies Equal**:
Since \( \omega_1 = \omega_2 \), we have:
\[ \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{L_2 \cdot 2C}} \]

6. **Square Both Sides**:
Squaring both sides to eliminate the square roots gives:
\[ \frac{1}{LC} = \frac{1}{2L_2C} \]

7. **Cross Multiply**:
Cross multiplying yields:
\[ 2L_2 = L \]

8. **Solve for the New Inductance**:
Rearranging the equation gives:
\[ L_2 = \frac{L}{2} \]

### Final Answer:
Thus, the new inductance \( L_2 \) should be \( \frac{L}{2} \) for the resonant frequency to remain unchanged.