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The lens formula and the lens maker's formula both start from one equation: refraction at a single curved surface. Students who learn the lens formulas directly often cannot handle a fish in a bowl, a bubble in a glass ball or a rod with a curved end, which use this equation alone. This post explains the formula, the sign rules that make it work, four worked examples and the slips to avoid.
Where does this appear in the syllabus?
The NCERT Class 12 chapter on ray optics (Reprint 2026-27) derives it in Section 9.5.1, Refraction at a spherical surface, and then applies it twice to get the lens maker's formula. The NTA Syllabus for JEE (Main) 2026 lists "Refraction of light at plane and spherical surfaces" in Unit 16, and the JEE (Advanced) 2026 Information Brochure lists "Reflection and refraction at plane and spherical surfaces". Check the syllabus for your own exam year.
What is the formula?
Light travels from a medium of refractive index n1 into one of index n2, through a spherical surface of radius of curvature R. For rays close to the axis, NCERT's Equation 9.16 is
n2/v − n1/u = (n2 − n1)/R.
Here u is the object distance, v the image distance, and n2 is always the medium the light enters. The derivation uses Snell's law with the small-angle approximation, so it holds for paraxial rays.
For the size of the image, the standard result for a single surface is m = (n1 v)/(n2 u). It follows from the same small-angle geometry.
Which sign convention should you use?
NCERT uses the Cartesian sign convention:
Measure all distances from the pole of the surface.
Distances in the direction the incident light travels are positive; distances against it are negative.
So a real object in front of the surface has negative u.
R is positive if the centre of curvature lies on the side the light is going to, and negative if it lies on the side the light comes from.
Worked example 1: NCERT's glass surface
A point source in air is 100 cm from a convex glass surface (n = 1.5, R = 20 cm), as in NCERT's Example 9.5.
n1 = 1, n2 = 1.5, u = −100 cm, R = +20 cm (the centre is inside the glass, where the light is going).
1.5/v − 1/(−100) = (1.5 − 1)/20 gives 1.5/v + 0.01 = 0.025.
1.5/v = 0.015, so v = +100 cm. A real image, 100 cm inside the glass.
Worked example 2: a bubble in a glass ball
An air bubble is inside a glass ball (n = 1.5) of radius 10 cm, 5 cm from the surface nearest your eye. Where does it appear?
Light goes from the bubble, through glass, out into air towards you. So n1 = 1.5, n2 = 1, and the positive direction points towards your eye.
The bubble is behind the surface as the light travels: u = −5 cm.
The centre of the ball is on the glass side, which the light is leaving: R = −10 cm.
1/v − 1.5/(−5) = (1 − 1.5)/(−10), so 1/v + 0.3 = 0.05.
1/v = −0.25, so v = −4 cm.
The bubble appears 4 cm inside the surface, nearer than it really is. Magnification m = (1.5 × −4)/(1 × −5) = 1.2, so it also looks 1.2 times larger.
Worked example 3: the flat-surface limit
For a flat surface, R becomes infinite and the right side is zero: n2/v = n1/u. A coin lies 12 cm under water (n = 4/3), viewed from directly above.
n1 = 4/3, n2 = 1, u = −12 cm.
1/v = (4/3)/(−12) = −1/9, so v = −9 cm.
The coin seems 9 cm deep. That is the apparent-depth rule, real depth ÷ n = 12 ÷ (4/3) = 9 cm, coming out of the same formula.
Worked example 4: the two focal lengths of a curved rod end
A long glass rod (n = 1.5) has a convex end of radius 10 cm, with light coming from air.
Parallel rays from the air side (u at infinity): 1.5/v = 0.5/10, so v = +30 cm. Rays focus 30 cm inside the rod.
For rays to leave the surface parallel inside the glass (v at infinity): −1/u = 0.05, so u = −20 cm. The object must be 20 cm in front.
A single surface has two different focal lengths, 30 cm and 20 cm here, and their ratio is n2/n1 = 1.5. A thin lens in air has equal focal lengths on both sides, which is why the lens formula looks simpler.
What sign slips do students make?
Fixing R as positive for "convex". Convex towards the light gives positive R; the same surface seen from inside gives negative R, as in example 2.
Swapping n1 and n2. n2 is always the medium the light enters.
Making u positive for a real object. With the Cartesian convention, a real object in front of the surface has negative u.
Using the mirror magnification −v/u. For refraction, m = (n1 v)/(n2 u).
Forgetting the paraxial assumption. The formula is for rays close to the axis.
Our post on common ray optics mistakes covers similar sign errors in mirrors and lenses.
A practice routine
Redo example 2 with the bubble viewed from the far side of the ball (15 cm from that surface). Check: u = −15 cm and R = −10 cm, so 1/v + 0.1 = 0.05, 1/v = −0.05 and v = −20 cm.
Derive the lens maker's formula by applying the equation twice, as NCERT does.
Solve five past JEE questions on fish tanks, glass spheres and curved rods, marking the direction of light first.
Write the R test from the callout at the top of every optics page in your notes.
When a sign keeps going wrong and you cannot see why, the Fermi Problem Library has a step-by-step solver that explains the reasoning behind each step.
Try now: draw example 2, mark the light's direction and the centre of curvature, and solve it again without looking.
