Lens maker’s formula - Equation

1. Place a thin lens (which is made of one convex surface and one concave surface) between two refractive indices.

2. Let n_a be the refractive index of one medium and and n_b be the refractive index of second medium.

3. Consider an object O placed on the principal axis of the thin lens.

4. Now consider a ray from object O incident on the convex surface  of the lens at point A which has a radius of curvature R₁.

5. If there is no concave surface, it forms an image at Q then we take,

Object distance PO = -u, image distance PQ = v = x, radius of curvature R = R₁

6. For convex surface, n₁ = n_a, n₂ = n_b

7. According to image formation on curved surface formula,

n₂/v – n₁/u = (n₂ – n₁)/R

By substituting the values we get,

n_b/x + n_a/u = (n_b - n_a)/R₁ -----------------(1)

8. After getting refracted at point A again the ray get refracted due to concave surface at point B and reaches on the principal axis at point I.

Note: The image Q of object O due to the convex surface is taken as the object for the concave surface.

Here,

Object distance PQ = u = x, Image distance = PI = v, radius of curvature = -R₂

For concave surface n₁ = n_b, n₂ = n_a

by substituting the values we get,

n_a/v – n_b/x = (n_a - n_b)/(-R₂) ---------------- (2)

Add eq (1) and (2) we get,

n_a/v + n_b/u = (n_b - n_a)(1/R₁ + 1/R₂)

devide both sides by n_a,

1/v + 1/u = (n_ba - 1)(1/R₁ + 1/R₂)

here n_ba = n_b/n_a, is called the refractive index of lens with respect to the surrounding medium.

By using sign convention we get,

1/v – 1/u = (n_ba - 1)(1/R₁ – 1/R₂)

but we k now,

1/f = 1/v – 1/u

therefore we get,

1/f = (n_ba - 1)(1/R₁ – 1/R₂)

if the surrounding medium is air then n_ba = n

1/f = (n - 1)(1/R₁ – 1/R₂) is the equation of Lens Maker’s Formula.

Note:

1. The convex lens act as converging lens (if refractive index of medium is less than refractive index of lens)

2. The convex lens acts as diverging lens (if refractive index of medium is greater than refractive index of lens)