Displacement of emerging light ray

[jssamp]

Homework Statement

A light ray is incident on a flat piece of glass with index of refraction n. Show that if the incident angle θ is small, the emerging ray is displaced a distance d = tθ(n - 1)/n form the incident ray, where t is the thickness of the glass and θ is in radians.

Homework Equations

n₁sinθ₁ = n₂sinθ₂

The Attempt at a Solution

Using Snell's law, right triangles, sin(a-b) identity, and small angle approximation, I have got it to:

d = tθ(n-(cosθ/cosθ_R))/n [θ_R is angle of refraction in glass]

how do I get cosθ/cosθ_R = 1?

 

[gneill]

Homework Statement

A light ray is incident on a flat piece of glass with index of refraction n. Show that if the incident angle θ is small, the emerging ray is displaced a distance d = tθ(n - 1)/n form the incident ray, where t is the thickness of the glass and θ is in radians.

Homework Equations

n₁sinθ₁ = n₂sinθ₂

The Attempt at a Solution

Using Snell's law, right triangles, sin(a-b) identity, and small angle approximation, I have got it to:

d = tθ(n-(cosθ/cosθ_R))/n [θ_R is angle of refraction in glass]

how do I get cosθ/cosθ_R = 1?

You should be able to get to the result in a more direct way. What is your understanding of "the small angle approximation"?

 

[jssamp]

sin(theta) = theta for theta < .2 radians

 

[gneill]

sin(theta) = theta for theta < .2 radians

Yes, anything else? (Hint: Another trig function has almost the same value for small angles)

 

[jssamp]

oh, yeah, cos(theta) = 1 - theta/2

 

[jssamp]

I might use that back when I used the sin(A-B)=sinAcosB-sinBcosA identity.

 

[jssamp]

I'll try it from the start with this added info. thanks gneill.

 

[gneill]

There's yet ANOTHER trig function that's ~θ when θ is small. You may find it even more convenient...

 

[jssamp]

OK, I got it solved! Thanks for the help, I'd never heard about small angle approximation for cos and tan until this problem. The tan approximation was the key piece I was missing.

Thanks for the help!

 

[gneill]

Glad to help