Current in LCR ac circuit will be maximum when omega is

Options

A. as large as possible
B. `sqrt(LC)`
C. `sqrt((1)/(LC))`
D. `sqrt(LCR)`

Answer

To determine the condition under which the current in an LCR AC circuit is maximum, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship of current in LCR circuit**:
The current \( I \) in an LCR circuit can be expressed as:
\[ I = \frac{V}{Z} \]
where \( Z \) is the impedance of the circuit.

2. **Impedance expression**:
The impedance \( Z \) for a series LCR circuit is given by:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
where \( X_L = \omega L \) (inductive reactance) and \( X_C = \frac{1}{\omega C} \) (capacitive reactance).

3. **Maximizing the current**:
To maximize the current \( I \), we need to minimize the impedance \( Z \). This means we need to minimize the term \( (X_L - X_C) \).

4. **Condition for minimum impedance**:
The term \( (X_L - X_C) \) will be zero when:
\[ X_L = X_C \]
Therefore, we set:
\[ \omega L = \frac{1}{\omega C} \]

5. **Solving for \( \omega \)**:
Rearranging the equation \( \omega L = \frac{1}{\omega C} \) gives:
\[ \omega^2 = \frac{1}{LC} \]
Taking the square root of both sides results in:
\[ \omega = \frac{1}{\sqrt{LC}} \]

6. **Conclusion**:
The current in the LCR AC circuit will be maximum when:
\[ \omega = \frac{1}{\sqrt{LC}} \]
This frequency is known as the resonance frequency.