Highest order of spectrum given two wavelengths

[MFAHH]

Homework Statement

I've attached the problem

Homework Equations

d*sinθ=mλ

The Attempt at a Solution

I've tried a couple of methods. I considered solving for m when θ=90 degrees => sinθ=1 because 90 degrees is the highest possible angle and will (in my mind at least) yield the highest order of spectrum by rounding the resultant value of m down to the nearest integer. But I don't know how to factor the two given wavelengths into all this.

I then considered fiddling about with approximations like, at small angles, tanθ=sinθ=θ=y/L but to no avail.

Can anyone help me on how to solve this? I don't even know how to picture the problem and draw a diagram.

 

[Simon Bridge]

I don't know how to factor the two given wavelengths into all this

To get a fringe at a position you need constructive interference at that position.
What is the condition for constructive interference?

Note: you probably have not been given the final equation for this situation, you will have to work it out for yourself.

 

[ehild]

The same problem is discussed in thread https://www.physicsforums.com/showthread.php?t=737573
You can also read about diffraction gratings here: http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/grating.html
ehild

 

[MFAHH]

Thanks for the replies.

To get a fringe at a position you need constructive interference at that position.
What is the condition for constructive interference?

Note: you probably have not been given the final equation for this situation, you will have to work it out for yourself.

Constructive interference would occur when m=some integer.

The same problem is discussed in thread https://www.physicsforums.com/showthread.php?t=737573
You can also read about diffraction gratings here: http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/grating.html
ehild

Thanks, but it appears that when I work through (with the assumption that θ=90) I get m=5 for the 410nm wave, whereas the OP of that topic arrived at m=4. Is the assumption I'm working with wrong (about θ being equal to 90 degrees)? And what's the reason for choosing m=3 over m=4?

Also, how do I go about working out the second part of the question?

 

[ehild]

Thanks for the replies.



Constructive interference would occur when m=some integer.



Thanks, but it appears that when I work through (with the assumption that θ=90) I get m=5 for the 410nm wave, whereas the OP of that topic arrived at m=4. Is the assumption I'm working with wrong (about θ being equal to 90 degrees)? And what's the reason for choosing m=3 over m=4?

Also, how do I go about working out the second part of the question?

d*sinθ=mλ, and θ<90°. The 5th order maximum does not appear for the 410 nm wave.
Also, the maximum order is 3 for the red light, and the problem asked the number of full spectra.

Determine the angles of both the violet and red light for m=1, 2, 3, 4 and arrange them in increasing sequence.

ehild

 

[Simon Bridge]

Constructive interference would occur when m=some integer.

Kinda - that's not the whole story.

This means you need to understand the physics:

Consider - if you had white light incident on the slits, you get a series of rainbow fringes.
Each fringe is a whole spectrum, but the highest order spectrum may be incomplete ... so you can end up with more blue fringes than red ones.

Originally I thought you were looking at something like water waves with a different frequency at each slit.

You should certainly do the exercise that ehild suggests at the end of post #5

Determine the angles of both the violet and red light for m=1, 2, 3, 4 and arrange them in increasing sequence. ​

 

[MFAHH]

Aha, thanks a bunch Simon Bridge and ehild, I'll finally got the answers!