Very confused: why is the path difference in double slit interference dsinθ?

[lillybeans]

Homework Statement

The more I think about path difference, the more confused I get.

First of all, HOW DO PEOPLE know that the path difference is dsinθ? Why do we draw a line coming from the first ray that is PERPENDICULAR to the second ray when determining path difference?

Please see diagram:

If the PD is the extra distance that ray B travels, then subtracting the PD would leave AP=OP, so now you have an isosceles triangle, and the two blue angles must equal. Now, if we do "dsinθ" to determine the PD, then that gives us a right triangle. If the orange angle is always 90 degrees, then the blue angles must also be 90 degrees. This doesn't make sense, because this suggests that the two rays are parallel. If they are parallel, they'll never converge to the same point and interfere. So my question is: is this "dsinθ" formula for PD simply a simplified assumption because the slits are so close together and the screen is so far away that the rays can be thought of as "nearly parallel"? But strictly speaking the orange angle is NOT actually 90 degrees in reality?

Please help. Thanks!

 

[Emilyjoint]

The dSin(theta) equation is only true for small angles which means roughly less than 10degrees

 

[lillybeans]

The dSin(theta) equation is only true for small angles which means roughly less than 10degrees

so the formula IS an approximation and the right angle is not mathematically-proven using geometry...

 

[Emilyjoint]

I think that is more or less correct.
The smaller the angle the more that little triangle looks like a 90degree triangle.
It is an approximation but it is surprising how close Sin(theta), Tan(theta) and theta in radians are for angles less than about 10 or 20degrees

 

[lillybeans]

I think that is more or less correct.
The smaller the angle the more that little triangle looks like a 90degree triangle.
It is an approximation but it is surprising how close Sin(theta), Tan(theta) and theta in radians are for angles less than about 10 or 20degrees

Thanks, I was quite confused as to why the orange angle is always 90 degrees, but I think that it makes more sense now that it's just an approximation.