Question

The waves from a radio station can reach a home receiver by two paths. One is a straight-line path from transmitter to home, a distance of 32.0 km. The second is by reflection from the ionosphere (a layer of ionized air molecules high in the atmosphere). Assume this reflection takes place at a point midway between receiver and transmitter, the wavelength broadcast by the radio station is 344 m, and no phase change occurs on reflection. Find the minimum height of the ionospheric layer that could produce destructive interference between the direct and reflected beams.

Answer 1

Step-by-step explanation:

The path of waves reaching directly and through reflection from ionosphere will form a isosceles triangle.

Let the height be h . This will be height of a triangle of equal side whose base is of length 32 km. we shall calculate the length of one side of this triangle .

this length be l

l² = 16² + h²

l = √(256 +h² )

2l = 2√(256 +h² )

path difference = 2l - 32 km

For destructive interference

path difference = wavelength /2 for minimum height .

2l - 32 = .344/2

2√(256 +h² )- 32 = . 172

2√(256 +h² ) = 32.172

4(256 +h² ) = 1035

(256 +h² ) = 258.76

h² = 2.76

h = 1.66 km