Question

(a) Obtain lens makers formula using the expression 

n₂/v - n₁/u = (n₂ - n₁)/R 

Here the ray of light propagating from a rarer medium of refractive index (n₁) to a denser medium of refractive index (n₂), is incident on the convex side of spherical refracting surface of radius of curvature R. 

(b) Draw a ray diagram to show the image formation by a concave mirror when the object is kept between its focus and the pole. Using this diagram, derive the magnification formula for the image formed.

Answer 1

(a) Lens Maker’s Formula: Suppose L is a thin lens. The refractive index of the material of lens is n₂  and it is placed in a medium of refractive index n₁. The optical centre of lens is C and X ' X is principal axis. The radii of curvature of the surfaces of the lens are R₁  and R₂  and their poles are P₁  and P₂. The thickness of lens is t, which is very small. O is a point object on the principal axis of the lens. The distance of O from pole P₁  is u. The first refracting surface forms the image of O at I' at a distance v' from P₁. From the refraction formula at spherical surface

The image I' acts as a virtual object for second surface and after refraction at second surface, the final image is formed at I. The distance of I from pole P₂  of second surface is v. The distance of virtual object(I') from pole P₂  is (v' -t). For refraction at second surface, the ray is going from second medium (refractive index n₂) to first medium (refractive index n₁), therefore from refraction formula at spherical surface

For a thin lens t is negligible as compared to v', therefore from (ii)

where ₁n₂ = n₂/n₁ is refractive index of second medium (i.e. medium of lens) with respect to first medium. 

If the object O is at infinity, the image will be formed at second focus i.e. 

if u = ∞, v = f₂ = f 

Therefore from equation (iv)

This is the formula of refraction for a thin lens. This formula is called Lens-Maker’s formula. If first medium is air and refractive index of material of lens be n, then ₁n₂ = n, therefore equation (v) may be written as

(b) Ray Diagram: The ray diagram of image formation for an object between focus (F) and pole (P) of a concave mirror is shown in fig.