Question

A radio transmitting station operating at a frequency of 120 MHz has two identical antennas that radiate in phase. Antenna B is 9.05 mm 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

Correct question:

A radio transmitting station operating at a frequency of 120 MHz has two identical antennas that radiate in phase. Antenna B is 9.05 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:

values of x in which constructive interference will occur at point P are; 0.775 m, 2.025 m, 3.275 m, 4.525 m, 5.775 m, 7.025 m, 8.275 m

Explanation:

Given;

frequency, F = 120 MH = 120 x 10⁶ Hz

distance between A and B = 9.05 m

make a sketch of this antenna

A-------------------P------------------B

         x                      9.05 - x

Now, we calculate the wavefront between the two antenna

λ = v/f

where;

v is speed of light = 3 x 10⁸ m/s

λ = (3 x 10⁸) / (120 x 10⁶)

λ = 2.5 m

Thus, the constructive interference = n(2.5)

For constructive interference to occur at point P, then

the path difference = constructive interference

Path difference = 9.05 - x - x

                          = 9.05 -2x

∴ 9.05 - 2x = n(2.5)

2x =  9.05 - n(2.5)

x = 4.525 - n(1.25)

Finally, determine the value of x for which n = -1, -2, -3 and 0, 1, 2, 3

when n = -1

x = 4.525 + 1(1.25)

x = 5.775m

when n = -2

x = 4.525 + 2(1.25)

x = 7.025m

when n = -3

x = 4.525 + 3(1.25)

x = 8.275m

when n = 0

x = 4.525 - 0(1.25)

x = 4.525m

when n = 1

x = 4.525 - 1(1.25)

x = 3.275m

when n = 2

x = 4.525 - 2(1.25)

x = 2.025m

when n = 3

x = 4.525 - 3(1.25)

x = 0.775m

Thus, values of x in which constructive interference will occur at point P are; 0.775 m, 2.025 m, 3.275 m, 4.525 m, 5.775 m, 7.025 m, 8.275 m