Question

Derive the lens formula, 1/f = 1/v - 1/u for a concave lens, using the necessary ray diagram.

Two lenses of power 10 D and −5 D are placed in contact. 

(i) Calculate the power of the new lens. 

(ii) Where should an object be held from the lens so as to obtain a virtual image of magnification 2?

Answer 1

Consider an object O placed at a distance u from a convex lens as shown in figure. Let its image I after two refractions from spherical surfaces of radii R₁ (positive) and R₂ (negative) be formed at a distance v from the lens. Let v₁ be the distance of image formed by refraction from the refracting surface of radius R₁ . This image acts as an object for the second surface. Using, 

This expression relates the image distance v of the image formed by a thin lens to the object distance u and to the thin lens properties (index of refraction and radii of curvature). It is valid only for paraxial rays and only when the lens thickness is much less then R₁ and R₂ . The focal length f of a thin lens is the image distance that corresponds to an object at infinity. So, putting u = ∞ and v = f in the above equation, we have

If the refractive index of the material of the lens is µ and it is placed in air, µ₂ = µ and  µ₁ = 1 so that Eq. (iv) becomes

This is called the lens maker’s formula because it can be used to determine the values of R₁ and R₂ that are needed for a given refractive index and a desired focal length f.

Combining Eqs. (iii) and (v), we get