MHB How Does Snell's Law Predict Light Behavior Across Multiple Layers?

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Snell's Law is applied to predict light behavior as it passes through multiple transparent layers, with specific refractive indices provided for each layer. The initial angle of incidence is 40.1 degrees, and using Snell's Law, the angle of refraction in air, θ5, is calculated to be approximately 56.9 degrees. To find the angle θ4 in the bottom material, the process involves calculating the angle of incidence between each layer sequentially. The discussion emphasizes the importance of applying Snell's Law at each boundary to determine the angles accurately. Further assistance is sought for navigating the calculations through the subsequent layers.
connormcole
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In the attached figure, light is incident at angle $${\theta}_{1} = 40.1^{\circ}$$ on a boundary between two transparent materials. Some of the light travels down through the next three layers of transparent materials, while some of it reflects upward and then escapes into the air. If $${n}_{1} = 1.30$$, $${n}_{2} = 1.40$$, $${n}_{3} = 1.32$$, and $${n}_{4} = 1.45$$, what is the value of (a) $${\theta}_{5}$$ in the air and (b) $${\theta}_{4}$$ in the bottom material?

I understand the concept at work here is Snell's law of refraction. I know that to calculate the angle $${\theta}_{5}$$ you start with $${n}_{air}\sin\left({{\theta}_{5}}\right) = {n}_{1}\sin\left({{\theta}_{1}}\right)$$, then simplify to $${\theta}_{5} = \arcsin\left({\frac{{n}_{1}}{{n}_{air}}\sin\left({{\theta}_{1}}\right)}\right)$$, $$\approx 56.9^{\circ}$$.

I'm not sure where to go from here though as this concept is still relatively new to me. Any help that can be offered would be greatly appreciated.
 

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connormcole said:
In the attached figure, light is incident at angle $${\theta}_{1} = 40.1^{\circ}$$ on a boundary between two transparent materials. Some of the light travels down through the next three layers of transparent materials, while some of it reflects upward and then escapes into the air. If $${n}_{1} = 1.30$$, $${n}_{2} = 1.40$$, $${n}_{3} = 1.32$$, and $${n}_{4} = 1.45$$, what is the value of (a) $${\theta}_{5}$$ in the air and (b) $${\theta}_{4}$$ in the bottom material?

I understand the concept at work here is Snell's law of refraction. I know that to calculate the angle $${\theta}_{5}$$ you start with $${n}_{air}\sin\left({{\theta}_{5}}\right) = {n}_{1}\sin\left({{\theta}_{1}}\right)$$, then simplify to $${\theta}_{5} = \arcsin\left({\frac{{n}_{1}}{{n}_{air}}\sin\left({{\theta}_{1}}\right)}\right)$$, $$\approx 56.9^{\circ}$$.

I'm not sure where to go from here though as this concept is still relatively new to me. Any help that can be offered would be greatly appreciated.
You need to do Snell's Law for each of the materials the ray is passing through. For example, your angle of 56.9 degrees gives you an incident angle of 90 - 56.9 = 33.1 degrees for the angle of incidence between the n2 and n3 layers.

-Dan
 
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