The resonant frequency of a series LCR circuit with `L=2.0 H,C =32 muF` and `R=10 Omega` is

Options

A. 20 Hz
B. 30 Hz
C. 40 Hz
D. 50 Hz

Explanation

To find the resonant frequency of a series LCR circuit, we can use the formula:

\[ f_0 = \frac{1}{2\pi \sqrt{LC}} \]

where:
- \( f_0 \) is the resonant frequency,
- \( L \) is the inductance in henries (H),
- \( C \) is the capacitance in farads (F).

### Step-by-step Solution:

1. **Identify the given values**:
- Inductance \( L = 2.0 \, H \)
- Capacitance \( C = 32 \, \mu F = 32 \times 10^{-6} \, F \)

2. **Substitute the values into the formula**:
\[ f_0 = \frac{1}{2\pi \sqrt{L \cdot C}} = \frac{1}{2\pi \sqrt{2.0 \cdot 32 \times 10^{-6}}} \]

3. **Calculate the product \( L \cdot C \)**:
\[ L \cdot C = 2.0 \cdot 32 \times 10^{-6} = 64 \times 10^{-6} \]

4. **Take the square root**:
\[ \sqrt{L \cdot C} = \sqrt{64 \times 10^{-6}} = 8 \times 10^{-3} \]

5. **Substitute back into the frequency formula**:
\[ f_0 = \frac{1}{2\pi (8 \times 10^{-3})} \]

6. **Calculate \( 2\pi (8 \times 10^{-3}) \)**:
\[ 2\pi (8 \times 10^{-3}) \approx 0.05027 \, \text{(using } \pi \approx 3.14159\text{)} \]

7. **Calculate \( f_0 \)**:
\[ f_0 \approx \frac{1}{0.05027} \approx 19.89 \, Hz \]

8. **Round to the nearest whole number**:
- The closest whole number to 19.89 Hz is 20 Hz.

### Final Answer:
The resonant frequency \( f_0 \) is approximately **20 Hz**.