Refraction at a Spherical Surface

SPS′ is a convex refracting surface separating two media, air and glass. C is its centre of curvature. P (pole) is a point on SPS′ almost symmetrically placed. CP is the principal axis. O is a point object. OA is an incident ray and AB is the refracted ray. Another ray OP falls on the surface normally and goes without any deviation after refraction. PC and AB appear to come from I. Hence I is the virtual image of O.

Let ∠OAN = i, the angle of incidence and ∠CAB = r, the angle of refraction. Using the proper sign convention,

• PO = –u
• PI = –v
• PC = +R

Let α, β, and γ be the angles subtended by OA, IA and CA, respectively with the principal axis and h the height of the normal dropped from A on the principal axis. In ∆OCA and ∆ICA,

i = α + γ (i is exterior angle)

r = β + γ (r is exterior angle)

From Snell’s law,

(sin i)/(sin r) = µ

where µ is the refractive index of the glass surface with respect to air. For a surface of small aperture, P will be close to A and so i and r will be very small (sin i ~ i, sin r ~ r). The above equation, therefore, gives

i = µr

Substituting the values of i and r

α + γ = µ(β + γ)

α – µβ = γ(µ – 1)

As α, β and γ are very small, tan α ~ α, and tan β ~ β, and tan γ ~ γ.

In ∆OAM

α ≈ tan α = AM/MO = AP/PO = h/(–u)

β ≈ tan β = AM/MI = AM/PI = h/(−v)

γ ≈ tan γ = AM/MC = AM/PC = h/R

Substituting for α, β and γ

h/(–u) - µh/v = (µ – 1)h/R

µ/v – 1/u = (µ – 1)/R

This relationship correlates the object and image distances to the refractive index µ and the radius of curvature of the refracting surface.