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 when the current in an LCR AC circuit is maximum, we can follow these steps:

### Step 1: Understand the LCR Circuit
An LCR circuit consists of an inductor (L), a capacitor (C), and a resistor (R) connected in series. The current flowing through the circuit depends on the impedance (Z) of the circuit.

### Step 2: Write the Formula for Current
The current (I) in the circuit can be expressed as:
\[ I = \frac{V}{Z} \]
where \( V \) is the voltage across the circuit and \( Z \) is the impedance.

### Step 3: Calculate Impedance (Z)
The impedance \( Z \) in an LCR circuit is given by:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
where:
- \( X_L = \omega L \) (Inductive Reactance)
- \( X_C = \frac{1}{\omega C} \) (Capacitive Reactance)

### Step 4: Condition for Maximum Current
The current will be maximum when the impedance \( Z \) is minimum. The impedance is minimized when the inductive reactance equals the capacitive reactance:
\[ X_L = X_C \]
This condition leads to resonance in the circuit.

### Step 5: Set Up the Resonance Condition
Setting \( X_L \) equal to \( X_C \):
\[ \omega L = \frac{1}{\omega C} \]

### Step 6: Solve for Angular Frequency (ω)
Rearranging the equation gives:
\[ \omega^2 = \frac{1}{LC} \]
Taking the square root of both sides, we find:
\[ \omega = \frac{1}{\sqrt{LC}} \]

### Conclusion
The current in the LCR AC circuit will be maximum when the angular frequency \( \omega \) is:
\[ \omega = \frac{1}{\sqrt{LC}} \]