Derive Prism formula and Thin Prism Formula.

The prism formula for a prism in general is given by-

μ = \(\frac{Sin[A+\delta_m/2]}{Sin(A/2)}\)

Here, μ is the refractive index of the glass (or the material from which the prism is constructed).

A is the apex angle of the prism.

δ_m is the minimum deviation produced in the given prism, the minimum deviation occurs when the angle of incidence is equal to the angle of emergence.

For a thin prism, the apex angle A is very small.

This implies that the refraction edge of the prism is also small. Lesser the refraction of the light, lesser will be the deviation it undergoes. Therefore in a thin prism the angle of deviation is also very small.

We know that, when the angle made in a triangle is very small, the sine of that angle can be approximated to equal to that angle. (in radians)

Sin θ_θ→0 = θ

Applying this for A and δ_m in the prism formula, we get-

μ = \(\frac{(A+\delta_m)/2}{(A/2)}\)

On rearranging this equation we get-

μ = \(A+\delta_m/A\)

Or

μA = A + δ_m

A(μ − 1) = δ_m

This is the required prism formula for a thin prism.