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The lens maker's formula is short, but it hides three sign decisions, a unit conversion and a trap about the surrounding medium. Combination problems add one more step. These are reliable marks in CBSE boards and JEE Main if you set them up carefully. This post covers the formula and its signs, power and combinations, lenses in liquids, five worked examples and a practice routine. For the thin lens formula and image positions, see our separate guide to ray optics sign conventions.
What does the lens maker's formula say?
The NCERT Class 12 Physics chapter on ray optics derives it by applying refraction at each of the two curved surfaces in turn. For a thin lens of refractive index n in air:
1/f = (n − 1)(1/R₁ − 1/R₂)
R₁ is the radius of the surface the light meets first and R₂ the second. With light travelling left to right, a radius is positive if the centre of curvature lies to the right of the surface (on the outgoing side) and negative if it lies to the left. NCERT notes that for a double convex lens R₁ is positive and R₂ negative, and that the same formula works for a concave lens, where R₁ is negative, R₂ positive and f comes out negative.
The JEE (Main) 2026 syllabus lists the thin lens formula and lens maker formula, power of a lens and combination of thin lenses in contact under Unit 16, Optics. Check the current year's syllabus and information bulletin for your exam; NEET-UG candidates should check the syllabus published by the National Medical Commission.
Worked example 1: a double convex lens
A double convex lens has n = 1.5, and its surfaces have radii 20 cm and 30 cm. Find f and the power.
Light meets the 20 cm surface first. Its centre is on the far side, so R₁ = +20 cm. The second surface curves the other way, so R₂ = −30 cm.
1/f = 0.5 × (1/20 − 1/(−30)) = 0.5 × (1/20 + 1/30) = 0.5 × 5/60 = 1/24.
f = +24 cm.
Power: P = 1/f with f in metres, so P = 1/0.24 ≈ +4.17 D.
The most common slip is step 2: writing 1/20 − 1/30 because both radii "look positive". The minus sign in R₂ turns the subtraction into an addition for a convex lens.
Worked example 2: does it matter which way round?
A plano-convex lens with n = 1.5 has one curved face of radius 15 cm.
Curved face first: R₁ = +15 cm, R₂ = ∞, so 1/f = 0.5 × (1/15 − 0) and f = 30 cm.
Flat face first: R₁ = ∞, R₂ = −15 cm, so 1/f = 0.5 × (0 + 1/15) and f = 30 cm.
For a thin lens the focal length is the same either way round. If your two answers differ, a sign has gone wrong.
Worked example 3: a double concave lens
A double concave lens with n = 1.5 has both radii 20 cm. Light meets a surface whose centre is on the incoming side, so R₁ = −20 cm and R₂ = +20 cm.
1/f = 0.5 × (−1/20 − 1/20) = 0.5 × (−1/10) = −1/20, so f = −20 cm and P = −5 D.
How do lenses in contact combine?
NCERT shows that for thin lenses in contact, 1/f = 1/f₁ + 1/f₂ + ..., or in terms of power, P = P₁ + P₂ + .... It stresses that this is an algebraic sum: convex lenses add positive power and concave lenses add negative power.
Worked example 4. A convex lens of focal length 20 cm is placed in contact with a concave lens of focal length 30 cm.
P₁ = 1/0.20 = +5 D; P₂ = 1/(−0.30) ≈ −3.33 D.
P = 5 − 3.33 = +1.67 D.
f = 1/1.67 m ≈ +0.60 m = +60 cm. Check with fractions: 1/20 − 1/30 = 1/60.
The combination is converging, because the convex lens is the stronger one. Combinations like this are how camera and telescope lenses are built; our guide to microscopes and telescopes uses them.
What happens when a lens is put in a liquid?
The (n − 1) factor is really the difference between the lens and its surroundings. NCERT's form for a medium of index n₁ and lens of index n₂ is n₁/f = (n₂ − n₁)(1/R₁ − 1/R₂). Its own example finds that a glass lens of focal length 20 cm in air has a focal length of about 78.2 cm in water.
Worked example 5. Put the lens from example 1 (f = 24 cm in air, so 1/R₁ − 1/R₂ = 1/12 per cm) into water, n₁ = 4/3.
(4/3)/f = (1.5 − 1.333) × (1/12) ≈ 0.1667/12 ≈ 0.01389, so f ≈ 1.333/0.01389 = 96 cm. The lens is four times weaker.
In a liquid with n₁ = 1.5, the bracket (n₂ − n₁) is zero: the lens has no power at all.
In a liquid with n₁ = 1.6, 1.6/f = (1.5 − 1.6)/12, so f ≈ −192 cm. The convex lens now diverges.
Why are these problems harder than they look?
Research on optics learning suggests the formula is not the only problem. Goldberg and McDermott (1987) interviewed university students before and after instruction in geometrical optics and found that many who had just finished the topic could not relate the rules and ray diagrams they had learned to a real lens, object and screen. Practise with a sketch of the lens and the light direction, not just the equation.
Common mistakes and how to avoid them
Signing R by the shape name. Sign each radius by where its centre lies relative to the light direction, not by "convex means positive".
Power in cm. P = 1/f needs f in metres; a 25 cm lens is +4 D, not +0.04 D.
Adding focal lengths. Add powers or reciprocals.
Forgetting the medium. In a liquid, use the relative index; a lens can weaken, vanish or flip.
Using the contact formula for separated lenses. It applies only to thin lenses in contact.
A practice routine for this week
Day 1: Six lens maker problems with different shapes, writing the sign of each radius with a one-line reason.
Day 2: Five combination problems, at least two with one convex and one concave lens.
Day 3: Four lens-in-liquid problems, including one where the lens changes from converging to diverging.
Day 4: A timed mixed set of eight from the NCERT exercises and past papers.
If the sign of R₂ still feels arbitrary, it helps to watch the rays bend. The Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read, so you can see how a lens behaves as you change it.
Tonight, redo worked example 1 with the lens turned round (30 cm surface first) and confirm you still get 24 cm.
