Derive the lens maker's formula for a thin double convex lens, and hence obtain the thin lens formula.
How to lay out the answer. Follow the frame, then write it in your own words.
1 · State the assumptions. A thin lens of refractive index n₂ in a medium n₁; a point object on the axis; paraxial rays; distances measured from the optical centre.
2 · Draw the ray diagram. [Thin double convex lens; O on the left; C₁ on the right and C₂ on the left; ray bending at the first surface towards I₁ (dotted) and again at the second surface to meet the axis at I; label n₁, n₂, R₁, R₂, u, v₁ and v]
3 · Refraction at the first surface. n₂/v₁ − n₁/u = (n₂ − n₁)/R₁ ...(1)
4 · Refraction at the second surface. I₁ is the object: n₁/v − n₂/v₁ = (n₁ − n₂)/R₂ ...(2)
5 · Add and simplify.
n₁/v − n₁/u = (n₂ − n₁)(1/R₁ − 1/R₂)
1/v − 1/u = (n₂₁ − 1)(1/R₁ − 1/R₂) ...(3)
6 · Get both formulas.
For u = −∞, v = f: 1/f = (n₂₁ − 1)(1/R₁ − 1/R₂), the lens maker's formula
Comparing with (3): 1/v − 1/u = 1/f, the thin lens formula
7 · Real life. Spectacle makers use it to choose the glass and the curvature of each face for the power they need.
8 · Close in one line. The focal length of a lens depends on its material, its surroundings and the curvature of its faces.
Scoring tip: Number equations (1), (2) and (3) so the examiner can follow you. Say that the lens is thin, so I₁ is the same distance from both surfaces.
