In L-C circuit, the charge oscillates with a natural frequency `omega_0` given by

Options

A. `omega_0 = 1/(LC)`
B. `omega_0 = 1/sqrt(LC)`
C. `omega_0 = 1/(2LC)`
D. `omega_0 = 1/sqrt(2LC)`

Answer

To find the natural frequency \(\omega_0\) of an LC circuit, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Components of the LC Circuit**:
- An LC circuit consists of an inductor (L) and a capacitor (C) connected together. The inductor stores energy in its magnetic field, while the capacitor stores energy in its electric field.

2. **Identify Reactances**:
- The capacitive reactance \(X_C\) is given by the formula:
\[ X_C = \frac{1}{\omega_0 C} \]
- The inductive reactance \(X_L\) is given by the formula:
\[ X_L = \omega_0 L \]

3. **Set the Reactances Equal**:
- At resonance in an LC circuit, the capacitive reactance equals the inductive reactance:
\[ X_C = X_L \]
- This leads to the equation:
\[ \frac{1}{\omega_0 C} = \omega_0 L \]

4. **Rearrange the Equation**:
- Multiply both sides by \(\omega_0 C\):
\[ 1 = \omega_0^2 LC \]

5. **Solve for \(\omega_0\)**:
- Rearranging the equation gives:
\[ \omega_0^2 = \frac{1}{LC} \]
- Taking the square root of both sides results in:
\[ \omega_0 = \frac{1}{\sqrt{LC}} \]

6. **Conclusion**:
- The natural frequency \(\omega_0\) of the LC circuit is given by:
\[ \omega_0 = \frac{1}{\sqrt{LC}} \]

### Final Answer:
The natural frequency \(\omega_0\) of the LC circuit is:
\[ \omega_0 = \frac{1}{\sqrt{LC}} \]