# Snells law

## Homework Statement

Find the index of refraction n of the rectangle using x, d, and θ.

http://img338.imageshack.us/img338/6079/snellslaw.jpg [Broken]

## Homework Equations

n1*sin(θ1)=n2*sin(θ2)

## The Attempt at a Solution

Is it possible to solve this problem given the relevant data? What I did was try to find θ2 in terms of x, d, and θ (now called θ1). What I came up with was:

$$cos(\theta_{2})=\frac{x*sin(\theta_{1}-\theta_{2})}{d}$$

From the two right triangles that can be constructed:

http://img11.imageshack.us/img11/6606/snellslaw2.jpg [Broken]

But either this isn't right or I don't see a simple way to proceed from here. I used wolfram to solve the above equation for θ2, but the result seems too complicated to be the answer. Can someone help, anyone with a simpler approach? Perhaps n1*sin(θ1)=n2*sin(θ2) is useless in this case?

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## Answers and Replies

ehild
Homework Helper
You do have to use Snell's law to get the relation among x, d, and θ1. Expand sin(θ1-θ2). Find sin(θ2). Write both sin(θ2) and cos(θ2) in terms of sin(θ1). Use them in the expression for d.

ehild

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What do you mean "Find sin(θ2)." Do you mean isolate it from sin(θ1-θ2)=sin(θ1)*cos(θ2)-cos(θ1)*sin(θ2)=d/a ?

ehild
Homework Helper
What do you mean "Find sin(θ2)." Do you mean isolate it from sin(θ1-θ2)=sin(θ1)*cos(θ2)-cos(θ1)*sin(θ2)=d/a ?

Find sin(θ2) from Snells law in terms of sin(θ1) and the refractive index of the slab relative to the surrounding medium.

ehild

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Ok, so here are the equations I'm using:
$$(1)n_{1}sin(\theta_{1})=n_{2}sin(\theta_{2})$$

$$(2)cos(\theta_{2})=\frac{x}{a}$$

$$(3)sin(\theta_{1}-\theta_{2})=\frac{d}{a}$$

I find $$sin(\theta_{2})$$ from (1):

$$\frac{n_{1}}{n_{2}}sin(\theta_{1})=sin(\theta_{2})$$

I expand (2):

$$sin(\theta_{1})cos(\theta_{2})-cos(\theta_{1})sin(\theta_{2})=\frac{d}{a}$$

Plug (1) and (3) into (2):

$$sin(\theta_{1})\frac{x}{a}-cos(\theta_{1})\frac{n_{1}}{n_{2}}sin(\theta_{1})=\frac{d}{a}$$

But..."a" is an unknown, I can't solve for n2 until a is eliminated. Am I doing something wrong?
(n1 is assumed to be 1)

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collinsmark
Homework Helper
Gold Member
Hello CINA,

Just out of curiosity, is there anything left out of the problem statement such as θ being small (perhaps less than π/18 [which corresponds to around 10o)?

If so, approximations exist that would make solving for an approximate answer much easier.

For small θ (where θ is expressed in radians),
sinθθ
cosθ ≈ 1​

I'm not sure if assuming a small θ applies to this particular problem though.

ehild
Homework Helper
But..."a" is an unknown, I can't solve for n2 until a is eliminated. Am I doing something wrong?
(n1 is assumed to be 1)

cos(θ2)=x/a . Write a in terms of x and cos(θ2) and use cos(θ2)=√(1-sin2(θ2))

ehild

ehild
Homework Helper
Hello CINA,

Just out of curiosity, is there anything left out of the problem statement such as θ being small (perhaps less than π/18 [which corresponds to around 10o)?

Hello, Collinsmark,

There is no indication in the problem that θ1 is small. As the refractive index of the slab is to be obtained, θ1 and d has to be accurately measured, so they can not be two small. The problem can be solved without assuming small angle of incidence.

ehild