Rank the material according to their indices of refraction

[MrMoose]

Homework Statement

In the figure below (see image in color), light travels from material 'a', through three layers of other materials with surfaces parallel to one another, and then back into another layer of material 'a'. The refractions (but not the associated reflections) at the surfaces are shown. Rank the materials according to their indices of refraction, greatest first.

Homework Equations

Laws of Refraction:

1. If the ray is bent so that it sits between the normal and incident beam, N2>N1
2. If the ray is bent so that it sits outside the normal and incident beam, N1>N2

The Attempt at a Solution

See image in black & white. Using the relevant equations, I was able to set the following boundaries:

Nb>Na
Nc<Nb
Nd>Nc
Na<Nd

The problem is that I still don't have enough information to order the indices of refraction. For example, the following could be true:

Nd > Nb > Nc > Na

But I really don't have any evidence that supports Na > Nc or that Nd > Nb.

The correct answer is: Nd > Nb > Na > Nc

Am I supposed to estimate the angles and draw the rest of the boundaries from that?

Thanks in advance, MrMoose

 

[Doc Al]

To compare any two regions, all you need do is compare the angles of those rays. Eyeball it.

 

[MrMoose]

Thanks Doc Al, I figured it was just an eyeball thing, so let me see if I see it... so to speak.

Given my original original boundary conditions, my only questions were on the relationship between Na and Nc, and Nd and Nb.

Right off the bat, I can see that θd < θb

Given the general equation: n2 * sin(θ2) = n1 * sin(θ1), I know the following:

If θ2<θ1, it follows that N2<N1, therefore

Nd >Nb

Making the judgement call on θa and θc is not quite so obvious though. I had to pull out my protractor. In the end, I did find that θa < θc. Therefore, it follows that:

Na > Nc

And the final answer is:

Nd > Nb > Na >Nc

 

[Doc Al]

Looks good to me! :thumbs: