Two radio antennae radiating in phase

[sghaussi]

hii guys! I'm having trouble with a homework and I was wondering if you could explain something to me. Here's the problem:


Two radio antennas A and B radiate in phase. Antenna B is a distance of X meters to the right of antenna A. Consider point Q along the extension of the line connecting the antennas, a horizontal distance of Y meters to the right of antenna B. The frequency, and hence the wavelength, of the emitted waves can be varied.

What is the longest wavelength for which there will be destructive interference at point Q?


I know I want to use the formula: (r2 - r1) = (m + 1/2)lamda

my r2 is the distance from B to Q which is Y meters
my r1 is the distance from A to Q which is Y + X meters

I'm trying to solve for lamda, but I don't know what m is. I know m has to do with "the longest wavelength" but I'm not sure how to calculate it. Or is it 0? I was guessing it would be 0 or 1. The book isn't clear. I hope you can help me!

 

[quasar987]

hii guys! I'm having trouble with a homework and I was wondering if you could explain something to me. Here's the problem:


Two radio antennas A and B radiate in phase. Antenna B is a distance of X meters to the right of antenna A. Consider point Q along the extension of the line connecting the antennas, a horizontal distance of Y meters to the right of antenna B. The frequency, and hence the wavelength, of the emitted waves can be varied.

What is the longest wavelength for which there will be destructive interference at point Q?


I know I want to use the formula: (r2 - r1) = (m + 1/2)lamda

my r2 is the distance from B to Q which is Y meters
my r1 is the distance from A to Q which is Y + X meters

I'm trying to solve for lamda, but I don't know what m is. I know m has to do with "the longest wavelength" but I'm not sure how to calculate it. Or is it 0? I was guessing it would be 0 or 1. The book isn't clear. I hope you can help me!

You'll want to set r_2 = Y+X and r_1=X, in order to get r_2 - r_1 > 0 (otherwise you'll get a negative wavelength).

Ok, where does the formula (r2 - r1) = (m + 1/2)lamda for destructive interference comes from? Let's try to understand it. It says that if the difference of the distances r_2 and r_1 is 1/2, 3/2, 5/2, etc. of the wavelength, then we'll have destructive interference. Why? Because then the waves will arrive at point Q completely out of phase (completely = phase difference of pi), like illustrated here: http://scienceworld.wolfram.com/physics/DestructiveInterference.html

Ok, that explanation sucked. But the formula says that we have destructive interference at point Q whenever $r_2-r_1 = (m+1/2)\lambda$ where $m=0,1,2,3,...$. Here, r_2 - r_1 = X, so the formula tells us that we have destructive interference whenever

$$X = (m+1/2)\lambda \Leftrightarrow \lambda = \frac{X}{m+1/2}$$

For which m between 0,1,2,3,... is lambda the biggest?

 

[sghaussi]

Thanks, that really clears things up, however I'm still confused a little. So by looking at the formula that equals lambda, I would assume that when m = 0, is when lambda is the greatest right? however the answer i am getting is still incorrect. Am i not understanding you correctly?

 

[quasar987]

Thanks, that really clears things up, however I'm still confused a little. So by looking at the formula that equals lambda, I would assume that when m = 0, is when lambda is the greatest right?

That's what I'd say too.

however the answer i am getting is still incorrect. Am i not understanding you correctly?

The answer is not 2X? What do they say it is?

 

[sghaussi]

okay.. my last comment.. ignore it, i wrote too soon! it was my fault i kept entering the wrong numbers in my calculator. thanks so much!

 

[quasar987]

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