Path difference between the waves

[toforfiltum]

Homework Statement

Homework Equations

The Attempt at a Solution

I formed a right angle triangle between the arrow downwards,d, and perpendicular line x from arrow. As such, sin θ = x/d.

x= d sinθ
path difference: d sinθ/λ

My answer is C, but answer is B. Where did cos θ come from?

 

[DrClaude]

Looking at the figure, in which case would you expect it to be no difference in path length?

 

[gneill]

Your description of how you formed your triangle is not very rigorous, so I may be misinterpreting what you've described. That said, I think you've allocated angle $\theta$ to the wrong corner of your triangle.

Perhaps you could sketch your construction on the image and post the result?

 

[toforfiltum]

Looking at the figure, in which case would you expect it to be no difference in path length?

Is it when the waves are received at angle perpendicular to the horizontal?

 

[DrClaude]

Is it when the waves are received at angle perpendicular to the horizontal?

Which corresponds to which value of θ?

 

[toforfiltum]

Which corresponds to which value of θ?

90°?

 

[toforfiltum]

Your description of how you formed your triangle is not very rigorous, so I may be misinterpreting what you've described. That said, I think you've allocated angle $\theta$ to the wrong corner of your triangle.

Perhaps you could sketch your construction on the image and post the result?

I did it this way, but I think now I'm wrong to assume that x is parallel to the wavelength, right?

 

[gneill]

You've drawn the correct triangle but as you say, you've chosen the wrong "leg" for x. You want the leg that lies along the wave's path.

 

[toforfiltum]

You've drawn the correct triangle but as you say, you've chosen the wrong "leg" for x. You want the leg that lies along the wave's path.
View attachment 91213

Ah, I see now why it's cos θ. But, why it is not D now? Shouldn't path difference be expressed in terms of fraction of wavelengths?

 

[gneill]

Ah, I see now why it's cos θ. But, why it is not D now? Shouldn't path difference be expressed in terms of fraction of wavelengths?

In this case the path difference is just the length, otherwise the problem would have specified to express it in terms of wavelengths.

 

[toforfiltum]

In this case the path difference is just the length, otherwise the problem would have specified to express it in terms of wavelengths.

Ok. Thanks!