Question

A radio transmitting station operating at a frequency of 125 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? Enter your answers in ascending order separated by a comma.

Answer 1

The answer is: 4.53 m, 4.03 m, 2.13 m, 0.93 m,..., are the values of x constructive interference occur at point P

Step-by-step explanation:

The wavelength is:

`\lambda =(v)/(f)`

Where

v = 3x10⁸m/s

f = 125 MHz = 125x10⁶Hz

Replacing:

`\lambda =(3x10^(8) )/(125x10^(6) ) =2.4m`

The expression of path difference is equal:

9.05 - x - x = 9.05 -2x (exp. 1)

nλ (exp. 2)

Matching both expressions:

9.05 - 2x = nλ

9.05 - 2x = 2.4λ

Integers for n between 0 and 9.05

For n = 0

`9.05-2x=0*2.4\\9.05-2x=0\\2x=9.05\\x=(9.05)/(2) =4.53m`

For n = 1

`9.05-2x=1*2.4\\9.05-2x=1\\2x=9.05-1\\x=(8.05)/(2) =4.03m`

For n = 2

`9.05-2x=2*2.4\\9.05-2x=4.8\\2x=9.05-4.8\\x=(4.25)/(2) =2.13m`

For n = 3

`9.05-2x=3*2.4\\9.05-2x=7.2\\2x=9.05-7.2\\x=(1.85)/(2) =0.93m`