The capacitance in an oscilatory `LC` circuit is increased by `1%`. The charge in inductance required to restore its frequency of oscillation is to

Options

A. decrease it by `0.5%`
B. increase it by `1%`
C. decrease it by `1%`
D. decrease it by `2%`

Explanation

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

### Step-by-Step Solution:

1. **Understanding the Frequency of Oscillation**:
The frequency of oscillation \( f \) in an LC circuit is given by the formula:
\[ f = \frac{1}{2\pi\sqrt{LC}} \]
where \( L \) is the inductance and \( C \) is the capacitance.

2. **Effect of Increasing Capacitance**:
According to the problem, the capacitance \( C \) is increased by 1%. Therefore, we can express the new capacitance \( C' \) as:
\[ C' = C + 0.01C = 1.01C \]

3. **New Frequency Calculation**:
Substituting \( C' \) into the frequency formula gives us the new frequency \( f' \):
\[ f' = \frac{1}{2\pi\sqrt{L \cdot 1.01C}} \]

4. **Setting the Frequencies Equal**:
To restore the original frequency \( f \), we need to adjust the inductance \( L \). Let the new inductance be \( L' \). The frequency with the new inductance and increased capacitance must equal the original frequency:
\[ f = f' \Rightarrow \frac{1}{2\pi\sqrt{LC}} = \frac{1}{2\pi\sqrt{L' \cdot 1.01C}} \]

5. **Squaring Both Sides**:
Squaring both sides to eliminate the square roots gives:
\[ \frac{1}{LC} = \frac{1}{L' \cdot 1.01C} \]

6. **Cross Multiplying**:
Cross multiplying leads to:
\[ L' \cdot 1.01C = LC \]

7. **Solving for New Inductance**:
Dividing both sides by \( 1.01C \) gives:
\[ L' = \frac{L}{1.01} \]

8. **Calculating the Change in Inductance**:
The change in inductance \( \Delta L \) can be calculated as:
\[ \Delta L = L' - L = \frac{L}{1.01} - L = L\left(\frac{1}{1.01} - 1\right) \]
Simplifying this expression:
\[ \Delta L = L\left(\frac{1 - 1.01}{1.01}\right) = L\left(\frac{-0.01}{1.01}\right) \]

9. **Percentage Change in Inductance**:
To find the percentage change in inductance:
\[ \text{Percentage Change} = \frac{\Delta L}{L} \times 100 = \frac{-0.01}{1.01} \times 100 \approx -0.9901\% \]
This can be approximated to about -1%.

### Conclusion:
The inductance must decrease by approximately 1% to restore the frequency of oscillation after the capacitance is increased by 1%.

Thus, the correct answer is option 3: decrease by 1%.