An `L-c` resonant circuit contains a `200 pF` capacitor and a `200muH` inductor. It is into oscillation coupled to an antenna. The wavelength of the radiated electromagnetic waves is

Options

A. `377 mm`
B. `377 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 can follow these steps:

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

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

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 \(L = 200 \, \mu H = 200 \times 10^{-6} \, H\)
- The capacitance \(C = 200 \, pF = 200 \times 10^{-12} \, F\)

### Step 3: Substitute the values into the frequency formula
Now, substituting the values into the formula:

\[ \nu = \frac{1}{2\pi\sqrt{(200 \times 10^{-6})(200 \times 10^{-12})}} \]

### Step 4: Calculate the product of \(L\) and \(C\)
Calculating \(L \cdot C\):

\[ L \cdot C = (200 \times 10^{-6}) \cdot (200 \times 10^{-12}) = 40 \times 10^{-18} = 4 \times 10^{-17} \, H \cdot F \]

### Step 5: Calculate the square root
Now, find the square root:

\[ \sqrt{L \cdot C} = \sqrt{4 \times 10^{-17}} = 2 \times 10^{-8.5} = 2 \times 10^{-8.5} \, H \cdot F \]

### Step 6: Calculate the frequency
Now substitute back into the frequency formula:

\[ \nu = \frac{1}{2\pi(2 \times 10^{-8.5})} \]

Calculating this gives:

\[ \nu \approx 0.079 \times 10^{7} \, Hz \]

### Step 7: Calculate the wavelength (λ)
The wavelength of the electromagnetic wave can be calculated using the formula:

\[ \lambda = \frac{C}{\nu} \]

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

### Step 8: Substitute the frequency into the wavelength formula
Substituting the values:

\[ \lambda = \frac{3 \times 10^{8}}{0.079 \times 10^{7}} \]

### Step 9: Calculate the wavelength
Calculating this gives:

\[ \lambda \approx 377 \, m \]

### Final Answer
The wavelength of the radiated electromagnetic wave is approximately **377 meters**.

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