White light is incident onto a 30∘ prism at the 40∘ angle shown in the figure. Violet light emerges perpendicular to the rear face of the prism. The index of refraction of violet light in this glass is 2.0% larger than the index of refraction of red light. At what angle ϕ does red light emerge from the rear face?

φ ≈ 1.0°

Explanation:

You'll need to use Snell's law:

n₁ sin θ₁ = n₂ sin θ₂

where n₁ is the index of refraction in the first medium,

θ₁ is the angle of incidence from normal,

n₂ is the index of refraction in the second medium,

and θ₂ is the angle of refraction from normal.

Draw a diagram of the path of the red light and violet light as they travel through the prism.

Let's start by looking at the violet light as it leaves the prism.  It exits the prism perpendicular to the rear face (angle of refraction = 0°).  So by Snell's law, it must be perpendicular to that face inside of the prism as well (angle of incidence = 0°).

Next, let's look at the violet light as it enters the prism.  Using trigonometry, we can show that the angle of incidence is 50° and the angle of refraction is 30°.  Plugging into Snell's law:

1 sin 50° = n sin 30°

n = 2 sin 50°

n ≈ 1.532

Now we can look at the red light as it enters the prism.  We know that for red light, the index of refraction is n/1.02 ≈ 1.502.

1 sin 50° = 1.502 sin θ₂

θ₂ = asin(0.51)

θ₂ ≈ 30.664°

Finally, let's look at the right light as it exits the prism.  Using trigonometry, we can show that θ₃ = θ₂ − 30° = 0.664°.  Plugging into Snell's law:

1.502 sin 0.664° = 1 sin φ

φ ≈ 0.997°

Rounded to two significant figures, the angle of refraction is approximately 1.0°.

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