Question

Two coherent sources of radio waves, A and B, are 5.00 meters apart. Each source emits waves with wavelength 6.00 meters. Consider points along the line connecting the two sources. Find a) At what distance from source A is there constructive interference between points A and B? (answer in meter) b)At what distances from source A is there destructive interference between points A and B? [ Note that there will be two separate interference fringes between point A and point B. Enter your answers in ascending order separated by a comma.]

Answer 1

a) 2.5 m

The condition for constructive interference to occur is

$|d_A-d_B|=m \lambda$ (1)

where

dA is the distance of the point from source A

dB is the distance of the point from source B

m is an integer number

$\lambda=6.0 m$ is the wavelength of the wave

We immediately notice that solutions to the equation are only possible if m=0: otherwise, the difference between dA and dB would be at least 6.0 m, which is impossible since the two sources have a separation of 5.0 m (and we are considering only points between the two sources). Therefore we have

$d_A - d_B = 0\\d_A = d_B = \frac{5.0 m}{2}=2.5 m$

So, constructive interference occurs halfway between the two sources.

b) 1 m, 4 m

The condition for destructive interference to occur is

$|d_A-d_B|=(m+\frac{1}{2}) \lambda$ (1)

As before, solutions are only possible for m=0, otherwise the point would not lie between the two sources.

So for m=0, we have:

$|d_A - d_B| = \frac{\lambda}{2} = 3$

So, the two possible solutions are (using $d_B = 5- d_A$)

$d_A = d_B + 3 \\d_A = (5-d_A)+3\\2d_A = 5+3 = 8\\d=4$

and

$d_B - d_A = 3\\(5-d_A)-d_A=3\\5-2d_A = 3\\2d_A = 2\\d_A = 1$

So, the solutions are dA= 1 m and dA=4 m.