Refraction at spherical surfaces

Drop a coin into a swimming pool and it looks closer to the surface than it really is. Look at a fish through the curved side of a round goldfish bowl and it looks bigger, or shifted, or in the wrong place entirely. Both tricks are refraction — light bending as it crosses from one transparent medium into another — but the goldfish bowl adds something the flat pool surface does not: curvature. A curved boundary does not just shift an image, it can magnify it, flip whether it is real or virtual, and change how far away it appears, all depending on how sharply the surface bends and which way it bends. Refraction at a spherical surface is the physics of exactly this situation: a single curved interface — one side glass, one side air, or one side glass and one side a denser glass — connecting an object on one side to an image on the other. It sounds like a narrow, specialised case, but it is in fact the single most useful building block in the whole of ray optics. Every simple lens has two curved surfaces, and the lens maker's formula that predicts what a lens will do is built by applying this one single-surface relation twice in a row, once at each face, and combining the results. Master this formula and the physics of lenses — camera lenses, spectacles, the eye's own cornea and lens — stops being a list of separate rules and becomes one idea used twice. This article derives the single-surface refraction formula from Snell's law under the paraxial (small-angle, near-axis) approximation, works it on both a convex glass surface and the classic "apparent depth" flat-surface special case, and flags the sign-convention traps that cost the most marks in CBSE board exams.

Key takeaways

A single spherical surface separating two media of refractive index $n_1$ (where the object sits) and $n_2$ (where the image forms) obeys $\dfrac{n_2}{v}-\dfrac{n_1}{u}=\dfrac{n_2-n_1}{R}$ , with $u$ , $v$ , $R$ measured from the pole using the Cartesian sign convention.
This single formula covers convex and concave surfaces, real and virtual objects and images — only the signs of $u$ , $v$ , $R$ change between cases.
A flat refracting surface is just the special case $R\to\infty$ , which turns the formula into the familiar apparent-depth rule.
The formula is derived from Snell's law, $n_1\sin i = n_2\sin r$ , using the small-angle approximation $\sin\theta\approx\tan\theta\approx\theta$ (radians) valid for paraxial rays close to the principal axis.
It is not an isolated fact to memorise: applying it once at each face of a thin lens, and combining the two results, is exactly how the lens maker's formula is built (that combination itself is not required — only the final formula is used later).
The commonest board-exam error is swapping which refractive index divides $u$ and which divides $v$ — it is always "index of the medium the ray is currently travelling in, over the corresponding distance."
Key idea: a curved interface bends light according to $\dfrac{n_2}{v}-\dfrac{n_1}{u}=\dfrac{n_2-n_1}{R}$ , a single sign-consistent relation that reduces to the apparent-depth rule for a flat surface and, applied twice, becomes the lens maker's formula for a thin lens.

The core insight

Refraction at one spherical surface obeys $\dfrac{n_2}{v}-\dfrac{n_1}{u}=\dfrac{n_2-n_1}{R}$ , a single paraxial relation (derived from Snell's law with small-angle approximation) that covers convex, concave and flat interfaces alike once $u$ , $v$ and $R$ are given the correct Cartesian sign.

Where it fits in the syllabus

In CBSE Class 12's Unit IX, "Ray Optics and Optical Instruments," refraction at a single spherical surface sits between the general Snell's-law/total-internal-reflection topics and the thin-lens formulas. The board expects the relation $\dfrac{n_2}{v}-\dfrac{n_1}{u}=\dfrac{n_2-n_1}{R}$ to be known, applied correctly with signs, and recognised as the stepping stone to the lens maker's formula — the full two-surface combination that produces the lens maker's formula is used as a given result, not re-derived from scratch in the prescribed syllabus. Aberrations of a single refracting surface (spherical or chromatic distortion) are outside the 2025-26 CBSE theory list and are not treated here.

The physical picture

Picture a spherical surface of radius of curvature $R$ , with its centre of curvature $C$ and its pole $P$ (where the surface crosses the principal axis) marking the geometry. On one side sits a medium of refractive index $n_1$ , holding the object; on the other side sits a medium of refractive index $n_2$ , where the refracted rays converge (or appear to diverge from) to form the image. A ray leaving the object along the axis passes through undeviated; a second ray leaving the object at a small angle strikes the curved surface and bends according to Snell's law, $n_1\sin i = n_2\sin r$ . For paraxial rays — rays that stay close to the axis, so every angle is small enough that $\sin\theta\approx\tan\theta\approx\theta$ in radians — all such bent rays cross the axis at (very nearly) the same single point, which we call the image. Applying Snell's law with this small-angle approximation, and defining distances from the pole $P$ using the Cartesian sign convention (all distances measured from $P$ ; distances in the direction of the incident light are positive, distances against it are negative), gives the single-surface refraction relation:

\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}

where: $u$ is the object distance from the pole, $v$ is the image distance from the pole, $R$ is the radius of curvature of the surface (positive if the centre of curvature $C$ lies on the outgoing-light side of the pole, negative if it lies on the incoming side), $n_1$ is the refractive index of the medium the incident ray is travelling in (containing the object), and $n_2$ is the refractive index of the medium the refracted ray travels into (containing the image). The physical reading is compact: the left side compares "index over distance" on the two sides of the pole, and the right side is fixed once and for all by the curvature and the pair of media — it does not depend on where the object is placed. Move the object and $u$ changes, but $(n_2-n_1)/R$ stays the same, so $v$ must adjust to keep the equation balanced. This single formula holds for convex or concave surfaces and for real or virtual objects and images, provided every quantity is substituted with its correct sign.

Figure

A convex spherical surface separating air on the left from glass on the right, with an object on the axis in air, a curved refracting surface at the pole, the centre of curvature marked on the glass side, and a refracted ray bending toward the axis to form a real image in the glass

When the surface is flat instead of curved, $R\to\infty$ and the right-hand side of the formula vanishes, giving $\dfrac{n_2}{v}=\dfrac{n_1}{u}$ — this is exactly the familiar apparent-depth rule that explains why a fish, a coin, or an air bubble seen through a flat denser-to-rarer interface looks shallower than it really is. Recognising this special case is often faster in an exam than re-deriving it: apparent depth is not a separate law, it is this same relation with $R$ sent to infinity.

Derivation AThe single spherical refracting surface formula from Snells law

The goal is to derive $\dfrac{n_2}{v}-\dfrac{n_1}{u}=\dfrac{n_2-n_1}{R}$ from Snell's law and simple triangle geometry, using nothing beyond the small-angle approximation $\sin\theta\approx\tan\theta\approx\theta$ (in radians) that defines the paraxial regime, so that the object distance $u$ , image distance $v$ and radius of curvature $R$ , all measured from the pole with the Cartesian sign convention, come out linked by one clean algebraic relation independent of exactly which paraxial ray is chosen.

Step 1. Set up the geometry. Let a ray leave an on-axis object $O$ at a small angle, strike the spherical surface at point $M$ close to the pole $P$ , and refract into the second medium to cross the axis at the image point $I$ . Let $C$ be the centre of curvature. The angle the incident ray $OM$ makes with the axis is $\alpha$ , the angle the normal $MC$ makes with the axis is $\gamma$ , and the angle the refracted ray $MI$ makes with the axis is $\beta$ . Because $M$ is close to the axis for a paraxial ray, all three angles are small, and the perpendicular height of $M$ above the axis, call it $h$ , is the same small quantity linking every angle to a distance: $\alpha\approx h/(-u)$ , $\gamma\approx h/R$ , $\beta\approx h/v$ , using the sign convention where $u$ is negative for a real object to the left of $P$ .

Step 2. Read off the angle of incidence and angle of refraction at $M$ from triangle geometry. In triangle $OMC$ , the exterior angle rule gives the angle of incidence $i=\alpha+\gamma$ . In triangle $ICM$ , the same exterior angle rule gives $\gamma=\beta+r$ , so the angle of refraction is $r=\gamma-\beta$ . These two relations simply say that the normal $MC$ splits the total bending between the "before" and "after" triangles.

Step 3. Apply Snell's law. For paraxial rays, Snell's law $n_1\sin i=n_2\sin r$ becomes, using the small-angle approximation, $n_1\,i=n_2\,r$ . Substituting the Step 2 expressions, $n_1(\alpha+\gamma)=n_2(\gamma-\beta)$ , which rearranges to $n_1\alpha+n_2\beta=(n_2-n_1)\gamma$ .

Step 4. Substitute the height relations from Step 1. Since $\alpha\approx h/(-u)=-h/u$ , $\beta\approx h/v$ , and $\gamma\approx h/R$ , the equation becomes $-n_1\dfrac{h}{u}+n_2\dfrac{h}{v}=(n_2-n_1)\dfrac{h}{R}$ . The common factor $h$ — the height of the ray above the axis — cancels completely, which is exactly why the result holds for any paraxial ray, not just one particular choice of height.

Step 5. Cancel $h$ and rearrange to the final form. Dividing through by $h$ gives $\dfrac{n_2}{v}-\dfrac{n_1}{u}=\dfrac{n_2-n_1}{R}$ , the single spherical refracting surface formula. Because the height cancelled, this relation is independent of which paraxial ray was used to derive it — every paraxial ray from the same object point converges to the same image point $I$ , which is precisely what makes $I$ a well-defined image rather than a blur.

Worked examples

Board level - image formed at a convex air-to-glass surface

A point object is placed 15 cm from the pole of a convex spherical surface of radius of curvature 20 cm, separating air ( $n=1$ ) from glass ( $n=1.5$ ). Light travels from air into the glass. Find the position and nature of the image.

Step 1. Assign signs. The object is in air, so $n_1=1$ ; the image forms in glass, so $n_2=1.5$ . The object is real and to the left of the pole, against the direction of incident light, so $u=-15$ cm. The surface bulges toward the object (convex as seen from the object side) with its centre of curvature on the outgoing (glass) side, so $R=+20$ cm.

Step 2. Substitute into $\dfrac{n_2}{v}-\dfrac{n_1}{u}=\dfrac{n_2-n_1}{R}$ : $\dfrac{1.5}{v}-\dfrac{1}{-15}=\dfrac{1.5-1}{20}$ , which gives $\dfrac{1.5}{v}+\dfrac{1}{15}=0.025$ .

Step 3. Solve for $v$ . $\dfrac{1.5}{v}=0.025-0.0667=-0.0417$ , so $v=\dfrac{1.5}{-0.0417}\approx -36$ cm.

Step 4. Interpret the sign. The negative value of $v$ means the image forms 36 cm on the same side as the object, in air, so it is a virtual image. Answer: the image is virtual, about 36 cm from the pole, on the object's side of the surface. The curved surface makes the object appear farther away than it actually is when viewed from the glass side — a single refracting surface can produce a virtual image exactly as a mirror can.

Competitive level - apparent depth of a trapped air bubble, checked against the general formula

A small air bubble is trapped 6 cm below the flat top surface of a thick glass slab ( $n=1.5$ ). At what apparent depth does the bubble appear to an observer looking straight down from air above, and how does this match the general formula's $R\to\infty$ limit?

Step 1. Identify the special case. The surface is flat, so $R\to\infty$ and the right-hand side of $\dfrac{n_2}{v}-\dfrac{n_1}{u}=\dfrac{n_2-n_1}{R}$ becomes zero, leaving $\dfrac{n_2}{v}=\dfrac{n_1}{u}$ . Light travels from the bubble in glass ( $n_1=1.5$ ) into the observer's air ( $n_2=1$ ), and the bubble sits at $u=-6$ cm, measured from the flat surface against the direction the light travels to reach the observer.

Step 2. Solve. $\dfrac{1}{v}=\dfrac{1.5}{-6}=-0.25$ , so $v=-4$ cm.

Step 3. Interpret and cross-check. The negative $v$ places the image 4 cm below the surface, on the same side as the real bubble, so the bubble appears shallower: 4 cm apparent depth versus 6 cm real depth. As a quick general check, apparent depth for a denser-to-rarer flat interface always satisfies apparent depth $=$ real depth $\times (n_2/n_1)$ ; here $6\times(1/1.5)=4$ cm, matching exactly. Answer: apparent depth $=4$ cm, the same effect that makes a coin at the bottom of a swimming pool look closer to the surface than it actually is. Timing note: for a flat-surface (apparent-depth) question, do not re-derive from Snell's law under exam pressure — recognise it instantly as $R\to\infty$ in the single formula, or use the shortcut apparent depth $=$ real depth $\times n_{\text{rarer}}/n_{\text{denser}}$ , and cross-check the two against each other as done above; this whole example should take under a minute.

Common traps

Forgetting that $n_1$ is always the refractive index of the medium containing the incident ray and the object, and $n_2$ is the medium the light travels into — swapping these gives a wrong, often reciprocal-looking, answer.
Carrying over the mirror-formula habit of always treating distances as negative — for a spherical refracting surface, $R$ can be positive or negative depending on which way the surface curves relative to the incident light, and must be judged case by case, not assumed.
Forgetting that the flat-surface apparent-depth case is simply $R\to\infty$ inside the same general formula, not an unrelated rule to memorise separately.
Mixing up which index divides $v$ and which divides $u$ : it is always $n_2/v-n_1/u$ , matching "index of the medium the light is currently travelling in, over the corresponding distance from the pole."
Forgetting to state whether the resulting image is real or virtual, and on which side — the sign of $v$ alone answers both questions, but only if $u$ and $R$ were assigned correctly first.

You should be able to…

State the single-spherical-surface refraction formula $\dfrac{n_2}{v}-\dfrac{n_1}{u}=\dfrac{n_2-n_1}{R}$ and correctly identify $n_1$ , $n_2$ , $u$ , $v$ , and $R$ with signs for a given ray diagram.
Reproduce, in outline, how the formula follows from Snell's law and the small-angle (paraxial) approximation, without needing every intermediate line from memory.
Apply the formula to find the position and nature (real or virtual) of the image for both curved and flat refracting surfaces.
Recover the apparent-depth rule as the $R\to\infty$ special case, and use it directly for flat-interface problems without re-deriving it each time.
Recognise this formula as the algebraic building block behind the lens maker's formula for a thin lens, even without repeating the two-surface combination yourself.

Ask the tutor: want to walk through how applying this formula twice, once per lens surface, builds the lens maker's formula, or need more guided sign-convention practice on curved refracting surfaces?

Why does the formula look different from the mirror formula I already know?

The mirror formula, $1/v+1/u=1/f$ , describes light that reflects and stays in the same medium, so there is only one refractive index to worry about (or none, since it cancels out). The spherical-surface refraction formula, $n_2/v-n_1/u=(n_2-n_1)/R$ , describes light crossing into a different medium, so both $n_1$ and $n_2$ appear explicitly, and there is a minus sign between the $v$ and $u$ terms instead of a plus sign. They look different because they describe genuinely different physical processes — reflection off a curved mirror versus refraction through a curved boundary — even though both come from the same paraxial small-angle approximation.

Do I need to memorise the derivation, or just the final formula?

For CBSE board purposes, you should be comfortable using the final formula $n_2/v-n_1/u=(n_2-n_1)/R$ correctly with signs, and you should understand roughly where it comes from (Snell's law plus the small-angle approximation), but a full step-by-step reproduction of the derivation is not typically the centrepiece of exam questions the way, say, the mirror formula's use is. Spend most of your practice time on sign-convention accuracy in numerical problems rather than on reciting every derivation step from memory.

Why does the same formula work for both convex and concave surfaces?

Because the formula was derived using only the general small-angle relations between the object distance, image distance, radius of curvature and the height of a paraxial ray — nothing in that derivation assumed the surface bulges one particular way. The shape information is entirely carried by the sign of $R$ : positive if the centre of curvature lies on the outgoing-light side of the pole, negative if it lies on the incoming side. So instead of separate formulas for convex and concave, you keep one formula and let the sign of $R$ (and of $u$ and $v$ ) do the work of describing the geometry.

How is this connected to the lens maker's formula I will meet next?

A thin lens has two curved surfaces, one where light enters and one where it exits. Applying the single-surface formula once at the first surface gives an intermediate image; treating that intermediate image as the "object" for the second surface and applying the formula again gives the final image. Carrying out that two-step substitution algebraically, and taking the lens to be thin enough that the two surfaces are essentially at the same point, produces the lens maker's formula, $1/f=(n-1)(1/R_1-1/R_2)$ . You are not asked to redo that combination from scratch — only to use the final lens maker's formula — but knowing that it comes from applying this exact relation twice makes the lens formula far less mysterious.