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

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

Wave Optics

Solution:

Distance between the towers is $40 km$.
Height of the line joining the hills is $d =50 m$.
Thus, the radial spread of the radio waves should not exceed $50 m$.
Since the hill is located halfway between the towers, Fresnel's distance can be obtained.
$ Z _{ P }=20 km $
Aperture is $a = d =50 m$
Fresnel's distance is given by the relation,
$ \begin{array}{l} Z_{ P }= a ^{2} / \lambda=2 \times 10^{4} \\ \lambda= a ^{2} / Z_{ P }=12.5 cm \end{array} $