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 21.0 km. The second path 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 and that the wavelength broadcast by the radio station is 350 m. Find the minimum height of the ionospheric layer that could produce destructive interference between the direct and reflected beams. (Assume that no phase changes occur on reflection.)

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 21.0 km. The second path 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 and that the wavelength broadcast by the radio station is 350 m. Find the minimum height of the ionospheric layer that could produce destructive interference between the direct and reflected beams. (Assume that no phase changes occur on reflection.)

Answer 1

To find the minimum height of the ionospheric layer that could produce destructive interference between the direct and reflected waves, we need to consider the path difference between the two waves.

Let's start by understanding the concept of destructive interference. Destructive interference occurs when the crest of one wave aligns with the trough of another wave, resulting in cancelation and a minimum amplitude. This happens when the path difference between the waves is equal to half the wavelength.

In this scenario, the direct wave travels straight from the transmitter to the receiver, while the reflected wave travels an extra distance due to the reflection. The path difference, Δd, between the two waves can be calculated as:

Δd = 2d

where d is the additional distance traveled by the reflected wave.

Now let's calculate d. We know that the radio station is 21.0 km away from the receiver. Since the reflection takes place midway between the receiver and transmitter, d is half of this distance:

d = 21.0 km / 2 = 10.5 km

But we need to convert this value to meters, as the wavelength is given in meters:

d = 10.5 km × 1000 m/km = 10,500 m

Now we can find the minimum height of the ionospheric layer. The path difference Δd is equal to half the wavelength:

Δd = λ/2

Substituting the given wavelength:

10,500 m = 350 m/2

Now solve for λ:

λ = (10,500 m) × 2 = 21,000 m

However, this is the total distance traveled by the reflected wave, which consists of two parts: the upward path and the downward path. The upward path will follow the same angle as the direct wave, so we need to consider only the vertical distance.

The minimum height of the ionospheric layer can be calculated using the equation:

h = λ/4

Substituting the calculated value of λ:

h = 21,000 m/4 = 5,250 m

Therefore, the minimum height of the ionospheric layer that could produce destructive interference between the direct and reflected waves is 5,250 meters.