Deriving an expression from geometry

[Levi Tate]

Homework Statement

I need to derive an expression for the displacement of light as a function of thickness of glass and the angles.

I will post a screen shot of the formula to be derived but it can also be found here

http://clas.wayne.edu/multimedia/usercontent/File/Physics%20and%20Astronomy/Labs/PHY5341/Lab4%20Brewsters%20Angle.pdf

(pg 3)

Homework Equations

The equation is given as the solution. I am not very good with geometry so I don't really even know where to begin.

The Attempt at a Solution

Using Snell's law,

n1sinθ1=n2sinθ2

Solving for θ2 = Arcsin[(n1/n2)sinθ1]

So then I just have θ2.

Help, please?

 

[ehild]

See attachment. You have to find the expression for the distance (d) between the original ray (1) and the displaced one (2). t is the thickness of the slab. There are two right triangles in the picture, with common hypotenuse (s ). Find s in terms of t and θ₂, find d in terms of s and θ₂-θ₁. Eliminate θ₂.

ehild

 

[Levi Tate]

Thanks a lot for your help mate. I tried this problem for about an hour and I'm completely stuck. I never went to high school to get that Euclidian intuition and I'm dyslexic so I don't have a good geometric intuition. I'm just stuck, I can't see the solution in sight.

But I thank you for your assistance very much.

 

[ehild]

Do you see the right triangles in the picture? What is the relation between t, s and θ2?

What is the angle of the blue triangle between side s and the hypotenuse?


ehild

 

[Nickie]

I understand the importance of being able to derive equations from basic principles and laws. In this case, we are looking to derive an expression for the displacement of light as a function of thickness of glass and angles.

To begin, we can start with the basic formula for refraction, Snell's law, which states that the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the refractive indices of the two media.

Using this, we can derive an expression for the angle of refraction, θ2, in terms of the angle of incidence, θ1, and the refractive indices of the two media, n1 and n2. This is shown in your attempt at a solution above.

Next, we need to consider the geometry of the situation. In this case, we have a beam of light entering a glass slab at an angle θ1, and then exiting at an angle θ2. The thickness of the glass slab can be represented as d, and we can use basic trigonometry to relate this to the angles θ1 and θ2.

By drawing a diagram and using the relationships between angles and sides in a right triangle, we can see that the displacement of the beam of light, x, is equal to d times the tangent of the angle θ2. In other words, x = d*tan(θ2).

Now, we can substitute in our expression for θ2 from Snell's law and simplify to get the final expression:

x = d*tan[Arcsin[(n1/n2)sinθ1]]

This is the desired expression for the displacement of light as a function of thickness of glass and angles. I hope this helps and clarifies the steps involved in deriving this formula.