Question

Derive the relation between u, v and R at a convex spherical refracting surface, when the ray travels from rarer to denser medium and virtual image is formed.

Or

Derive the expression

\(-\frac{\mu_1}{u} +\frac{\mu_2}{v} = \frac{\mu_2 - \mu_1}{R}\)

when refraction occurs from rarer to denser medium at convex spherical refracting surface (µ₁ < µ₂) and virtual image is formed.

Answer 1

Let a point object 0 be situated on the principal axis of the convex spherical surface. Let P be pole and C be centre of curvature and PC = R be the radius of curvature. Let µ₁ and µ₂ be refractive indices of rarer and denser medium respectively as shown in the figure.

The incident ray OA refracts at A and bends towards the normal along AQ and when produced in the backward direction, it meets at point I on the principal axis. Then I will be the virtual image of point O.

From Snell's law for small i and r

 

\(\frac{sin\ i}{sin\ r} = \frac{i}{r} = \frac{\mu_2}{\mu_1}\)

or µ₁i = µ₂r ............(1)

In ∆ AOC,

i = α + γ

In ∆ AIC,

r = ß + γ

Putting the values of i and r in Eq. (1), we get

µ1 (α + γ) = µ₂ (ß + γ)

or µ₁α + µ₁γ = µ₂ß + µ₂γ

or µ1α - µ₂ß = (µ₂ - µ₁) γ

Since α, ß and γ are small, so they can be replaced by their tangents.

Hence 

µ₁ tan α - µ₂ tan ß = (µ₂ - µ₁) tan γ

Using sign conventions,

PO = -u, PI = v and PC = R, we get