Question

A radio transmitting station operating at a frequency of 120 MHz has two identical antennas that radiate in phase. Antenna B is 9.00 m to the right of antenna A. Consider point P between the antennas and along the line connecting them, a horizontal distance x to the right of antenna A. For what values of x will constructive interference occur at point P?

Answer 1

x = 4.5 m, 5.75 m, 6.5 m, 8.25 m

Step-by-step explanation:

`f` = frequency of radio transmitting station = 120 MHz = 120 x 10⁶ Hz

`v` = speed of the radio wave = 3 x 10⁸ ms⁻¹

`\lambda` = wavelength

Wavelength is given as

`\lambda = (v)/(f) \\\lambda = (3*10^(8))/(120*10^(6)) \\\lambda = 2.5 m`

`d` = distance between the two antennas = 9.00 m

`x` = Distance of point P from the antenna A

`d - x` = Distance of point P from the antenna B

`\Delta x` = Path difference

Path difference is given as

`\Delta x = x - (d - x)`

`\Delta x = 2 x - d`

For constructive interference, Path difference is given as

`\Delta x = n \lambda`

`2 x - d = n \lambda\\2 x - 9 = 2.5 n`

For n = 0

`2 x - 9 = 2.5 n\\2 x - 9 = 2.5 (0)\\x = 4.5 m`

For n = 1

`2 x - 9 = 2.5 n\\2 x - 9 = 2.5 (1)\\x = 5.75 m`

For n = 2

`2 x - 9 = 2.5 n\\2 x - 9 = 2.5 (2)\\x = 6.5 m`

For n = 3

`2 x - 9 = 2.5 n\\2 x - 9 = 2.5 (3)\\x = 8.25 m`