Question

Derive the relation for the refractive index (µ) of a prism in terms of angle of minimum derivation (δ_m) and angle of prism (A).

Answer 1

At the position of minimum deviation (δ = δ_m):

The angle of incidence equals the angle of emergence (i) = e

The angle of refraction at the first face equals the angle of incidence at the second face:

r₁ = r₂ = r

From A = r₁ + r₂, we get:

A = r + r

⇒ A = 2r

⇒ r = `A/2`

From δ_m = i + e − A, we get:

δ_m = i + i − A

⇒ δ_m = 2i − A

⇒ i = `(A + delta_m)/2`

Applying Snell’s Law at the first surface:

µ = `(sin i)/(sin r_1)`

= `(sin i)/(sin r)`

Substituting the values of i and r:

µ = `(sin((A + delta_m)/2))/(sin(A/2))`