A 50 MHz sky wave sky wave takes 4.04 ms to reach a receiver via re-transmission from a satellite 600 km above earth's surface. Assuming re-transmission time by satellite negligible, find the distance between source and receiver. If communication between the two was to be done by Line of sight (LOS) method,what should be the size of transmitting antenna ?

Options

A. 606km
B. 170km
C. 340km
D. 280km

Explanation

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the Speed of the Signal
The speed of the signal (which is a sky wave in this case) is equal to the speed of light, which is approximately:
\[ c = 3 \times 10^8 \text{ m/s} \]

### Step 2: Convert Time to Seconds
The time taken for the signal to reach the receiver is given as 4.04 milliseconds. We need to convert this time into seconds:
\[ t = 4.04 \text{ ms} = 4.04 \times 10^{-3} \text{ s} \]

### Step 3: Calculate the Distance Traveled by the Signal
The distance traveled by the signal is given by the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
Since the signal travels to the satellite and back to the receiver, the total distance \(D\) is:
\[ D = c \times t = (3 \times 10^8 \text{ m/s}) \times (4.04 \times 10^{-3} \text{ s}) \]
Calculating this gives:
\[ D = 1.212 \times 10^6 \text{ m} = 1212 \text{ km} \]

### Step 4: Calculate the Distance from the Receiver to the Satellite
Since the signal travels to the satellite and back, the one-way distance \(x\) from the receiver to the satellite is:
\[ x = \frac{D}{2} = \frac{1212 \text{ km}}{2} = 606 \text{ km} \]

### Step 5: Use Pythagorean Theorem to Find the Horizontal Distance
In a line-of-sight communication system, we can use the Pythagorean theorem to find the horizontal distance \(d\) between the source and receiver. The height of the satellite \(h_s\) is given as 600 km. Thus:
\[ d = \sqrt{x^2 - h_s^2} \]
Substituting the values:
\[ d = \sqrt{(606 \text{ km})^2 - (600 \text{ km})^2} \]

### Step 6: Calculate the Value of \(d\)
Calculating the squares:
\[ d = \sqrt{(606^2) - (600^2)} = \sqrt{367236 - 360000} = \sqrt{7236} \]
Calculating the square root:
\[ d \approx 85 \text{ km} \]

### Step 7: Calculate the Total Distance Between Source and Receiver
Since the total distance between the source and receiver is twice the horizontal distance:
\[ \text{Total Distance} = 2d = 2 \times 85 \text{ km} = 170 \text{ km} \]

### Step 8: Determine the Size of the Transmitting Antenna
For line-of-sight communication, the size of the transmitting antenna can be estimated using the formula:
\[ \text{Size of Antenna} \approx \frac{\lambda}{2} \]
Where \(\lambda\) is the wavelength, calculated as:
\[ \lambda = \frac{c}{f} \]
Given \(f = 50 \text{ MHz} = 50 \times 10^6 \text{ Hz}\):
\[ \lambda = \frac{3 \times 10^8 \text{ m/s}}{50 \times 10^6 \text{ Hz}} = 6 \text{ m} \]
Thus, the size of the antenna:
\[ \text{Size of Antenna} \approx \frac{6 \text{ m}}{2} = 3 \text{ m} \]

### Final Answers
- The distance between the source and receiver is **170 km**.
- The size of the transmitting antenna should be approximately **3 m**.