In an L C R circuit having L = 8 H, C = 0 `5muF` and `R=100Omega` in series, the resonance frequency in rad/s is

Options

A. 600 rad//sec
B. 600 Hz
C. 500 rad//sec
D. 500 Hz

Answer

To find the resonance frequency in an LCR circuit, we can use the formula for the resonance angular frequency, which is given by:

\[ \omega = \frac{1}{\sqrt{LC}} \]

where:
- \(L\) is the inductance in henries (H),
- \(C\) is the capacitance in farads (F).

### Step-by-step Solution:

1. **Identify the values of L and C**:
- Given \(L = 8 \, \text{H}\)
- Given \(C = 0.5 \, \mu\text{F} = 0.5 \times 10^{-6} \, \text{F}\)

2. **Substitute the values into the resonance frequency formula**:
\[ \omega = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{8 \times 0.5 \times 10^{-6}}} \]

3. **Calculate the product \(LC\)**:
\[ LC = 8 \times 0.5 \times 10^{-6} = 4 \times 10^{-6} \, \text{H}\cdot\text{F} \]

4. **Take the square root of \(LC\)**:
\[ \sqrt{LC} = \sqrt{4 \times 10^{-6}} = 2 \times 10^{-3} \, \text{s} \]

5. **Calculate \(\omega\)**:
\[ \omega = \frac{1}{2 \times 10^{-3}} = 500 \, \text{rad/s} \]

### Final Answer:
The resonance frequency in rad/s is \(500 \, \text{rad/s}\).