In communication with help of antenna if height is doubled then the range covered which gas initially r would become

Options

A. `sqrt2r`
B. 3r
C. 4r
D. 5r

Answer

To solve the problem of how the range covered by an antenna changes when its height is doubled, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The range \( R \) covered by an antenna is related to its height \( h \). The relationship can be derived using the geometry of the situation where the antenna height and the Earth's radius form a right triangle.

2. **Using Pythagorean Theorem**:
In a right triangle formed by the antenna height and the Earth's radius, we can apply the Pythagorean theorem:
\[ R^2 + R_e^2 = (h + R_e)^2 \]
where \( R \) is the range, \( R_e \) is the radius of the Earth, and \( h \) is the height of the antenna.

3. **Simplifying the Equation**:
Expanding the equation gives:
\[ R^2 + R_e^2 = h^2 + 2hR_e + R_e^2 \]
Cancelling \( R_e^2 \) from both sides leads to:
\[ R^2 = h^2 + 2hR_e \]

4. **Approximation for Small Heights**:
If the height \( h \) is much smaller than the Earth's radius \( R_e \), we can approximate:
\[ R \approx \sqrt{2hR_e} \]
This shows that the range \( R \) is proportional to the square root of the height \( h \):
\[ R \propto \sqrt{h} \]

5. **Doubling the Height**:
If the height is doubled (i.e., \( h \) becomes \( 2h \)), we can find the new range \( R_1 \):
\[ R_1 \propto \sqrt{2h} \]

6. **Finding the Ratio**:
To find the relationship between the new range \( R_1 \) and the original range \( R \):
\[ \frac{R_1}{R} = \frac{\sqrt{2h}}{\sqrt{h}} = \sqrt{2} \]
Therefore, we can express \( R_1 \) in terms of \( R \):
\[ R_1 = \sqrt{2} R \]

7. **Conclusion**:
The new range \( R_1 \) when the height is doubled is \( \sqrt{2} \) times the original range \( R \).

### Final Answer:
The range covered when the height is doubled becomes \( R_1 = \sqrt{2} R \).

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