Question

Two towers on top of two hills are 40 km apart. The line joining them passes 50 m above a hill halfway between the towers. What is the longest wavelength of radio waves, which can be sent between the towers without appreciable diffraction effects?

Answer 1

Hint: In this question using the concept of Fresnel distance .i.e. minimum distance a beam of light can travel before it deviates from the straight-line path due to diffraction take ${Z_F}$= 20,000 m and use the formula ${Z_F} = \dfrac{{{a^2}}}{\lambda }$ to find the answer.

Complete step-by-step solution -

According to the given information the distance between the two towers = 40 km
We know that for a distance smaller than Fresnel’s distance (${Z_F}$), spreading due to diffraction is smaller as compared to the size of the waves.
But the hill is halfway between the towers so distance up to which much of diffraction will not be observable should be $\dfrac{{40}}{2} = 20km = 20,000m$
We know that Fresnel’s distance is the minimum distance a beam of light can travel before it deviates from a straight line path due to diffraction becomes significant.
Hence $Z_F$ = 20,000 m
The radial spread of the beam over the hill 20 km away must not exceed 50 m for the hill not to obstruct the spreading radio beam.
Since we know that ${Z_F} = \dfrac{{{a^2}}}{\lambda }$
Where a = 50 m and ${Z_F}$= 20,000 m
Therefore $\lambda = \dfrac{{{a^2}}}{{{Z_F}}}$
$\Rightarrow$ $\lambda = \dfrac{{{{50}^2}}}{{20,000}} = 0.125m$
Therefore the longest wavelength of radio waves can be sent between the towers without appreciable diffraction = 0.125 m.

Note: Here we find a word radio wave that can be described as radio waves are a form of electromagnetic radiation with wavelengths longer than infrared light in the electromagnetic spectrum. The frequencies of the radio waves are as high as 300 gigahertz (GHz) and as low as 30 hertz (Hz). A radio wave's wavelength can be anything from shorter than a grain of rice to larger than the Earth's diameter. As with all other electromagnetic waves, radio waves travel in the vacuum at the speed of light. They are produced by accelerated electrical charges, such as time-varying electrical currents. Lightning and astronomical objects emit naturally occurring radio waves.