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:

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

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

Given:
- \( L = 2.0 \, H \)
- \( C = 32 \, \mu F = 32 \times 10^{-6} \, F \)

Now, let's calculate the resonant frequency step by step.

### Step 1: Convert capacitance to farads
We have already converted the capacitance:
\[ C = 32 \, \mu F = 32 \times 10^{-6} \, F \]

### Step 2: Substitute values into the formula
Now we can substitute \( L \) and \( C \) into the formula for resonant frequency:
\[ f_r = \frac{1}{2\pi\sqrt{LC}} = \frac{1}{2\pi\sqrt{2.0 \times (32 \times 10^{-6})}} \]

### Step 3: Calculate \( LC \)
First, calculate \( LC \):
\[ LC = 2.0 \times (32 \times 10^{-6}) = 64 \times 10^{-6} \, H \cdot F \]

### Step 4: Calculate the square root of \( LC \)
Now, calculate the square root:
\[ \sqrt{LC} = \sqrt{64 \times 10^{-6}} = 8 \times 10^{-3} \, \text{s} \]

### Step 5: Substitute back into the frequency formula
Now substitute \( \sqrt{LC} \) back into the frequency formula:
\[ f_r = \frac{1}{2\pi(8 \times 10^{-3})} \]

### Step 6: Calculate \( f_r \)
Now calculate \( f_r \):
\[ f_r = \frac{1}{2 \times 3.14 \times 8 \times 10^{-3}} \approx \frac{1}{0.05024} \approx 19.87 \, Hz \]

### Step 7: Round to appropriate significant figures
Rounding to two significant figures, we get:
\[ f_r \approx 20 \, Hz \]

Thus, the resonant frequency of the series LCR circuit is approximately **20 Hz**.