Question

Derive the Lens Maker’s Formula for a thin convex lens.

Hint:

Lens Maker’s Formula gives focal length from lens geometry. Use it when refractive index and radii of curvature are known.

Answer 1

Derivation of Lens Maker’s Formula for a Thin Convex Lens:

The Lens Maker’s Formula gives the relationship between the focal length of a lens, the refractive index of the material of the lens, and the radii of curvature of its two surfaces.

Step 1: Refraction at the First Spherical Surface
Consider a thin convex lens of refractive index μ placed in air. Let R₁ and R₂ be the radii of curvature of the first and second surfaces respectively.

For refraction at a spherical surface, the relation is:
n₂/v − n₁/u = (n₂ − n₁)/R

For the first surface:
n₁ = 1 (air), n₂ = μ
Object distance = u
Image distance after first refraction = v₁
Radius of curvature = R₁

So,
μ/v₁ − 1/u = (μ − 1)/R₁    (1)

Step 2: Refraction at the Second Spherical Surface
The image formed by the first surface acts as a virtual object for the second surface. For the second surface:
n₁ = μ, n₂ = 1 (air)
Object distance = −v₁
Final image distance = v
Radius of curvature = R₂

Using the formula again:
1/v − μ/(−v₁) = (1 − μ)/R₂

This becomes:
1/v + μ/v₁ = (1 − μ)/R₂    (2)

Step 3: Adding Equations (1) and (2)
From (1):
μ/v₁ = (μ − 1)/R₁ + 1/u

Substitute μ/v₁ in (2):
1/v + (μ − 1)/R₁ + 1/u = (1 − μ)/R₂

Rearranging,
1/v − 1/u = (μ − 1) (1/R₁ − 1/R₂)

Step 4: For Focal Length
For focal length f, object is placed at infinity (u = ∞), so 1/u = 0.

Therefore,
1/f = (μ − 1) (1/R₁ − 1/R₂)

Final Result (Lens Maker’s Formula):
1/f = (μ − 1) (1/R₁ − 1/R₂)

Conclusion:
This formula relates the focal length of a thin convex lens to the refractive index of its material and the radii of curvature of its two surfaces. It shows that the focal length depends on both the material and the shape of the lens.