# Lens Maker Formula > 1/f = (μ − 1)(1/R1 − 1/R2), behaviour in water, power in dioptres and the silvered-lens equivalent mirror — JEE Physics lens optics. - Canonical URL: https://prepelephant.com/topics/jee/physics/lens-maker-formula - Exam / course: JEE · Subject: Physics - Publisher: PrepElephant (https://prepelephant.com) — Prepared and reviewed by the PrepElephant Academic Review Team - First published: 2026-10-02 - Last updated: 2026-10-02 - How to cite: "Lens Maker Formula", PrepElephant, https://prepelephant.com/topics/jee/physics/lens-maker-formula ## Direct answer The focal length of a thin lens in air is set by the glass and its two curvatures through 1/f = (μ − 1)(1/R1 − 1/R2), the lens maker's formula, which combined with 1/v − 1/u = 1/f runs all image formation. Only the relative refractive index matters: immerse the same lens in water and (μ − 1) shrinks from 0.5 to about 0.13, stretching the focal length roughly fourfold and weakening the power in dioptres (P = 1/f in metres) in proportion. The formula also powers the two great JEE extensions — combining thin lenses in contact (powers add) and the silvered lens, which behaves as an equivalent mirror because light traverses the lens twice and reflects once. Sign discipline on R1 and R2, in the Cartesian convention, is where most of the marks are actually decided. ## What you must remember - **The formula:** 1/f = (μ − 1)(1/R1 − 1/R2); for a symmetric biconvex lens (R1 = +R, R2 = −R), it collapses to 1/f = 2(μ − 1)/R, and a plano-convex lens gives 1/f = (μ − 1)/R. - **Medium dependence:** replace (μ − 1) by (μ_lens/μ_medium − 1); a glass lens (μ = 1.5) in water (μ = 1.33) has its power cut to about a quarter, and a lens inside a liquid of equal index vanishes optically (f → ∞). - **Power and combination:** P = 1/f(metres) in dioptres; thin lenses in contact obey P = P1 + P2 with signs. - **Silvered lens recipe:** light passes through the lens, reflects off the silvered face (a mirror of radius R_s), and passes through the lens again, so the equivalent mirror obeys 1/F = 2/f_lens + 1/f_mirror — the object of the recipe being to convert the stack into one equivalent mirror. - **Bending light the other way:** in the Cartesian convention, R1 is positive for a convex front face and R2 negative for a convex back face, and the two enter the formula with opposite signs. - **Thin-lens limits:** the formula assumes paraxial rays, a thin centre (thickness << R), and one medium on both sides — violate the last and each face must be treated separately. - **Numerical anchor:** biconvex crown glass (μ = 1.5) with R = 20 cm gives f = 20 cm — the exam-grade number pair. ## From formula to silvered systems Build the standard biconvex lens: μ = 1.5, both radii 20 cm. Then 1/f = 0.5 × (1/0.2 + 1/0.2) = 5 m⁻¹, so f = 20 cm, power 5 D. Drop it into water: the relative index becomes 1.5/1.33 = 1.128, and the curvatures contribute 10 m⁻¹, so 1/f′ = 1.28 and f′ ≈ 78 cm. The same glass that focused sunlight to a point in air barely bends it underwater — the reason a water-filled mask changes how everything looks. Now silver the back face of that lens. Light crosses the lens (+P), reflects from the curved silvered surface acting as a mirror (P_m = 2/R_s = 10 D for R_s = 20 cm), then crosses the lens again (+P). The equivalent single mirror has power 1/F = 2 × 5 + 10 = 20 D, so F = 5 cm — a strongly converging mirror where the bare lens was 20 cm. The recipe is mechanical once stated: count two lens transits and one reflection, add the powers with signs, then solve with the mirror equation. ## Where students slip Sign conventions cause roughly half of all lost marks: R1 and R2 must be substituted with signs from the Cartesian scheme (for a biconvex lens in incident light, R1 is positive and R2 negative, so the difference becomes a sum), and students who take both radii as magnitudes confuse plano-convex and biconvex answers. The medium question is the second trap: the lens in water is weaker, not stronger — a converging glass lens can even become diverging inside a denser liquid, a favourite assertion–reason twist. In silvered-lens problems, forgetting that light crosses the lens twice (writing 1/F = 1/f + 1/f_m instead of 2/f + 1/f_m) is the standard error. Finally, power questions demand metres — dioptres computed with f in centimetres shift answers by a factor of 100, and the options are always waiting. ## Frequently asked questions ### What does the lens maker's formula state? 1/f = (μ − 1)(1/R1 − 1/R2) for a thin lens in air, with the radii signed per the Cartesian convention — the lens's geometry and material alone fix its focal length. ### How does immersing a lens in water change its focal length? The factor (μ − 1) becomes (μ_lens/μ_water − 1), which is about a quarter as large for glass in water, so the focal length stretches roughly four times and the power drops proportionally. ### What happens when one face of a lens is silvered? The assembly acts as an equivalent mirror: light traverses the lens, reflects from the silvered curved surface, and traverses the lens again, so 1/F = 2/f_lens + 1/f_mirror with proper signs. ### How do thin lenses in contact combine? Their powers add algebraically, P = P1 + P2, so a converging and a diverging lens in contact can produce any net focal length including none at all. ### Why does the formula fail for thick lenses? It assumes paraxial rays and a centre thickness negligible against the radii; a thick lens needs surface-by-surface refraction, and its two principal planes no longer coincide at one optical centre.