In the figure, light is incident at angle ? 1 = 22.8? on a boundary between two

Question

In the figure, light is incident at angle ?1 = 22.8? on a boundary between two transparent materials. Some of the light then travels down through the next three layers of transparent materials, while some of it reflects upward and then escapes into the air. The index of refractions are n1 = 1.33, n2 = 1.40, n3 = 1.32, n4 = 1.47 and nair = 1.00. What is the value of ?4?    What is the value of ?5?

Please show all work and explain your methods.

If the image can't be seen here is the URL: https://loncapa2.physics.sc.edu/res/sc/gblanpied/courses/usclib/hrw7/hrwpictures/34-45.jpg

Explanation / Answer

The question is bases on snell's law :

n1*sin(?1) = n2*sin(?2)

For this figure :

applying snell's law multiple times we get :

n1*sin(?1) = n2*sin(?2)

and n2*sin(?2) = n3*sin(?3)

and n3*sin(?3) = n4*sin(?4)

and therefore: n1*sin(?1) = n4*sin(?4)
=> n1=1.33 , n4=1.47 , ?1 =22.8

therefore we get: ?4 = arcsin(n1*sin(?1)/n4) = arcsin(1.33*sin(22.8)/1.47) = 20.52

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