Question

Describe Refraction by a Lens. Derive an expression for lens-maker’s formula.

Answer 1

Figure .(a) shows geometry of image formation by a double convex lens. The image formation can be seen in terms of two steps: 

(i) The first refracting surface forms the image I₁ of the object O [Fig. (b)]. The image I₁ acts as a virtual object for the second surface that forms the image at I [Fig. (c)]. Applying Eq. (15) to the first interface ABC, we get,

A similar procedure applied to the second interface ADC gives,

For a thin lens, BI₁ = DI₁. Adding Eqs. (1) and (2), we get, 

Suppose the object is at infinity, i.e., OB → ∞ and DI = f, Eq. (2) gives

The point where image of an object placed at infinity is formed is called the focus F, of the lens and the distance f gives its focal length. A lens has two foci, F and F′, on either side of it (Fig.). By the sign convention,

So Eq. (3) can be written as

Eqn. (3) is known as the lens maker’s formula. It is useful to design lenses of desired focal length using surfaces of suitable radii of curvature.