Find the resonant frequency of a series circuit consist of an inductance 200 `muH`, a capacitance of `0.0005muF` and a resistance of 10 `Omega`.

Options

A. 480 kHz
B. 503 kHz
C. 406 kHz
D. 607 kHz

Answer

To find the resonant frequency of a series circuit consisting of an inductance (L), a capacitance (C), and a resistance (R), we can follow these steps:

### Step 1: Write down the formula for resonant frequency
The resonant frequency (f₀) of a series RLC circuit is given by the formula:
\[ f_0 = \frac{1}{2\pi\sqrt{LC}} \]
where:
- \( L \) is the inductance in henries (H)
- \( C \) is the capacitance in farads (F)

### Step 2: Convert the given values to standard units
- Inductance \( L = 200 \, \mu H = 200 \times 10^{-6} \, H = 2 \times 10^{-4} \, H \)
- Capacitance \( C = 0.0005 \, \mu F = 0.0005 \times 10^{-6} \, F = 5 \times 10^{-10} \, F \)

### Step 3: Substitute the values into the formula
Now substitute the values of \( L \) and \( C \) into the resonant frequency formula:
\[ f_0 = \frac{1}{2\pi\sqrt{(2 \times 10^{-4})(5 \times 10^{-10})}} \]

### Step 4: Calculate the product \( LC \)
First, calculate the product \( LC \):
\[ LC = (2 \times 10^{-4})(5 \times 10^{-10}) = 10 \times 10^{-14} = 1 \times 10^{-13} \]

### Step 5: Calculate the square root of \( LC \)
Now, find the square root:
\[ \sqrt{LC} = \sqrt{1 \times 10^{-13}} = 1 \times 10^{-6.5} = 10^{-6.5} \]

### Step 6: Substitute back into the frequency formula
Now substitute back into the frequency formula:
\[ f_0 = \frac{1}{2\pi(10^{-6.5})} \]

### Step 7: Calculate \( 2\pi \)
Calculate \( 2\pi \):
\[ 2\pi \approx 6.2832 \]

### Step 8: Calculate the resonant frequency
Now calculate \( f_0 \):
\[ f_0 = \frac{1}{6.2832 \times 10^{-6.5}} \approx \frac{1}{6.2832 \times 3.162 \times 10^{-7}} \approx \frac{1}{1.985 \times 10^{-6}} \approx 503 \, kHz \]

### Final Answer
Thus, the resonant frequency \( f_0 \) is approximately:
\[ f_0 \approx 503 \, kHz \]