A condenser of capacity 20`mu`F is first charged and then discharged through a 10mH inductance. Neglecting the resistance of the coil, the frequency of the resulting vibrations will be

Options

A. 356 cycles/s
B. 35.6 cycles/s
C. `365 xx 10^(3)` cycles/s
D. 3.56 cycles/s

Explanation

To find the frequency of the resulting vibrations when a capacitor is discharged through an inductor, we can use the formula for the resonant frequency of an LC circuit. The steps to solve the problem are as follows:

### Step-by-Step Solution:

1. **Identify the given values**:
- Capacitance (C) = 20 µF = 20 × 10^(-6) F
- Inductance (L) = 10 mH = 10 × 10^(-3) H

2. **Use the formula for resonant frequency**:
The resonant frequency (f) of an LC circuit is given by the formula:
\[ f = \frac{1}{2 \pi \sqrt{L \cdot C}} \]

3. **Substitute the values into the formula**:
\[ f = \frac{1}{2 \pi \sqrt{(10 \times 10^{-3}) \cdot (20 \times 10^{-6})}} \]

4. **Calculate the product of L and C**:
\[ L \cdot C = (10 \times 10^{-3}) \cdot (20 \times 10^{-6}) = 200 \times 10^{-9} = 2 \times 10^{-7} \]

5. **Take the square root**:
\[ \sqrt{L \cdot C} = \sqrt{2 \times 10^{-7}} = \sqrt{2} \times 10^{-3.5} \approx 1.414 \times 10^{-3.5} \]

6. **Calculate the frequency**:
\[ f = \frac{1}{2 \pi \cdot (1.414 \times 10^{-3.5})} \]
\[ f \approx \frac{1}{2 \pi \cdot 1.414 \times 10^{-3.5}} \approx \frac{1}{8.88 \times 10^{-3.5}} \approx 356 \text{ Hz} \]

7. **Final result**:
The frequency of the resulting vibrations is approximately 356 Hz.