Diffraction Grating Overlapping Orders

[Ailiniel]

Homework Statement

The same diffraction grating is used with two different wavelengths of light, λA and λB. The fourth-order principal maximum of light A exactly overlaps the third-order principal maximum of light B. Find the ratio λA/λB.

Homework Equations

sin theta = m lambda / width of the slits

The Attempt at a Solution

sin theta 1 = sin theta 2 since they overlap.

3/4

I don't understand how can 4th order principal and 3rd order principle can have the same sin theta? Because, unless I am misunderstanding something, I don't think it's possible for different orders to completely overlap on top of each other so that they meet at the same point on the opposite screen of the diffraction grating.

 

[Simon Bridge]

The spacing between fringes depends on the wavelength.
This would happen if, say, light A created fringes 3deg apart and light B 4deg apart - the third fringe of B would be at 12deg which is the same as the 4th fringe of A.

In this case - you are correct:
[itex]4\lambda_A = 3\lambda_B \Rightarrow \frac{\lambda_A}{\lambda_B}=\frac{3}{4}[/itex]

This result demonstrates the conditions required for this to happen.

eg. if λ_A=330nm then λ_B=440nm

Ideally I should show you:

... the green 2nd order matches with the purple 3rd order. If we tweaked the wavelength only slightly it would be as exact as you like.