Question

A spherical surface of radius of curvature R, separates a rarer and a denser medium as shown in the figure. Complete the path of the incident ray of light, showing the formation of a real image. Hence derive the relation connecting object distance ‘u’, image distance ‘v’, radius of curvature R and the refractive indices n₁ and n₂ of two media.

Briefly explain how the focal length of a convex lens changes, with increase in wavelength of incident light.

Answer 1

Sign Conventions

• All the distances are to be measured from the pole of spherical surface.
• The distances measured in the same direction as the incident light are taken as positive.
• Distances measured in the direction opposite to the direction of the incident light are taken as negative.

Asumptions:

• Object is a point object lying on the principal axis.
• The aperture of refracting edge is small.
• The indcident ray from the object strikes the surface at a point very close to point so that angles of incidence and refraction are very small.

Derivation: Consider a point object O 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 of the surface. Let µ₁ and µ₂ be refractive indexes of rarer and denser medium respectively as shown in the figure.

Let incident ray OA refracts at point A and bends towards the normal CAN and real image is formed at I.

From Snell’s law, for small i and r, we have

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

or µ₁ i = µ₂ r .............(1)

In ∆ AOC,

i = α + γ

In ∆ AIC,

γ = r + ß

or r = γ - ß

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

µ₁ (α + γ) = µ₂ (γ - ß)

or µ₁α + µ₁γ = µ₂γ - µ₂ß

or µ₁α + µ₂ß = (µ₂ - µ₁) γ ..........(2)

Since α, ß and γ are small, so they can be replaced by their tangents. So equation (2) becomes

Using sign conventions,

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

we get

When wavelength is increased, the value of n decreases and hence the focal length of the lens increases with increase in wavelength.