Refraction in a prism

Question

I'm having problems with this optics problem:

Glass prism ABC with refractive index n1 = 1.6, vertex A angle α = 40° and vertex B angle β = 60°,

placed in a container with two different liquids washing over its sides with refractive indices n2 = 1.1 and n3 = 1.33 respectively (see diagram).

A ray parallel to the prism base CB is incident on the prism wall AB. Then, the ray reaches the wall AC

and exits the prism through it.

a. Calculate the angle δ of the ray that falls into the prism deviates from the original direction.

b. Draw an accurate diagram of the path of the ray through the prism.

Answer 1

A lot of geometry (Here's my attempt - please check the math...) The first application of Snell's law gives you the refraction angle. Because of the slopes of the sides of the prism and the relative indices of refraction, you know that the beam will deflect downwards both times. The angle of incidence to the first surface is 30° - easy to see if you draw a horizontal and a normal at the point of entry (angle β is 60 and it is the complement to the incidence angle)

θ₁ = sin^-1(n₃sinθ_inc/n₁) = sin^-1(1.33sin(30)/1.6) = 24.559 with respect to the normal and 5.44113 below the horizontal.

The angle of incidence with the 2nd side: Draw a normal, a horizontal and a line at 5.44 below horizontal at the point of incidence. The angle at C, γ, is 80°. The angle of the normal is 10° above the horizontal. The angle of incidence is 10 + 5.44113 = 15.44113

θ₂ = sin^-1(n₁sin(15.4413)/n₂) = sin^-1(1.6sin(15.4413)/1.1) = 22.7860 below the normal which is 12.79 below the horizontal. (This should be delta)

Please consider a tutor. Take care.