Reflection of Light at Curved Surfaces

1. What This Chapter Covers

The textbook opens by admitting you already know the first law of reflection from classes 7 and 8. What you do not yet know is how to apply it to a curved surface, and that gap is the whole chapter.

The book's own opening questions are worth keeping in front of you, because the chapter answers each one:

• Is the image in a bulged surface the same as in a plane mirror?
• Is the mirror used in automobiles a plane mirror, and why does it show small images?
• Why does our image appear thin or bulged in some mirrors?
• Can we see an inverted image in any mirror?
• Can we focus sunlight to a point using a mirror instead of a magnifying glass?
• Are the angle of reflection and angle of incidence still equal for curved surfaces?

Sections 1.1 to 1.7 answer these in order. The chapter is allotted 6 periods in June and runs from textbook page 2 to page 42.

The one idea everything rests on

The first law of reflection says a ray incident at some angle to the normal reflects at an equal angle to that normal. This law is true for every surface, flat or curved. Nothing about it changes here.

The difficulty is entirely practical. On a plane surface, finding the normal is trivial. On a curved surface it is not obvious at all. So the chapter's real first job is to find the normal — and everything else is a consequence.

2. Finding the Normal on a Curved Surface (Textbook 1.1)

Activity 1 — the foam and pins

Take a thin piece of foam or rubber, such as a slipper sole. Push pins into it along a straight line, as in figure 1(a). Every pin stands perpendicular to the foam, so if the foam were a mirror, each pin would mark the normal at that point.

Now bend the foam inwards, as in figure 1(b). The pins still mark the normals, but they now lean towards each other — they converge on a single point.

Bend the foam outwards instead, as in figure 1(c). Now the pins lean away from each other; they diverge.

What that tells you about the two mirrors

A concave mirror behaves like the sole bent inwards. A convex mirror behaves like the sole bent outwards. This is the mapping to hold on to, because it converts a question about mirrors into a question about pins you can actually picture.

For a concave mirror, then, all the normals converge to one point. That point has a name: the centre of curvature (C).

And that gives the rule the rest of the chapter uses constantly:

The normal at any point on a spherical mirror is simply the line joining that point to the centre of curvature.

Why this is geometrically true

The textbook asks you to recall a result from circles and tangents: a radius is always perpendicular to the tangent drawn at that point on the circle. A spherical mirror is part of a sphere, and the line from C to a point on the mirror is a radius. So it is perpendicular to the mirror surface there — which is exactly what "normal" means.

3. Pole, Principal Axis and Radius of Curvature

Three more terms are defined in the same section, and all of them are measured from the mirror itself:

In figure 2(b) the incident angle i is measured from the normal, and the reflected angle is r. By the first law, i = r — unchanged from the flat-mirror case.

A drawing convention appears here too, and it is easy to miss. When it is not obvious which face of the mirror reflects, the book shades the non-reflecting (coated) side with short lines. Read the shading to find the working surface.

4. Focus and Focal Length (Textbook 1.1.1)

The problem of getting parallel rays

To test the drawings, you need a genuine beam of parallel rays. The book works out how to get one experimentally rather than just asserting it.

Two pins are stuck upright on a thermocole block (figure 3). With a light source held close, the shadows diverge from the base of the pins. Move the source further away and the angle of divergence shrinks. Move it far enough and the shadows become parallel.

But as the candle retreats, the light gets dim. So a parallel beam needs a source that is both very distant and bright enough — which points to one obvious candidate.

Activity 2 — the Sun and a concave mirror

Hold a concave mirror so sunlight falls on it. Move a small piece of paper in front of the mirror until you find the smallest, brightest spot. That spot is an image of the Sun. Keep the paper small so it does not block the incoming rays.

Rays arriving parallel to the principal axis converge to this one point after reflection. That point is the focus, or focal point (F).

The distance from the pole to this spot is the focal length (f). The chapter states the relation between it and the radius of curvature:

R = 2f

What the paper shows as you move it

The book asks you to move the paper closer than the focal length and then away. The image of the Sun first keeps getting smaller, reaches its minimum at the focal point, and then enlarges again beyond it. This is a useful check that you have actually found F and not just a bright patch.

The convex case

For a convex mirror the same parallel rays diverge after reflection (figure 5). Extend the reflected rays backwards and they meet at F. So a convex mirror has a focus too, but it sits behind the mirror and no light actually passes through it.

5. The Lab Activity — Letting the Mirror Tell You

Before any ray diagrams, the textbook makes you collect data. This ordering is deliberate: the diagrams are checked against observations, not the other way round.

Aim. Observe the types of images formed, and measure object distance and image distance.

Materials. A candle, paper, a concave mirror of known focal length, a V-stand, and a measuring tape or metre scale.

Procedure. Mount the mirror on the V-stand and set up the candle and scale as in figure 6. Move the candle along the axis from about 10 cm to 80 cm. For each position, move the paper screen until the image is sharp, and record both distances.

Two practical cautions the book gives: keep the flame above the axis and the paper below the axis, so the screen does not block the light forming the image.

Record whether each image is enlarged or diminished, and inverted or erect. At some positions you will get no image at all — the instruction is to note that down too, because those failures matter.

The result, reorganised

Once the focal point and centre of curvature are marked, the raw distances can be regrouped by where the object sits relative to F and C. That regrouping is the payoff of the whole experiment:

The book adds one more instruction that students usually skip: while hunting for the image on the screen, also look directly into the mirror and note what you see. The two observations disagree for the first row, and that disagreement is the point.

6. Ray Diagrams for a Concave Mirror (Textbook 1.2)

Why two rays are enough

Take at least two rays leaving the same point on the object in different directions, reflect them, and find where they meet. That intersection is the image of that point.

Figure 7 shows two rays from the tip of the flame meeting at a point A. The book then asks the sharp question: why only at A?

Hold the screen at any other point — say B — and the rays strike the screen at different places. Draw more rays from the same tip and they all pass through A, but they miss each other at B. So at A the image is sharp; anywhere else the overlapping images blur it. This is the same effect you saw hunting for the Sun's image with the paper.

The four convenient rays

Constructing the normal and measuring angles for an arbitrary ray each time would be tedious. Instead the chapter identifies rays whose reflected paths are already known:

R2 is simply the converse of R1. R3 works because a ray along the normal always retraces its path — and a line through C is the normal.

The four rays whose reflected paths are known in advance. Choosing two of them replaces measuring angles.

Locating the base of the image

Trace two rays from the tip to get point A, and two from the bottom to get point B. The book notes that B turns out to be the same distance from the mirror as A, so the image stands vertical and inverted.

There is a shortcut for the base. Any ray leaving a point on the axis and travelling along the axis reflects straight back along the axis. So the base of the image must also lie on the axis. Drop a perpendicular from A to the axis and you have it (figure 12).

The case with no image on the screen

Place the object closer than the focal length (figure 13). R1 is easy. R2 is impossible — a ray through F would never reach the mirror. R3 looks impossible too, so the book makes a small adjustment: take a ray from the tip heading in a direction that would pass through C if extended backwards. That ray is still normal to the surface, so it still reflects back along itself.

Now the two reflected rays diverge. They never meet, so no screen position gives a sharp image — exactly matching the blank rows in your lab table. Moving the screen further away would not have helped.

Yet you still see an image when you look into the mirror. Extend the diverging reflected rays backwards until they meet, as with a plane mirror. That intersection (figure 14) gives an image that is erect and enlarged.

This is a virtual image: it cannot be caught on a screen, because no light actually passes through it. An image formed by the actual intersection of reflected rays is a real image, and that one can be caught on a screen.

Two everyday consequences

A concave mirror enlarges an erect image when the object is nearer than the focal length — which is why shaving mirrors and dentists' mirrors are concave. And it converges parallel rays to a point, which is why TV dish antennas have that shape.

7. Ray Diagrams for a Convex Mirror (Textbook 1.3)

The same method works, with the rays restated for a surface that diverges. The chapter gives three rules:

Rule 2 is again the converse of rule 1. Notice the repeated phrase "appears to come from" — for a convex mirror, F and C are behind the mirror, so the rays only seem to originate there.

Applying rules 1 and 3 to an object AB anywhere on the axis (figure 20) gives an image that is erect, diminished and virtual, always located between P and F behind the mirror.

That last word "always" is what makes convex mirrors useful. Whatever the object distance, the image stays small and upright, so the mirror shows a wide field of view — which answers the opening question about rear-view mirrors in vehicles.

8. Deriving the Mirror Formula (Textbook 1.4)

Figure 21 sets up the derivation. A ray from the tip B of object AB travels parallel to the axis, strikes the mirror at X, and passes through F. A second ray from B passes through C, strikes at Y, and returns along the same direction. The two reflected rays meet at B′, so A′B′ is the image.

Step 1. Triangles ABC and A′B′C are similar, so

AB / A′B′ = AC / A′C ..... (1)

Step 2. Draw P′X perpendicular to the principal axis. Triangles P′XF and A′B′F are similar, so

P′X / A′B′ = P′F / A′F ..... (2)

Step 3. From the figure, P′X = AB, so equation (2) becomes

AB / A′B′ = P′F / A′F ..... (3)

Step 4. Comparing (1) and (3):

AC / A′C = P′F / A′F ..... (4)

Step 5. For paraxial rays — rays travelling very close to the principal axis — P′ coincides with P, so P′F = PF:

AC / A′C = PF / A′F ..... (5)

Step 6. Read the segments off the figure:

AC = PA − PC, A′C = PC − PA′, A′F = PA′ − PF

Substituting into (5):

(PA − PC) / (PC − PA′) = PF / (PA′ − PF) ..... (6)

Step 7. Now write PA = u, PC = R = 2f, PA′ = v, PF = f:

(u − 2f) / (2f − v) = f / (v − f)

Cross-multiplying: (u − 2f)(v − f) = f(2f − v)

uv − uf − 2vf + 2f² = 2f² − vf

uv = 2f² − vf + uf + 2vf − 2f²

uv = uf + vf ..... (7)

Step 8. Divide throughout by uvf:

uv/uvf = uf/uvf + vf/uvf

which gives the mirror formula:

1/f = 1/v + 1/u

The chapter is explicit that this formula is only usable together with a sign convention, which is why the next section exists.

9. Sign Convention (Textbook 1.5)

The textbook states three rules:

1. All distances are measured from the pole.
2. Distances measured in the direction of the incident light are positive; those measured opposite to it are negative.
3. The height of the object (hₒ) and the height of the image (hᵢ) are positive measured upwards from the axis, negative measured downwards.

Rule 1 is the one worth repeating to yourself. It is also asked directly as a multiple-choice question at the end of the chapter: all distances related to spherical mirrors are measured from the pole — not the focus, not the object.

10. Magnification (Textbook 1.6)

Magnification compares the size of the image with the size of the object. The chapter restricts the discussion to height only.

In figure 22, a ray from O′ hits the pole at angle θ and reflects at the same angle θ. Triangles POO′ and PII′ are similar, so

II′ / OO′ = PI / PO ..... (1)

Applying the sign convention: PO = −u, PI = −v, OO′ = hₒ, II′ = −hᵢ. Substituting,

−hᵢ / hₒ = −v / −u

which rearranges to the two forms of magnification:

m = hᵢ / hₒ and m = −v / u

The textbook's worked example

Problem. An object 4 cm in size is placed 25 cm in front of a concave mirror of focal length 15 cm. At what distance should a screen be placed to get a sharp image? Find the nature and size of the image.

Applying the sign convention: f = −15 cm, u = −25 cm, hₒ = +4 cm.

Substituting into the mirror formula:

1/(−15) = 1/v + 1/(−25)

1/v = 1/25 − 1/15

1/v = −2/75, so v = −37.5 cm

So the screen goes 37.5 cm from the pole, and since v is negative the image is real.

For the size:

m = hᵢ/hₒ = −v/u, so hᵢ/4 = −(−37.5)/(−25)

hᵢ = −(37.5 × 4)/25 = −6 cm

The negative height means the image is inverted, and 6 cm against an object of 4 cm means it is enlarged. Both conclusions agree with the lab table: the object at 25 cm sits between F (15 cm) and C (30 cm), and that row predicts an enlarged, inverted, real image beyond C.

11. Putting It to Work — the Solar Cooker (Textbook 1.7)

The chapter closes with a build, prompted by the story of Archimedes burning ships with mirrors.

Start small: a concave mirror focuses parallel sunlight at F sharply enough to scorch paper (figure 23). The book asks you to try the same with a convex mirror and observe the difference — the convex mirror diverges the light and never concentrates it.

To scale it up, make a wooden or iron frame shaped like a TV dish. Cut acrylic mirror sheet into 8 or 12 isosceles triangles, each with height equal to the radius of the dish, so their bases together make up the circumference. Stick them to the frame (figure 24).

Point the concave side at the Sun, find the focal point, and place a vessel there. The book states it gets hot enough to cook rice.

One refinement is noted at the end: in real applications such as car headlights, concave mirrors are parabolic rather than spherical (figure 25).

Key words from the chapter

Centre of curvature, radius of curvature, principal axis, pole, focus or focal point, focal length, object distance, image distance, virtual image, real image, magnification.

12. Summary

The chapter is built on one geometric fact: the normal at a point on a spherical mirror is the line joining it to the centre of curvature. Activity 1 makes this visible with pins in bent foam, converging for a concave surface and diverging for a convex one.

From that, the standard terms follow — pole, principal axis, radius of curvature — along with the focus, found experimentally in Activity 2 using the Sun as a distant, bright source of parallel rays, and related to the radius by R = 2f.

The lab activity comes before the theory on purpose. Measuring object and image distances for a concave mirror, then regrouping the results by position relative to F and C, produces the six-case table. The positions that yield no image on the screen are as informative as the ones that do.

Ray diagrams then explain the table. Four rays have predictable reflected paths, and any two of them locate an image point. When the reflected rays diverge, extending them backwards gives a virtual image — erect and enlarged for a concave mirror, and always erect, diminished and virtual for a convex mirror, which is why convex mirrors serve as rear-view mirrors.

Finally, similar triangles give the mirror formula 1/f = 1/v + 1/u and magnification m = hᵢ/hₒ = −v/u, both valid only under the stated sign convention with all distances measured from the pole. The worked example shows the two working together, and the solar cooker shows the converging property put to use.