Solving for Index of Refraction Using x, d, and θ

[CINA]

Homework Statement

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

http://img338.imageshack.us/img338/6079/snellslaw.jpg

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

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?

 

[ehild]

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

 

[CINA]

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]

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

 

[CINA]

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)

 

[collinsmark]

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 10^o)?

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

For small θ (where θ is expressed in radians),

sinθ ≈ θ
cosθ ≈ 1​

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

 

[ehild]

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]

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 10^o)?

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