Refraction

Description:

Consider a spherical surface which separates medium-1 and medium-2.

Consider a point object ‘O’ on the principle axis in medium-2 which is denser.

One Incident ray goes un-deviated through medium-1 along principle axis.

Consider a second ray OP, incident on surface, which makes an angle of incidence ‘i’ at point P. After refraction, it bends away from the normal as it is moving from denser to rarer medium.

These two rays do not intersect actually so real image of the object is not formed, but if both the ways are drawn backwards they intersect at point ‘I’ which is a virtual image.

We have from Snell’s law −

μ₁₂ = sin i/sin r

∴ μ₁/μ₂ = sin i/sin r

From our assumptions made earlier we get −

sin i = i and sin r = r

Therefore,

∴ μ₁/μ₂ = i/r

μ₁r = μ₂i

From figure we can obtain −

i = α - γ and r = β - γ

∴ μ₁(β - γ) = μ₂(α - γ)

∴μ₁β - μ₁γ = μ₂α - μ₂γ

∴ μ₁β - μ₂α = μ₁γ - μ₂γ

Again using assumptions, we get −

∴ μ₁tan β - μ₂tan α = tan γ(μ₁ - μ₂)

Suppose in figure point M is very close to P.

From figure −

tan β = PM/IM and tan α = PM/OM and tan γ = PM/CM

Substituting values, we get −

μ₁PM/IM - μ₂PM/OM = PM/CM(μ₁ - μ₂)

Divide the equation by PM

μ₁1/IM - μ₂1/OM = 1/CM(μ₁ - μ₂)

Substituting values of IM, OM, CM in optical terms from figure −

μ₁/-v - μ₂/-u = μ₁ - μ₂/-R

∴ μ₂/u - μ₁/v = μ₂ - μ₁/R

It can also be written as −

μ₁/v - μ₂/u = μ₁ - μ₂/R

Where,

μ₁ = Refractive index of medium-1(rarer).

μ₂ = Refractive index of medium-2(denser).

v = Image distance.

u = Object distance.

R = Radius of curvature.

This equation is for the case when object is placed in denser medium and image is also formed in the denser medium and it is a virtual image and curved surface is concave.