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

Answer:

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

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

$\lambda = \frac{v}{f} \\\lambda = \frac{3\times10^{8}}{120\times10^{6}} \\\lambda = 2.5 m$

$d$ = 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

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