Refraction through a prism

A prism bends light twice, once at each face, and both bends turn the ray the same way, towards the base. The total turn is the angle of deviation δ.

For a prism of angle A, with angle of incidence i, angle of emergence e, and inside angles r₁ and r₂:

• r₁ + r₂ = A
• δ = i + e – A

As i is increased, δ first falls, then rises. Its smallest value is the angle of minimum deviation D_m. At that point the ray passes through symmetrically: i = e and r₁ = r₂ = A/2, and inside the prism it runs parallel to the base.

This gives a way to measure the refractive index of the glass:

n = sin((A + D_m)/2) / sin(A/2)

Example: A = 60° and n = √2. Then sin((60° + D_m)/2) = √2 × sin 30° = 0.707, so (60° + D_m)/2 = 45° and D_m = 30°.

For a thin prism (small A), δ = (n – 1)A.

In a liquid, use the relative index n_glass/n_liquid, which is closer to 1, so the prism bends light less.

If the ray meets the second face at more than the critical angle, it does not come out: it is totally reflected inside.

Dispersion: n is slightly different for each colour, larger for violet than for red. So violet is deviated most and red least, and white light spreads into a spectrum. A rainbow forms the same way, in raindrops.