In an LCR series circuit the capacitance is changed from `C` to `4C` For the same resonant fequency the inductance should be changed from `L` to .

Options

A. `(L)/(3)`
B. `(L)/(2)`
C. `(L)/(4)`
D. `(L)/(8)`

Explanation

To solve the problem, we need to understand the relationship between the inductance (L), capacitance (C), and the resonant frequency in an LCR series circuit. The resonant frequency (ω₀) is given by the formula:

\[ \omega_0 = \frac{1}{\sqrt{LC}} \]

### Step-by-Step Solution:

1. **Identify the Initial Resonant Frequency**:
The initial resonant frequency is given by:
\[ \omega_{0} = \frac{1}{\sqrt{LC}} \]

2. **Change in Capacitance**:
The capacitance is changed from \(C\) to \(4C\). We denote the new inductance as \(L'\).

3. **Set Up the Equation for New Resonant Frequency**:
For the new configuration with capacitance \(4C\) and inductance \(L'\), the resonant frequency remains the same:
\[ \omega_{0} = \frac{1}{\sqrt{L' \cdot 4C}} \]

4. **Equate the Two Expressions for Resonant Frequency**:
Since the resonant frequencies are equal, we can set the two equations equal to each other:
\[ \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{L' \cdot 4C}} \]

5. **Cross-Multiply to Eliminate the Square Roots**:
Cross-multiplying gives:
\[ \sqrt{L' \cdot 4C} = \sqrt{LC} \]

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

7. **Solve for the New Inductance \(L'\)**:
Dividing both sides by \(4C\) gives:
\[ L' = \frac{L}{4} \]

### Final Answer:
The inductance should be changed from \(L\) to \(\frac{L}{4}\).

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