JEE Mains Physics · Ray Optics

Refraction at a Spherical Surface and the Lens-Maker's Formula

One curved boundary refracts by μ₂/v − μ₁/u = (μ₂ − μ₁)/R; two such boundaries make a thin lens, whose focal length follows from 1/f = (μ − 1)(1/R₁ − 1/R₂).

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Why this matters

Twenty-one PYQs, fifteen of them multiple choice, and seven from 2026. Eleven image an object through one curved surface: a parallel beam entering a glass ball, an object and its image equally far from the surface, a meniscus between two liquids, two concave faces facing each other. Ten use the lens-maker's formula to link the radii, the refractive index and the focal length, including lenses built by joining two plano lenses.

Concept 1 of 2: Refraction at a single spherical surface

A single curved boundary between two media forms an image, just as a lens does, but the two sides have different refractive indices. Each side's distance is weighted by its own index. The sign of R follows the same convention as every other distance: positive when the centre of curvature lies on the side the light goes to.

Definition

$\dfrac{\mu_2}{v} - \dfrac{\mu_1}{u} = \dfrac{\mu_2 - \mu_1}{R}$ : $\mu_1$ is the medium the light comes from, $\mu_2$ the medium it enters.
$R > 0$ if the centre of curvature is on the outgoing side (a convex face met from outside); $R < 0$ if it is on the incoming side.
Magnification $m = \dfrac{\mu_1 v}{\mu_2 u}$ .
Parallel beam ( $u = \infty$ ): $v = \dfrac{\mu_2 R}{\mu_2 - \mu_1}$ .
A sphere or any thick piece: image through the first surface, move the origin to the second surface, and use that image as the object there.
Light going from the denser to the rarer side uses the same formula with $\mu_1 > \mu_2$ ; never rearrange it by hand.

Single spherical surface

\frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R}, \qquad m = \frac{\mu_1 v}{\mu_2 u}

Worked example

A point object in air is 30 cm in front of a convex glass surface (

\mu = 1.5

) whose radius of curvature is 20 cm. Find the position and nature of the image, and the magnification.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 23 January 2025 · Q5Moderate

Example 1 · Ray Optics · Refraction at a Spherical Surface and the Lens-Maker's Formula

A spherical surface of radius of curvature

R

, separates air from glass (refractive index

= 1.5

). The centre of curvature is in the glass medium. A point object ' O ' placed in air on the optic axis of the surface, so that its real image is formed at ' I ' inside glass. The line OI intersects the spherical surface at

P

and

PO = PI

. The distance

PO

equals to-

Each distance carries its own index

The image distance is divided into μ₂ and the object distance into μ₁. Writing 1/v − 1/u, as for a lens, ignores the two media.

The sign of R depends on where the centre is

A concave face met from outside has its centre on the incoming side, so R is negative. Using +R for every curved face can turn a virtual image into a real one.

The magnification has the indices too

For one surface m = μ₁v/(μ₂u), not v/u. Leaving out the indices gives a wrong image height.

Drill 10 more on refraction at a single spherical surface

Concept 2 of 2: Lens-maker's formula

A thin lens is two curved surfaces close together. Adding their effects gives the lens-maker's formula. The focal length depends on how strongly the glass bends light, μ − 1, and on how curved the faces are, 1/R₁ − 1/R₂. With the sign convention, a biconvex lens has R₁ > 0 and R₂ < 0, so the two curvatures add.

Definition

$\dfrac{1}{f} = (\mu - 1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$ , where $R_1$ is the face the light meets first, and the lens is in air. Power $P = 1/f$ , in dioptres when f is in metres.
Equiconvex, radius R: $f = \dfrac{R}{2(\mu - 1)}$ . Plano-convex: $f = \dfrac{R}{\mu - 1}$ , because the flat face has $1/R = 0$ .
Biconvex with radii a and b: $\dfrac{1}{f} = (\mu - 1)\left(\dfrac{1}{a} + \dfrac{1}{b}\right)$ . Keeping the power fixed means keeping $\dfrac{1}{a} + \dfrac{1}{b}$ fixed.
μ from the speed of light in the glass: $\mu = c/v$ .
A curved face of aperture radius r rising by t (the centre thickness of a plano-convex lens): $R^{2} = r^{2} + (R - t)^{2}$ , so $R \approx \dfrac{r^{2}}{2t}$ .
Lenses joined face to face act as thin lenses in contact: find each one's power from this formula and add the powers.
A thin layer of liquid trapped between glass faces is itself a lens; include its power.

Lens-maker's formula

\frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right), \qquad P = \frac{1}{f}

Worked example

A biconvex lens made of glass of refractive index 1.6 has radii of curvature 15 cm and 30 cm. (a) Find its focal length and power. (b) A plano-convex lens of the same glass is to have the same power. What radius should its curved face have?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 28 Jan 2026 Shift 1 · Q11Moderate

Example 2 · Ray Optics · Refraction at a Spherical Surface and the Lens-Maker's Formula

The magnitudes of power of a biconvex lens (refractive index 1.5) and that of a Plano-concave lens (refractive index

= 1.7

) are same. If the curvature of plano-concave lens exactly matches with the curvature of back surface of the biconvex lens, then ratio of radius of curvature of front and back surface of the biconvex lens is

\_\_\_\_

R₂ of a biconvex lens is negative

Writing both radii as positive in 1/R₁ − 1/R₂ subtracts the curvatures instead of adding them, and f comes out far too long.

A flat face has 1/R = 0, not R = 0

A plane surface has an infinite radius. Its term 1/R vanishes, so a plano-convex lens has f = R/(μ − 1).

μ − 1 is for a lens in air

In another medium the factor becomes μ(lens)/μ(medium) − 1. Using μ − 1 for a lens in water gives a focal length that is far too short.

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Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Refraction at a single spherical surface
Single spherical surface
$\frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R}, \qquad m = \frac{\mu_1 v}{\mu_2 u}$
Lens-maker's formula
Lens-maker's formula
$\frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right), \qquad P = \frac{1}{f}$

Watch out for (6)

Each distance carries its own index
→ Refraction at a single spherical surface
The sign of R depends on where the centre is
→ Refraction at a single spherical surface
The magnification has the indices too
→ Refraction at a single spherical surface
R₂ of a biconvex lens is negative
→ Lens-maker's formula
A flat face has 1/R = 0, not R = 0
→ Lens-maker's formula
μ − 1 is for a lens in air
→ Lens-maker's formula

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