In a `LCR` circuit having `L=8.0` henry, `C=0.5 mu F` and `R=100` ohm in series. The resonance frequency in per second is

Options

A. 500 rad/s
B. 600 rad/s
C. 800 rad/s
D. 1000 rad/s

Answer

To find the resonance frequency in an LCR circuit, we can use the formula:

\[ f_0 = \frac{1}{2\pi\sqrt{LC}} \]

Where:
- \( f_0 \) is the resonance frequency in hertz (Hz),
- \( L \) is the inductance in henries (H),
- \( C \) is the capacitance in farads (F).

### Step-by-Step Solution:

1. **Identify the given values**:
- Inductance \( L = 8.0 \, \text{H} \)
- Capacitance \( C = 0.5 \, \mu\text{F} = 0.5 \times 10^{-6} \, \text{F} \)
- Resistance \( R = 100 \, \Omega \) (not needed for resonance frequency calculation)

2. **Substitute the values into the formula**:
\[ f_0 = \frac{1}{2\pi\sqrt{LC}} = \frac{1}{2\pi\sqrt{8.0 \times 0.5 \times 10^{-6}}} \]

3. **Calculate \( LC \)**:
\[ LC = 8.0 \times 0.5 \times 10^{-6} = 4.0 \times 10^{-6} \, \text{H}\cdot\text{F} \]

4. **Calculate \( \sqrt{LC} \)**:
\[ \sqrt{LC} = \sqrt{4.0 \times 10^{-6}} = 2.0 \times 10^{-3} \, \text{s} \]

5. **Substitute \( \sqrt{LC} \) back into the resonance frequency formula**:
\[ f_0 = \frac{1}{2\pi(2.0 \times 10^{-3})} \]

6. **Calculate \( 2\pi(2.0 \times 10^{-3}) \)**:
\[ 2\pi(2.0 \times 10^{-3}) \approx 0.0125664 \, \text{s} \]

7. **Calculate the resonance frequency**:
\[ f_0 \approx \frac{1}{0.0125664} \approx 79.577 \, \text{Hz} \]

8. **Convert to radians per second**:
Since \( 1 \, \text{Hz} = 2\pi \, \text{rad/s} \):
\[ f_0 \approx 79.577 \times 2\pi \approx 500 \, \text{rad/s} \]

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

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