A given `LCR` series circuit satisfes the condition for resonance with a given `AC` source. If the angular frequency of the `AC` source is increased by `100%` then in order to establish resonance, and without changing the value of inductance, capacitance must be

Options

A. (a)Increased by 100%
B. (b)Reduced by 50%
C. (c)Increased by 75%
D. (d)Reduced by 75%

Explanation

To solve the problem step by step, we need to understand the concept of resonance in an LCR circuit and how the change in angular frequency affects the capacitance required for resonance.

### Step-by-Step Solution:

1. **Understanding Resonance Condition**:
In an LCR series 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. **Initial Condition**:
At resonance, we have:
\[ \omega L = \frac{1}{\omega C} \]
Rearranging gives:
\[ \omega^2 LC = 1 \quad \text{(1)} \]

3. **Change in Angular Frequency**:
The problem states that the angular frequency of the AC source is increased by 100%. Therefore, the new angular frequency (\(\omega'\)) is:
\[ \omega' = 2\omega \]

4. **New Resonance Condition**:
For the new frequency, the resonance condition becomes:
\[ \omega' L = \frac{1}{\omega' C'} \]
Substituting \(\omega' = 2\omega\) gives:
\[ 2\omega L = \frac{1}{2\omega C'} \]

5. **Rearranging the New Condition**:
Rearranging the above equation yields:
\[ 4\omega^2 LC' = 1 \quad \text{(2)} \]

6. **Relating the Two Conditions**:
From equation (1), we know \( \omega^2 LC = 1\). Substituting this into equation (2):
\[ 4 \cdot 1 = 1 \Rightarrow C' = \frac{C}{4} \]

7. **Conclusion**:
To establish resonance with the new frequency, the capacitance must be reduced to:
\[ C' = \frac{C}{4} \]
This means the capacitance must be decreased to one-fourth of its original value.