A 0.5μF capacitor is discharged through a 10 millihenry inductor. Find the frequency of discharged.

Answer

To find the frequency of discharge of a capacitor through an inductor, we can use the formula for the resonant frequency of an LC circuit:

\[ f = \frac{1}{2\pi \sqrt{LC}} \]

where:
- \( f \) is the frequency in hertz (Hz),
- \( L \) is the inductance in henries (H),
- \( C \) is the capacitance in farads (F).

### Step-by-Step Solution:

1. **Identify the values given in the problem:**
- Capacitance, \( C = 0.5 \, \mu F = 0.5 \times 10^{-6} \, F \)
- Inductance, \( L = 10 \, mH = 10 \times 10^{-3} \, H \)

2. **Convert the units if necessary:**
- The values are already in the correct units for the formula.

3. **Substitute the values into the formula:**
\[ f = \frac{1}{2\pi \sqrt{L \cdot C}} = \frac{1}{2\pi \sqrt{(10 \times 10^{-3}) \cdot (0.5 \times 10^{-6})}} \]

4. **Calculate the product \( L \cdot C \):**
\[ L \cdot C = (10 \times 10^{-3}) \cdot (0.5 \times 10^{-6}) = 5 \times 10^{-9} \, H \cdot F \]

5. **Calculate the square root:**
\[ \sqrt{L \cdot C} = \sqrt{5 \times 10^{-9}} \approx 2.236 \times 10^{-5} \, s \]

6. **Substitute the square root back into the frequency formula:**
\[ f = \frac{1}{2\pi (2.236 \times 10^{-5})} \]

7. **Calculate the frequency:**
\[ f \approx \frac{1}{6.2832 \times 2.236 \times 10^{-5}} \approx \frac{1}{1.404 \times 10^{-4}} \approx 7115.56 \, Hz \]

8. **Final Result:**
\[ f \approx 7115.56 \, Hz \]

### Final Answer:
The frequency of discharge is approximately **7115.56 Hz**.