Reflection Through Curved Surface

Description:

While studying refraction through curved spherical surface we will consider some assumptions −

• Object is always considered as point object which is placed on principle axis.

• For small angle: ∠i = sin i = tan i

• Curvature of surface is so small that perpendicular on principle axis from any point of the surface will be at pole.

• Medium m-1 will always be rarer and m-2 will always be denser.

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

Consider a point object ‘O’ on the principle axis in medium-1.

One Incident ray goes un-deviated through medium-2 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 towards the normal as it is moving from rarer to denser medium and meets axis at point I. Thus, point I is real image of point object O.

We have from Snell’s law −

μ = sin i/sin r

From our assumptions made earlier we get −

sin i = i and sin r = r

Therefore,

i = μ r

From figure we can obtain −

i = α + γ and r = γ - β

∴ α + γ = μ(γ - β)

∴ α + γ = μγ - μβ

∴ α + μβ = γ(μ - 1)

Again using assumptions, we get −

∴ tan α + μ tan β = tan γ(μ - 1)

Suppose in figure point M is very close to P.

From figure −

tan α = PM/OM and tan β = PM/MI and tan γ = PM/MC

Substituting values, we get −

PM/OM + μ PM/MI = PM/MC(μ - 1)

Divide the equation by PM

1/OM + μ 1/MI = 1/MC(μ - 1)

Substituting values of OM, MI, MC in optical terms from figure –

1/-u + μ1/+v = 1/R(μ - 1)

∴ μ/v - 1/u = (μ -1) 1/R

Where,

μ = Refractive index.

v = Image distance.

u = Object distance.

R = Radius of curvature.

This equation is for the case when object is placed in rarer medium and image is formed in the denser medium and it is a real image and curved surface is convex.