An antenna is mounted on a 400 m tall building. What will be the wavelength of signal that can be radiated effectively by the transmission tower upto a range of 44 km?

We need to determine the ideal wavelength ($$\lambda$$) of a signal that can be radiated effectively by a transmission tower to cover a specified transmission range.

1. Analyze the Given Parameters

• Height of the building ($$H$$): $$400\text{ m}$$
• Transmission range ($$d$$): $$44\text{ km} = 44 \times 10^3\text{ m}$$
• Radius of the Earth ($$R$$): $$6400\text{ km} = 6.4 \times 10^6\text{ m}$$

2. Calculate the Required Height of the Antenna ($$h_T$$)

The transmission range of a tower is related to its height by the standard horizon line-of-sight propagation formula:

$$d = \sqrt{2Rh_T}$$

Squaring both sides to isolate the height of the transmitting antenna ($$h_T$$):

$$d^2 = 2Rh_T \implies h_T = \frac{d^2}{2R}$$

Substitute the given numerical values into the equation:

$$h_T = \frac{(44 \times 10^3)^2}{2 \times (6.4 \times 10^6)}$$

$$h_T = \frac{1936 \times 10^6}{12.8 \times 10^6} = \frac{1936}{12.8} = 151.25\text{ m}$$

The actual antenna length required to cover that range is $$151.25\text{ m}$$ (which is safely smaller than the building height of $$400\text{ m}$$, allowing it to be perfectly mounted on top).

3. Determine the Safe Wavelength ($\lambda$) Using Radiation Efficiency

For an antenna to radiate an electromagnetic signal effectively with high resonance, its physical length ($$l$$) must be at least a quarter of the signal's wavelength ($$\lambda$$):

$$l = \frac{\lambda}{4} \implies \lambda = 4l$$

Using the calculated functional length of the antenna ($$l = h_T = 151.25\text{ m}$$):

$$\lambda = 4 \times 151.25\text{ m} = 605\text{ m}$$