Refraction of Light at Curved Surfaces
1. What This Chapter Covers
The chapter opens with things you have handled: the spectacles some people use for reading, and the small magnifying glass a watch repairer uses. Have you touched one? Is the surface plane or curved? Is it thicker in the middle or at the edge?
Refraction at a plane surface was covered in the previous class. This chapter does the curved case, and it is the most mathematical chapter in the volume — 9 periods in July, textbook pages 124 to 170, containing four derivations and seven worked examples.
The structure is worth seeing in advance, because each step is used to build the next:
- Refraction at one curved surface gives the curved-surface formula.
- A lens is two curved surfaces, so applying that formula twice gives the lens maker's formula.
- Along the way, similar triangles give the lens formula and magnification.
2. Activity 1 — the Arrow Behind the Tumbler
Draw a 4 cm arrow with a black sketch pen on a thick sheet of paper. Take an empty cylindrical transparent vessel, such as a glass tumbler, and set it on a table. Have a friend hold the sheet behind the vessel, arrow horizontal, while you look from the other side.
You see a diminished image of the arrow.
Now have your friend fill the vessel with water and look again from the same position. Now the image is inverted.
Why the two cases differ
With the vessel empty, light from the arrow refracts at the curved interface, travels through the glass, enters the air inside, refracts again at the opposite curved surface, and comes out. Travelling through two media this way, it forms a diminished image.
With the vessel full, light enters through the curved surface, travels through water, comes out of the glass, and forms an inverted image. There is now a curved interface between two different media, air and water. The book asks you to assume the refractive indices of water and glass are the same, while noting honestly that in fact they are not.
3. Refraction at a Curved Surface (Textbook 4.1)
The terminology parallels the mirror chapter exactly:
| Term | Meaning |
|---|---|
| Centre of curvature (C) | Centre of the sphere of which the curved surface is a part |
| Normal | Any line drawn from C to a point on the curved surface |
| Pole (P) | The centre of the curved surface |
| Principal axis | The line joining the centre of curvature and the pole |
The key difference from a plane surface is that the direction of the normal changes from point to point.
The basic bending rule
As with plane surfaces, a ray bends towards the normal going from a rarer to a denser medium, and bends away from the normal going from denser to rarer.
Two rays do not bend at all. By Snell's law, a ray travelling along the normal does not deviate. So a ray along the principal axis and a ray through the centre of curvature both pass straight through.
The four cases of a parallel ray
For an incident ray parallel to the principal axis there are four combinations, and the book draws all of them:
| Case | Surface | Direction | Refracted ray |
|---|---|---|---|
| 1 | Convex | Rarer to denser | Converges to a point on the axis |
| 2 | Convex | Denser to rarer | Diverges away from the axis |
| 3 | Concave | Denser to rarer | Converges to a point on the axis |
| 4 | Concave | Rarer to denser | Diverges away from the axis |
In cases 1 and 3 the refracted ray actually reaches a point on the principal axis. In cases 2 and 4 it moves away, but extending it backwards makes it cut the axis. Either way, that intersection point is the focal point (F).
The observation this explains is one you have seen: a lemon in a glass of water looks bigger than it really is when viewed from the side. What you are seeing is an image, not the lemon.
4. The Curved-Surface Formula (Textbook 4.1.1, 4.1.2)
Setting up
Consider a curved surface separating two media of refractive indices n₁ and n₂, with a point object at O on the principal axis.
The ray along the principal axis passes through the pole undeviated. A second ray, making an angle α with the axis, meets the surface at A with angle of incidence θ₁, bends, and travels through the second medium along AI with angle of refraction θ₂. The two refracted rays meet at I, and the image forms there.
Let γ be the angle the second refracted ray makes with the axis, and β the angle between the normal and the axis. Then PO = u, PI = v, PC = R.
The derivation
In triangle ACO, θ₁ = α + β. In triangle ACI, β = θ₂ + γ, so θ₂ = β − γ.
By Snell's law, n₁ sin θ₁ = n₂ sin θ₂, so
n₁ sin(α + β) = n₂ sin(β − γ) ..... (1)
The paraxial approximation. If the rays travel very close to the principal axis they can be treated as parallel — these are paraxial rays — and the angles α, β and γ all become very small. Then sin(α + β) = α + β and sin(β − γ) = β − γ, so
n₁(α + β) = n₂(β − γ), giving n₁α + n₁β = n₂β − n₂γ ..... (2)
Since the angles are small, tan α = AN/NO = α, tan β = AN/NC = β, and tan γ = AN/NI = γ. Substituting,
n₁(AN/NO) + n₁(AN/NC) = n₂(AN/NC) − n₂(AN/NI) ..... (3)
Because the rays are close to the axis, N coincides with the pole P, so NI, NO and NC become PI, PO and PC:
n₁/PO + n₁/PC = n₂/PC − n₂/PI
n₁/PO + n₂/PI = (n₂ − n₁)/PC ..... (4)
Sign convention
The same five rules apply to everything in this chapter, curved surfaces and lenses alike:
- All distances are measured from the pole, or optic centre.
- Distances along the direction of the incident ray are positive.
- Distances opposite to the incident ray are negative.
- Heights measured vertically above the axis are positive.
- Heights measured vertically below the axis are negative.
Applying PO = −u, PI = v, PC = R to equation (4) gives the general result:
n₂/v − n₁/u = (n₂ − n₁)/R
The plane-surface special case
A plane surface is one whose radius of curvature approaches infinity, so 1/R becomes zero and the formula collapses to
n₂/v = n₁/u
with u and v measured from the plane interface.
One derivation carries the chapter. The lens maker's formula is the curved-surface formula applied at each of a lens's two surfaces and added.
5. The Three Worked Examples on Curved Surfaces
Example 1 — the bird and the fish
A bird flies vertically down towards a pond at constant speed, with a fish directly below it. To the fish, does the bird appear farther, closer, faster or slower?
Using the plane-surface form n₂/v = n₁/u with n₁ = 1 for air, n₂ = n for water, object distance u = −x and image distance v = y:
1/(−x) = n/(−y), giving y = nx
Since n is greater than 1, y is greater than x. So the bird appears farther away than it actually is. And because in a given time the bird really covers x while appearing to cover y, and y is larger, the fish also sees it moving faster than its actual speed.
Options (a) and (c) are correct.
Example 2 — the transparent sphere
A transparent sphere of radius R and refractive index n is in air. At what distance from its surface must a point object be placed so that a real image forms at the same distance from the second surface?
From the symmetry, the rays must pass through the sphere parallel to the principal axis. So after refraction at the first surface, v = infinity. With u = −x, n₁ = 1, n₂ = n, R = R:
n/infinity − 1/(−x) = (n − 1)/R
1/x = (n − 1)/R, so x = R/(n − 1)
Example 3 — the dot at the centre
A glass sphere has a small opaque dot at its centre. Does the dot appear to be where it actually is?
Here n₁ = n for glass, n₂ = 1 for air, u = −R and R = −R. Substituting into the curved-surface formula and solving gives v = −R.
The image distance equals the object distance, so the apparent position is the same as the actual position — and notably, the result is independent of the refractive index of the sphere's material.
6. Lenses (Textbook 4.2)
A lens is formed when a transparent material is bounded by two surfaces, of which one or both are spherical — so a lens has at least one curved surface.
| Type | Description |
|---|---|
| Biconvex | Two spherical surfaces bulging outwards; thick at the middle |
| Plano-convex | One flat and one outward-curved surface |
| Concavo-convex | One inward and one outward curved surface |
| Biconcave | Two spherical surfaces curved inwards; thin at the middle, thicker at the edges |
| Plano-concave | One flat and one inward-curved surface |
| Convexo-concave | One outward and one inward curved surface |
The first three are grouped as converging lenses, the last three as diverging lenses. Throughout the chapter only thin lenses are considered, meaning the thickness is treated as negligible.
Each curved surface is part of a sphere, so a lens has two centres of curvature, C₁ and C₂, and two radii, R₁ and R₂. The line joining C₁ and C₂ is the principal axis, and the midpoint of a thin lens is its optic centre (P).
Focal length (4.2.1)
A parallel beam incident on a lens either converges to a point or seems to emanate from a point on the principal axis. That point is the focus (F), and every lens has two focal points. The distance from the focus to the optic centre is the focal length, f.
A crucial labelling warning the book gives explicitly: in the ray diagrams, C₁ and C₂ are not centres of curvature — they are simply the points at a distance of 2f from the optic centre.
7. How Rays Behave at a Lens (Textbook 4.2.2)
Because a thin lens is treated as a single surface element, the net refraction is shown at one surface only. Four ray behaviours follow:
| Situation | Incident ray | After refraction |
|---|---|---|
| I | Along the principal axis | Undeviated |
| II | Through the optic centre | Undeviated |
| III | Parallel to the principal axis | Passes through the focus, or appears to diverge from it |
| IV | Through the focus | Travels parallel to the principal axis |
Situation IV is justified by the principle of least time, and it is the converse of situation III.
If parallel rays arrive at an angle to the principal axis, they converge to, or appear to diverge from, a point on the focal plane — the plane perpendicular to the principal axis at the focus.
8. Drawing Ray Diagrams and the Six Cases (Textbook 4.2.3)
The procedure is:
- Select a point on the object placed on the principal axis.
- Draw two of the four rays from situations I to IV.
- Extend them to intersect; that point is the image position.
- Drop a normal from the intersection to the principal axis.
- The length of that normal is the size of the image.
For a convex lens, the six standard positions give:
| Position of object | Position of image | Characteristics |
|---|---|---|
| At infinity | At the focal point F₁ | Point image |
| Beyond C₂ (beyond 2f) | Between F₁ and C₁ | Inverted, diminished, real |
| At C₂ (at 2f) | At C₁ | Inverted, same size, real |
| Between F₂ and C₂ | Beyond C₁ | Inverted, magnified, real |
| At F₂ | At infinity | Size and nature cannot be discussed |
| Between F₂ and P | Beyond F₂, same side as object | Erect, magnified, virtual |
Two conclusions the chapter draws from the last row. A virtual image can be seen directly with the eye, whereas the real images can only be viewed if captured on a screen. And a magnified virtual image forms on the same side as the object. This behaviour is what a microscope is built on, and it only occurs when the object is nearer than the focal length.
For a concave lens the result is much simpler: whatever the object position on the principal axis, you get an erect, virtual, diminished image between the focal point and the optic centre.
Examples 4 and 5 — constructing rays in awkward cases
Example 4 gives a construction for a point source S on the axis beyond F₂, using the focal plane:
- Draw a perpendicular to the principal axis through the focus F₁.
- Draw a ray from S in any direction to meet the lens at P′.
- Draw a line parallel to that ray through the optic centre P; it cuts the perpendicular at F₀.
- Draw a line from P′ through F₀ to meet the principal axis at I.
- I is the image of S.
Example 5 asks you to complete ray paths through given lenses using the same steps. The value of this construction is that it works for a ray in any direction, not only the four convenient ones.
9. The Lab Activity
Aim. Observe the types of images and measure object and image distances from a lens.
Materials. A candle, paper, a convex lens of known focal length, a V-stand, and a measuring tape or metre scale.
Procedure. Place the V-stand in the middle of a table about 2 m long and set the convex lens on it. Light the candle and have a friend take it far away along the principal axis. Adjust a screen — a white paper perpendicular to the axis — on the other side until an image appears. Measure both distances.
Then place the candle at 60 cm, get a clear image, and record u and v. Repeat for 50 cm, 40 cm, 30 cm and so on.
The questions the activity is really asking are the interesting ones: could you get an image on the screen for every object distance, and if not, what is the minimum limiting object distance for a real image? Where no image appears on the screen, look through the lens with your eye from the screen's position — you will see a magnified virtual image on the same side as the object, which is why the screen showed nothing.
10. The Lens Formula (Textbook 4.3)
Consider an object OO′ on the principal axis in front of a convex lens, forming a real image II′ on the other side.
The ray from O′ parallel to the axis passes through F₁ after refraction. A second ray from O′ through the optic centre P is undeviated. They meet at I′, the image of O′.
From similar triangles PP′F₁ and F₁I′I:
PP′/I′I = PF₁/F₁I ..... (1)
Since F₁I = PI − PF₁:
PP′/I′I = PF₁/(PI − PF₁) ..... (2)
From the second pair of similar triangles OO′P and PI′I, and using OO′ = PP′:
PP′/I′I = PO/PI ..... (3)
Comparing (2) and (3):
PO/PI = PF₁/(PI − PF₁)
Rearranging and dividing through by PI gives
1/PO = 1/PF₁ − 1/PI, so 1/PO + 1/PI = 1/PF₁ ..... (4)
Applying the sign convention with PO = −u, PI = v, PF₁ = f:
1/v − 1/u = 1/f
This is the lens formula, usable for any lens provided the sign convention is applied.
11. Magnification (Textbook 4.4)
Triangles OO′P and II′P are similar, so I′I/OO′ = PI/PO. Substituting the sign convention values gives
m = hᵢ/hₒ = v/u
Note the contrast with mirrors, where magnification carries a minus sign. For lenses the chapter writes it as v/u with no minus.
Example 6 — lamp and screen one metre apart
An electric lamp and a screen sit 1 m apart. In what positions will a convex lens of focal length 21 cm give a sharp image?
Let x be the lamp-to-lens distance, so u = −x and v = 100 − x, with f = 21. Substituting into the lens formula:
1/(100 − x) + 1/x = 1/21
This rearranges to the quadratic x² − 100x + 2100 = 0, which factorises as (x − 70)(x − 30) = 0.
So x = 70 cm or x = 30 cm. There are two lens positions that give a sharp image — a result worth remembering, since a quadratic here always signals two valid placements.
12. Focal Length Depends on the Surroundings (Activity 2)
Take the lens from the lab activity and note its average focal length. Take a cylindrical vessel whose depth is roughly four times that focal length, put a black stone at the bottom, and pour in water so the depth above the stone exceeds the focal length. Using a circular lens holder, dip the lens horizontally above the stone, setting the stone-to-lens distance equal to or less than the focal length measured in air. Do this in the open.
You can still see the image of the stone even though you have dipped the lens beyond what should be its focal length in air. Raise it further and eventually the image disappears.
The conclusion: the focal length of the lens has increased in water, so focal length depends on the surrounding medium.
The chapter states the general consequence. A convex lens converges when the surrounding medium has a lower refractive index than the lens, and diverges when the surrounding medium has a greater refractive index. This is why an air bubble in water behaves like a diverging lens.
13. The Lens Maker's Formula (Textbook 4.5)
This is the curved-surface formula applied twice.
Take a point object O on the principal axis of a thin lens, with surrounding medium of refractive index nₐ and lens material n_b.
At the first surface, radius R₁, the ray refracts at A. Assume it would form an image at Q if the second surface were absent. With PO = −u, v = x, R = R₁, n₁ = nₐ, n₂ = n_b:
n_b/x + nₐ/u = (n_b − nₐ)/R₁ ..... (1)
At the second surface, radius R₂, the ray refracts again at B and reaches I. The image Q from the first surface acts as the object for the second. Now the lens material is medium 1 and the surroundings are medium 2, so the suffixes interchange: n₁ = n_b, n₂ = nₐ, with u = x, v = v, R = −R₂:
nₐ/v − n_b/x = (nₐ − n_b)/(−R₂) ..... (2)
Adding (1) and (2) eliminates x:
nₐ/v + nₐ/u = (n_b − nₐ)(1/R₁ + 1/R₂)
Dividing by nₐ and writing n_ba = n_b/nₐ for the refractive index of the lens relative to its surroundings:
1/v + 1/u = (n_ba − 1)(1/R₁ + 1/R₂)
Applying the sign convention to generalise, and using 1/v − 1/u = 1/f:
1/f = (n_ba − 1)(1/R₁ − 1/R₂) ..... (3)
If the surrounding medium is air, the relative refractive index becomes the absolute refractive index n of the lens:
1/f = (n − 1)(1/R₁ − 1/R₂)
This is the lens maker's formula, and this last form may be used only when the lens is in air.
Example 7 — a double concave lens
Find the focal length of a double concave lens in air with radii R₁ = 30 cm and R₂ = 60 cm, taking n = 1.5.
By the sign convention, R₁ = −30 cm and R₂ = +60 cm:
1/f = (1.5 − 1)(1/(−30) − 1/60)
Solving gives f = −40 cm. The minus sign indicates the lens is divergent, which is the expected result for a double concave lens.
Key words from the chapter
Lens, focal length, focus, optic centre, principal axis, radius of curvature, centre of curvature, focal plane, convergence, divergence.
14. Summary
Activity 1 shows the two behaviours the chapter explains: an empty tumbler gives a diminished image of an arrow, a water-filled one gives an inverted image.
At a curved surface, the normal is the line to the centre of curvature and its direction changes from point to point. Rays bend towards the normal going rarer to denser and away from it going denser to rarer, while rays along the axis or through the centre of curvature pass undeviated. A parallel ray gives four cases, and in each the refracted ray, extended backwards if necessary, cuts the axis at the focal point.
Snell's law under the paraxial approximation gives the curved-surface formula n₂/v − n₁/u = (n₂ − n₁)/R, which reduces to n₂/v = n₁/u for a plane surface. Three worked examples apply it: the bird appearing farther and faster to a fish, the object distance R/(n − 1) for a sphere, and the dot at a sphere's centre appearing exactly where it is, independent of refractive index.
A lens is bounded by at least one curved surface, with six named types split into converging and diverging. For thin lenses, four ray behaviours — along the axis, through the optic centre, parallel to the axis, and through the focus — let any two rays locate an image. A convex lens gives six standard cases, only the last of which is virtual, erect and magnified; a concave lens always gives an erect, virtual, diminished image.
Similar triangles give the lens formula 1/v − 1/u = 1/f and magnification m = hᵢ/hₒ = v/u. The lamp-and-screen example shows the resulting quadratic has two solutions, so two lens positions work. Activity 2 shows focal length depends on the surrounding medium, which is why an air bubble in water diverges light.
Finally, applying the curved-surface formula at both surfaces and adding gives the lens maker's formula 1/f = (n − 1)(1/R₁ − 1/R₂), valid for any thin lens in air, with a negative f signalling a divergent lens.
