What is the Refractive Index for Total Internal Reflection in a 45-45-90 Prism?

[Cummings]

The Question:
A 45-45-90 prism is immersed in water. What is the minimum index or fefraction the prism must have if it is to reflect totally a ray incident normally on one of its faces.

Now, we have found this question a bit hard to understand.
We assume that light is hitting one of the sides at 90 degrees (normally) and so will pass into the prism without refraction. Thus, it hits the other side of the prism at an angle of incidence of 45 degrees.

This is where we get stuck, we are assuming that it is totally internaly reflecting and by using snells law nSin(45) = Sin(90) we find that n, the refractive index of the prism is 1.88

This complies with
the fact that to be totally internaly reflected the light must travel from a high refractive index (1.88) to a low refractive index (1 for air)

If we are on the right track, please let us know!

 

[Doc Al]

This is where we get stuck, we are assuming that it is totally internaly reflecting and by using snells law nSin(45) = Sin(90) we find that n, the refractive index of the prism is 1.88

You didn't get n=1.88 using that equation. I assume you meant to write Snell's law as: n_1 sin\theta_1 = n_2 sin\theta_2. And, yes, the critical angle is found by setting \theta_2 = 90.

This complies with
the fact that to be totally internaly reflected the light must travel from a high refractive index (1.88) to a low refractive index (1 for air)

If you assumed going from n=1.88 to air (n=1) I am very curious how you got your answer. (The prism is immersed in water, not air.)

 

[relinquished™]

This is where we get stuck, we are assuming that it is totally internaly reflecting and by using snells law nSin(45) = Sin(90) we find that n, the refractive index of the prism is 1.88

If we are on the right track, please let us know!

I think it should be n1*Sin(45) = n2*sin(90), not nSin(45) = Sin(90)

which is

n1*sin(45) = 1.333
n1 = 1.333/sin 45

n1 is approx 1.88

 

[Cummings]

Yes, i made a few erros in that post. Had to make it to the next class of uni.

Yes, the prism is in water. relinquished managed to type in what i failed to. He got the same answer to.

I just for some reason replaced water with air when i typed this up.

So, would 1.88 be the minumum index of refraction?

 

[Doc Al]

I just for some reason replaced water with air when i typed this up.

I knew you must have, since your answer was correct.

So, would 1.88 be the minumum index of refraction?

Yep.