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?
Exercise
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Distance between the two towers $= 40\ km$
Size of aperture, $a = 50\ m$
Distance of aperture from tower, $Z_f$
Fresnel distance is the half of the distance between the towers.
$\text{i.e},\ \text{Z}_\text{f}=\frac{40}{2}=20\text{km}=20\times10^3\ \text{m}.$
Therefore, now using the formula,
Fresnel distance, $\text{Z}_\text{F}=\frac{\text{a}^2}{\lambda}$
$\text{we have}\ \lambda=\frac{\text{a}^2}{\text{Z}_\text{F}}=\frac{(50)^2}{20\times10^3}$
$\lambda=125\times10^{-3}\ \text{m}=12.5\ \text{cm}.$
This is the required longest wavelength of radio waves, which can be sent in between the towers without considerable diffraction effects.
Size of aperture, $a = 50\ m$
Distance of aperture from tower, $Z_f$
Fresnel distance is the half of the distance between the towers.
$\text{i.e},\ \text{Z}_\text{f}=\frac{40}{2}=20\text{km}=20\times10^3\ \text{m}.$
Therefore, now using the formula,
Fresnel distance, $\text{Z}_\text{F}=\frac{\text{a}^2}{\lambda}$
$\text{we have}\ \lambda=\frac{\text{a}^2}{\text{Z}_\text{F}}=\frac{(50)^2}{20\times10^3}$
$\lambda=125\times10^{-3}\ \text{m}=12.5\ \text{cm}.$
This is the required longest wavelength of radio waves, which can be sent in between the towers without considerable diffraction effects.
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