An LC series circuit with an inductance L and a capacitance C has an oscillation frequency f. Two isolated induc tors, each with inductance L, and two capacitors, each with capacitance C, are all wired in series and the circuit is com pleted. The oscillation frequency is

Options

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

Answer

To solve the problem, we need to determine the oscillation frequency of an LC series circuit when two inductors and two capacitors are connected in series.

### Step-by-Step Solution:

1. **Understanding the LC Circuit**:
An LC circuit consists of an inductor (L) and a capacitor (C). The oscillation frequency (f) of a simple LC circuit is given by the formula:
\[ f = \frac{1}{2\pi\sqrt{LC}} \]

2. **Analyzing the New Configuration**:
In the new configuration, we have two inductors (each with inductance L) and two capacitors (each with capacitance C) connected in series.

3. **Calculating Effective Inductance**:
When inductors are connected in series, their inductances add up. Therefore, the total inductance \(L_{total}\) for two inductors in series is:
\[ L_{total} = L + L = 2L \]

4. **Calculating Effective Capacitance**:
When capacitors are connected in series, the total capacitance \(C_{total}\) is given by the formula:
\[ \frac{1}{C_{total}} = \frac{1}{C} + \frac{1}{C} = \frac{2}{C} \]
Thus, the total capacitance becomes:
\[ C_{total} = \frac{C}{2} \]

5. **Finding the New Oscillation Frequency**:
Now, we can substitute the effective inductance and capacitance into the frequency formula:
\[ f_{new} = \frac{1}{2\pi\sqrt{L_{total} \cdot C_{total}}} = \frac{1}{2\pi\sqrt{(2L) \cdot \left(\frac{C}{2}\right)}} \]
Simplifying this expression:
\[ f_{new} = \frac{1}{2\pi\sqrt{L \cdot C}} = f \]

6. **Conclusion**:
The oscillation frequency of the new circuit remains the same as the original circuit, which is \(f\).

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

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