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?

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Respuesta :

JemdetNasr JemdetNasr · Answer 1 · 2019-12-24T06:39:34+01:00

Answer:

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

Explanation:

[tex]f[/tex] = frequency of radio transmitting station = 120 MHz = 120 x 10⁶ Hz

[tex]v[/tex] = speed of the radio wave = 3 x 10⁸ ms⁻¹

[tex]\lambda[/tex] = wavelength

Wavelength is given as

[tex]\lambda = \frac{v}{f} \\\lambda = \frac{3\times10^{8}}{120\times10^{6}} \\\lambda = 2.5 m[/tex]

[tex]d[/tex] = distance between the two antennas = 9.00 m

[tex]x[/tex] = Distance of point P from the antenna A

[tex]d - x[/tex] = Distance of point P from the antenna B

[tex]\Delta x[/tex] = Path difference

Path difference is given as

[tex]\Delta x = x - (d - x)[/tex]

[tex]\Delta x = 2 x - d[/tex]

For constructive interference, Path difference is given as

[tex]\Delta x = n \lambda[/tex]

[tex]2 x - d = n \lambda\\2 x - 9 = 2.5 n[/tex]

For n = 0

[tex]2 x - 9 = 2.5 n\\2 x - 9 = 2.5 (0)\\x = 4.5 m[/tex]

For n = 1

[tex]2 x - 9 = 2.5 n\\2 x - 9 = 2.5 (1)\\x = 5.75 m[/tex]

For n = 2

[tex]2 x - 9 = 2.5 n\\2 x - 9 = 2.5 (2)\\x = 6.5 m[/tex]

For n = 3

[tex]2 x - 9 = 2.5 n\\2 x - 9 = 2.5 (3)\\x = 8.25 m[/tex]