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. 600 radian/s
B. 600 Hz
C. 500 radian/s
D. 500 Hz

Answer

To find the resonance frequency in an LCR circuit, we can use the formula for the resonant frequency \( \omega \):

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

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

### Step 1: Identify the values of \( L \) and \( C \)

From the question, we have:
- \( L = 8.0 \, \text{H} \)
- \( C = 0.5 \, \mu \text{F} = 0.5 \times 10^{-6} \, \text{F} \)

### Step 2: Substitute the values into the formula

Substituting the values into the resonance frequency formula:

\[ \omega = \frac{1}{\sqrt{8.0 \times 0.5 \times 10^{-6}}} \]

### Step 3: Calculate the product inside the square root

Calculating the product:

\[ 8.0 \times 0.5 = 4.0 \]

So, we have:

\[ \omega = \frac{1}{\sqrt{4.0 \times 10^{-6}}} \]

### Step 4: Simplify the square root

Calculating the square root:

\[ \sqrt{4.0 \times 10^{-6}} = \sqrt{4.0} \times \sqrt{10^{-6}} = 2.0 \times 10^{-3} \]

### Step 5: Calculate \( \omega \)

Now substituting back into the equation for \( \omega \):

\[ \omega = \frac{1}{2.0 \times 10^{-3}} = 500 \, \text{rad/s} \]

### Final Answer

Thus, the resonance frequency in per second is:

\[ \omega = 500 \, \text{rad/s} \]

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