Calculated the phase velocity of electromagnetic wave having electron density and frequency for D-layer : (N = 400 electrion/cc and v = 300 kHz)

Options

A. `3 xx 10^(8)` m/s
B. `3.75 xx 10^(8)` m/s
C. `6.8 xx 10^(8)` m/s
D. `1.1 xx 10^(9)` m/s

Answer

To calculate the phase velocity of an electromagnetic wave in the D-layer with the given electron density and frequency, we can follow these steps:

### Step 1: Convert Electron Density
The electron density \( N \) is given as 400 electrons per cubic centimeter. We need to convert this to electrons per cubic meter.

\[ N = 400 \, \text{electrons/cc} = 400 \times 10^6 \, \text{electrons/m}^3 = 4 \times 10^8 \, \text{electrons/m}^3 \]

### Step 2: Convert Frequency
The frequency \( v \) is given as 300 kHz. We need to convert this to hertz.

\[ v = 300 \, \text{kHz} = 300 \times 10^3 \, \text{Hz} = 3 \times 10^5 \, \text{Hz} \]

### Step 3: Calculate the Refractive Index
The refractive index \( \mu \) for the D-layer can be calculated using the formula:

\[ \mu = \sqrt{1 - \frac{81.45 \times N}{v^2}} \]

Substituting the values of \( N \) and \( v \):

\[ \mu = \sqrt{1 - \frac{81.45 \times 4 \times 10^8}{(3 \times 10^5)^2}} \]

Calculating \( v^2 \):

\[ (3 \times 10^5)^2 = 9 \times 10^{10} \]

Now substituting this back into the refractive index equation:

\[ \mu = \sqrt{1 - \frac{81.45 \times 4 \times 10^8}{9 \times 10^{10}}} \]

Calculating the fraction:

\[ \frac{81.45 \times 4 \times 10^8}{9 \times 10^{10}} = \frac{325.8 \times 10^8}{9 \times 10^{10}} = \frac{325.8}{9} \times 10^{-2} \approx 36.2 \times 10^{-2} = 0.362 \]

Now substituting this value back into the equation for \( \mu \):

\[ \mu = \sqrt{1 - 0.362} = \sqrt{0.638} \approx 0.798 \]

### Step 4: Calculate the Phase Velocity
The phase velocity \( v_p \) of the electromagnetic wave can be calculated using the formula:

\[ v_p = \frac{c}{\mu} \]

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

Substituting the values:

\[ v_p = \frac{3 \times 10^8}{0.798} \approx 3.75 \times 10^8 \, \text{m/s} \]

### Final Answer
The phase velocity of the electromagnetic wave in the D-layer is approximately:

\[ v_p \approx 3.75 \times 10^8 \, \text{m/s} \]

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