An LC series circuit has an oscillation frequency f. Two isolated inductors, each with inductance L and two capacitors each with capacitance C, all are connected in series and circuit is completed. The oscillation frequency is

Options

A. `f/4`
B. `f/2`
C. f
D. 4 f

Answer

To find the oscillation frequency of the LC series circuit with two inductors and two capacitors connected in series, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components**: We have two inductors, each with inductance \( L \), and two capacitors, each with capacitance \( C \).

2. **Determine the Equivalent Inductance**:
- When inductors are connected in series, the total inductance \( L_{\text{eq}} \) is the sum of the individual inductances.
- Therefore, for two inductors in series:
\[ L_{\text{eq}} = L_1 + L_2 = L + L = 2L \]

3. **Determine the Equivalent Capacitance**:
- When capacitors are connected in series, the total capacitance \( C_{\text{eq}} \) can be calculated using the formula:
\[ \frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} \]
- For two identical capacitors in series:
\[ \frac{1}{C_{\text{eq}}} = \frac{1}{C} + \frac{1}{C} = \frac{2}{C} \implies C_{\text{eq}} = \frac{C}{2} \]

4. **Calculate the New Resonant Frequency**:
- The resonant frequency \( f \) of an LC circuit is given by:
\[ f = \frac{1}{2\pi \sqrt{L_{\text{eq}} C_{\text{eq}}}} \]
- Substituting the equivalent inductance and capacitance:
\[ f = \frac{1}{2\pi \sqrt{2L \cdot \frac{C}{2}}} \]
- Simplifying this expression:
\[ f = \frac{1}{2\pi \sqrt{L \cdot C}} = f_0 \]
- Here, \( f_0 \) is the original frequency of the circuit.

5. **Conclusion**: The oscillation frequency remains the same as the original frequency \( f \).

### Final Answer:
The oscillation frequency of the circuit is \( f \).