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 of how the inductance \( L \) should change when the capacitance \( C \) is changed to \( 2C \) while keeping the resonant frequency unchanged, we can follow these steps:

### Step-by-Step Solution:

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

2. **Initial Resonant Frequency**:
Let the initial values be \( L \) and \( C \). The initial resonant frequency is:
\[ \omega_1 = \frac{1}{\sqrt{LC}} \]

3. **Change in Capacitance**:
The capacitance is changed from \( C \) to \( 2C \). We need to find the new inductance \( L_2 \) such that the resonant frequency remains the same.

4. **New Resonant Frequency**:
The new resonant frequency with the updated capacitance \( 2C \) and new inductance \( L_2 \) is:
\[ \omega_2 = \frac{1}{\sqrt{L_2 \cdot 2C}} \]

5. **Set the Frequencies Equal**:
Since we want the resonant frequency to remain unchanged, we set \( \omega_1 = \omega_2 \):
\[ \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{L_2 \cdot 2C}} \]

6. **Cross-Multiply**:
Cross-multiplying gives us:
\[ \sqrt{L_2 \cdot 2C} = \sqrt{LC} \]

7. **Square Both Sides**:
Squaring both sides results in:
\[ L_2 \cdot 2C = LC \]

8. **Solve for \( L_2 \)**:
Dividing both sides by \( 2C \) gives:
\[ L_2 = \frac{L}{2} \]

### Final Answer:
Thus, the inductance should be changed from \( L \) to \( \frac{L}{2} \).