A L-C resonant circuit contains a 200 pF capacitor and a `100 muG` inductor. It is set into oscillation coupled to an antenna. The wavelength of the radiated electromagnetic waves is

Options

A. `272cm`
B. `0.0272 m`
C. `377 cm`
D. `3.77 cm`

Explanation

To find the wavelength of the radiated electromagnetic waves from an L-C resonant circuit, we will follow these steps:

### Step 1: Calculate the resonant frequency (f) of the LC circuit
The resonant frequency of an LC circuit is given by the formula:

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

Where:
- \(L\) is the inductance in henries (H)
- \(C\) is the capacitance in farads (F)

### Step 2: Convert the given values to standard units
- The inductance \(L = 100 \mu H = 100 \times 10^{-6} H = 10^{-4} H\)
- The capacitance \(C = 200 pF = 200 \times 10^{-12} F = 2 \times 10^{-10} F\)

### Step 3: Substitute the values into the resonant frequency formula
Substituting the values of \(L\) and \(C\):

\[ f = \frac{1}{2\pi\sqrt{(10^{-4})(2 \times 10^{-10})}} \]

Calculating the product inside the square root:

\[ LC = 10^{-4} \times 2 \times 10^{-10} = 2 \times 10^{-14} \]

Now, taking the square root:

\[ \sqrt{LC} = \sqrt{2 \times 10^{-14}} = \sqrt{2} \times 10^{-7} \]

Now substituting back into the frequency formula:

\[ f = \frac{1}{2\pi(\sqrt{2} \times 10^{-7})} \]

Calculating \(2\pi\sqrt{2}\):

\[ 2\pi\sqrt{2} \approx 8.885 \]

Thus,

\[ f \approx \frac{1}{8.885 \times 10^{-7}} \approx 1.126 \times 10^{6} Hz \approx 112.6 \times 10^{3} Hz \]

### Step 4: Calculate the wavelength (λ)
The wavelength of the electromagnetic wave can be calculated using the formula:

\[ \lambda = \frac{c}{f} \]

Where:
- \(c\) is the speed of light, approximately \(3 \times 10^{8} m/s\)

Substituting the values:

\[ \lambda = \frac{3 \times 10^{8}}{1.126 \times 10^{6}} \approx 266.5 m \]

### Step 5: Convert the wavelength to centimeters
To convert meters to centimeters, multiply by 100:

\[ \lambda \approx 266.5 \times 100 \approx 26650 cm \]

### Step 6: Round to the nearest option
Among the given options, the nearest value to \(266.5 cm\) is \(272 cm\).

Thus, the final answer is:

**The wavelength of the radiated electromagnetic waves is approximately 272 cm.**