A ground receiver station is receiving a signal at (a) 5 MHz and (b) 100MHz.transmitted from a ground transmitter at a height of 400 m located at a distance of 125 km on whether coming via space wave of sky wave propagation or satellite transponder Radius of earth=`6.4 xx10^(6)` m, maximum number density of electrons in lonosphere =`10^(12) m^(-3)`

Answer

To determine whether the signals at 5 MHz and 100 MHz are received via space wave or sky wave propagation, we will follow these steps:

### Step 1: Calculate the maximum distance for space wave propagation
The formula for the maximum distance \( d \) for space wave propagation is given by:

\[ d = \sqrt{2 \cdot r \cdot h} \]

where:
- \( r \) is the radius of the Earth (given as \( 6.4 \times 10^6 \) m),
- \( h \) is the height of the transmitter (given as 400 m).

### Calculation:
1. Substitute the values into the formula:
\[ d = \sqrt{2 \cdot (6.4 \times 10^6) \cdot 400} \]
2. Calculate:
\[ d = \sqrt{2 \cdot 6.4 \times 10^6 \cdot 400} = \sqrt{5.12 \times 10^9} \approx 71.55 \times 10^3 \text{ m} = 71.55 \text{ km} \]

### Step 2: Compare the calculated distance with the given distance
The distance between the transmitter and receiver is given as 125 km.

- Since 125 km > 71.55 km, the space wave propagation is not possible.

### Step 3: Determine the critical frequency for sky wave propagation
The critical frequency \( \nu_c \) can be calculated using the formula:

\[ \nu_c = 9 \sqrt{n_{\text{max}}} \]

where:
- \( n_{\text{max}} \) is the maximum number density of electrons in the ionosphere (given as \( 10^{12} \, \text{m}^{-3} \)).

### Calculation:
1. Substitute the value of \( n_{\text{max}} \):
\[ \nu_c = 9 \sqrt{10^{12}} = 9 \times 10^6 \, \text{Hz} = 9 \, \text{MHz} \]

### Step 4: Compare the frequencies
- The received frequency at the ground station is 5 MHz for the first case and 100 MHz for the second case.
- Since 5 MHz < 9 MHz, sky wave propagation is possible for the 5 MHz signal.
- For the 100 MHz signal, since 100 MHz > 9 MHz, sky wave propagation is not possible.

### Conclusion:
- The signal at **5 MHz** can be received via **sky wave propagation**.
- The signal at **100 MHz** cannot be received via **sky wave propagation**.