URGENT HELP with Single slit application problem is please?

Question

Me and my friends ave homework due, so please help us!

You are driving down the road in your car and you notice that your favorite radio station that you are listening to, 95.7 MHz, is going in and out. You look out and over and you see two long buildings with a gap in them. You know that you are a distance of 100 meters from the buildings. As you pass directly across from the gap in the buildings, you notice that the readio reception is the strongest. From there it continually get worse and worse untill you have traveled 17.6 meters where there is no reception. It then comes back in but goes out agian after traveling 37.1 meters from the center point (19.5 m further down the road), and it goes out agian at 61.1 meters from where you started measuring.

What is the width of the gap between the buildings?

Answer 1

Hi there,

There are a few factors that you have not been given. You need to know the velocity of light which is 299 792 km per second. Radio waves propogate at the same speed.

Therefore the wave length of the radio wave is 299,792 / 95,7 meters

= 3,1326 m. Where the radio waves are in phase, i.e. directly opposite the gap there is constructive interference and where there is no signal there is destructive interference.

So at 17,6 m there is destructive interference so the waves are exactly a half wave length out of phase.

One half wave = 3,1326 / 2  m

= 1,5336 meters

Now I've drawn a diagram that I can't show you, but let the distance between the buildings be 2 X  meters

From the geometry of my diagram we have two triangles with a common height of 100 meters. From this we get: -

(d1)^2  = 10^4 + (X+ 17,6)^2   and

(d2)^2  + 10^4  + ( X-17,6)^2

Also d1 - d2  = 1,5336  (the difference in wave path)

So (10^4 + (X+17,6)^2) - (10^4 +( X -17,6)^2)  = 1.5336

Therefore X^2 + 35,2X +309,76 -X^2 +35X - 309,76 = 1,5336

This simpilfies to: -

70,4X = 1,5336

So X = 0,0219 m

Therefore the gap = 2X

= 0,0438 m

Seems a very small gap so I'll need to check the calcs.

(The null spot another 19,5 m on is second oder destructive interfernce)