An alternating voltage, of angular frequency `omega` is induced in electric circuit consistin of inductance L and capacitance C, connected in parallel. Then across the inductance coil

Options

A. current is maximum when `omega^(2)=(1)/(LC)`
B. current is minimum when `omega^(2)=(1)/(LC)`
C. voltage is minimum when `omega^(2)=(1)/(LC)`
D. voltage is maximum when `omega^(2)=(1)/(LC)`

Answer

To solve the problem, we need to analyze the behavior of an LC circuit (inductor and capacitor in parallel) when an alternating voltage of angular frequency \( \omega \) is applied. We will determine the conditions under which the current is minimum and maximum across the inductance coil.

### Step-by-Step Solution:

1. **Understanding the LC Circuit**:
- In a parallel LC circuit, the inductor (L) and capacitor (C) are connected in parallel to an alternating voltage source.
- The circuit can resonate at a particular frequency, which is determined by the values of L and C.

2. **Resonant Frequency**:
- The resonant frequency \( \omega_0 \) of the LC circuit is given by the formula:
\[ \omega_0 = \frac{1}{\sqrt{LC}} \]
- At this frequency, the inductive reactance \( X_L \) and capacitive reactance \( X_C \) are equal.

3. **Condition for Minimum Current**:
- The current in the circuit is minimum when the circuit is at resonance. This occurs when:
\[ \omega^2 = \frac{1}{LC} \]
- At this point, the impedance of the circuit is at its maximum, leading to minimum current flow.

4. **Voltage and Current Relationship**:
- At resonance, the voltage across the inductor is maximum while the current is minimum. This is because the inductive and capacitive reactances cancel each other out, leading to a situation where the total current is minimized.

5. **Conclusion**:
- Therefore, we can conclude:
- The current is minimum when:
\[ \omega^2 = \frac{1}{LC} \]
- The voltage across the inductance coil is maximum at this condition.

### Final Results:
- **Minimum Current Condition**: \( \omega^2 = \frac{1}{LC} \)
- **Maximum Voltage Condition**: Voltage is maximum when current is minimum.