Reflection & Mirrors
Welcome to Ray & Wave Optics. The Oscillations & Waves topic followed waves on ropes, on water and in air. Light is a wave too, but for most of this topic its wavelength does not matter. In one uniform material, such as air, water or glass, light travels in straight lines, and a straight line with an arrow on it, called a ray, shows the way it goes. Mirrors, windows and glass blocks can all be handled with rays and a protractor. This is ray optics, also called geometrical optics, and it fills the first three pages: this page follows light that bounces off a surface, the second follows light that crosses into glass or water and bends, and the third follows light trapped inside glass and white light split into colours. The last page returns to light as a wave, with interference and diffraction.
The topic is small but steady: 9 of the 192 questions in the four papers so far, two or three in every paper. Three of them belong to this page, one each in December 2025 (Q31), March 2026 (Q28) and April 2026 (Q30). All three test the law of reflection, and none needs more than one subtraction or one doubling. Each adds a twist, though, and each offers the wrong answer the twist leads to: the angle to the mirror's surface taken as the angle of incidence, the angle of reflection given where the angle between the two rays was asked, and a refraction angle given in a question about reflection. All three are worked through below.
Rays and the normal
Evidence that light travels in straight lines is all around you: the sharp edge of a shadow, a beam of sunlight crossing a dusty room. A narrow beam is drawn as a single ray, a straight line with an arrow showing the direction the light travels.
When a ray meets a surface, the first line to draw is the normal: a line at right angles to the surface, through the point where the ray hits it. Every angle in ray optics is measured from the normal, not from the surface. There are two reasons. The normal is defined the same way for every surface: a curved mirror turns from point to point, but at each point there is exactly one line at right angles to it. And the laws of reflection and refraction take their simplest form with angles measured from the normal, as this page and the next show.
The angle between the incoming ray and the normal is the angle of incidence, i. A ray that travels along the normal has i = 0, and a ray that only skims the surface has i close to 90°. A question may give the angle between the ray and the surface instead; call it θ. The normal and the surface are at 90° to each other and the ray lies between them, so the two angles share that 90°:
i = 90° − θ.
A ray at 30° to the surface has i = 60°, and a ray at 70° to the surface has i = 20°. Only a ray at 45° makes the same angle with both.
The law of reflection
Lay a plane mirror on a sheet of paper and shine a narrow beam from a ray box at it. Mark the incident and reflected beams with pencil dots, draw the normal where they meet the mirror, and measure both angles from it with a protractor. Then repeat at other angles. Every run gives the same three results. The reflected ray is on the other side of the normal from the incident ray. Both rays and the normal lie in one plane, here the plane of the paper: the reflected beam never leaves the paper. And the angle between the reflected ray and the normal, the angle of reflection r, always equals the angle of incidence: 20° in, 20° out; 50° in, 50° out.
Mechanics gives a picture of why. Roll a ball at an angle against a smooth, rigid wall, so that it bounces off without losing speed. A smooth wall can push only along its normal, so the part of the ball's velocity along the wall does not change. The speed does not change either, so the part along the normal keeps its size and only reverses. The ball leaves with the same two parts, one of them flipped, and so it makes the same angle with the normal as it came in with. Light obeys the same rule, and so do the water and sound waves of the Wave Graphs & Wave Behaviour page.
The figure shows a second fact. The reflected ray makes the angle 90° − r with the surface, and since r = i, that is the incident ray's angle θ again. The two rays make equal angles with the normal, and equal angles with the surface too. So when a question gives an angle to the surface, convert it once, with i = 90° − θ, and the law of reflection does the rest. That was the whole of the March question.
A light ray strikes a mirror at 30° as shown in the figure. The angle of reflection is ( ).
The figure shows a horizontal plane mirror and a ray coming in from the upper left. The 30° is marked between the ray and the surface of the mirror, and a dotted normal stands at the point where the ray hits. No reflected ray is drawn.
A. 0° B. 60° C. 45° D. 30°
Step 1, the angle of incidence: the 30° is measured from the surface, not from the normal, so i = 90° − 30° = 60°.
Step 2, the law of reflection: r = i = 60°. The answer is B.
D, 30°, is the angle on the figure copied straight across. The reflected ray does make 30° with the surface, but that is not its angle of reflection. A, 0°, is the angle of reflection only for a ray that travels along the normal and comes straight back. C, 45°, does not follow from any common slip; it would be right only for a ray at 45° to the surface.
The angle between the two rays
The incident ray is i from the normal on one side, and the reflected ray is r = i from it on the other, so the angle between the two rays is i + r = 2i. The normal cuts that angle exactly in half. If the question gives the angle to the surface instead, convert first. Each ray makes θ with the surface, so the angle between them is 180° − 2θ, which is the same as 2(90° − θ) = 2i.
When a light ray hits a glass-air interface at an angle of incidence 30°, the angle between the reflected and incident rays is ( ).
A. 90° B. 60° C. 45° D. 30°
Step 1, the angle of reflection: this time the angle is given from the normal, so there is nothing to convert, and r = i = 30°.
Step 2, the angle between the rays: the two rays lie on opposite sides of the normal, so they are 30° + 30° = 60° apart. The answer is B.
D, 30°, is the angle of reflection, the answer to Step 1 rather than to the question. A, 90°, would need an angle of incidence of 45°; it is also what you get by adding the ray's 30° from the normal to its 60° from the surface, two angles measured from different lines. C, 45°, does not follow from any common slip. The stem does not say from which side the light meets the glass-air surface, and it does not need to: reflection obeys the same law on both sides, as the next section explains.
Reflection at glass and water
Mirrors are not the only surfaces that reflect. A window at night shows a faint picture of the room, and a still lake shows the trees on its bank. Whenever light reaches the boundary between two transparent materials, such as air and glass, part of it is reflected back into the first material and the rest crosses into the second. The part that crosses is refracted: it changes direction, and by how much depends on the two materials, which is the subject of the next page. The reflected part never enters the second material. It obeys the law of reflection exactly as at a mirror, r = i, whatever the two materials are, and whichever side the light comes from.
Light travels from air into glass. If the angle of incidence is 45°, the angle of reflection is ( )
A. 90° B. 60° C. 45° D. 30°
Step 1, which ray: the question asks for the angle of reflection, so it is about the light that bounces back into the air. "From air into glass" describes where the rest of the light goes, and the question asks nothing about it.
Step 2, the law of reflection: r = i = 45°. The answer is C. No refractive index is given, and none is needed.
D, 30°, is a refraction angle. The light that enters the glass bends towards the normal, and in a glass with refractive index √2 it would travel at exactly 30° to the normal, as the next page shows. That answers a question about the other ray. A, 90°, is 2 × 45°, the angle between the incident and reflected rays: December's question, not this one. B, 60°, does not follow from any common slip.
Light paths are also reversible. Send a ray back along the reflected ray, and it leaves along the old incident ray: swapping the two rays still leaves equal angles on the two sides of the normal. The same holds for refraction on the next page. One everyday consequence: if you can see someone's eyes in a mirror, they can see yours.
Images in a plane mirror
Look into a flat mirror, a plane mirror, and you see the room again behind the glass. The law of reflection explains where that image is and why it looks exactly like the real thing.
Take a point object O a distance d in front of a plane mirror, and draw the line from O that meets the mirror at right angles, at N. Light leaves O in every direction. Follow one ray to a point P on the mirror. The normal at P is parallel to ON, since both are at right angles to the mirror, and the ray reflects with r = i. To an eye that receives it, the reflected ray seems to come from somewhere behind the mirror, along its own line extended backwards. Extend it until it meets the line ON, also extended behind the mirror, at a point I.
Now compare the triangles ONP and INP. They share the side NP, and each has a right angle at N. At P, the incident ray makes the angle 90° − i with the mirror. The reflected ray makes 90° − r with the mirror on the other side of the normal, and its extension behind the mirror makes the same angle with the mirror, as vertically opposite angles. Since r = i, the two angles at P are equal. So the two triangles are congruent, and IN = ON = d.
Nothing in that argument depended on which ray was chosen. Every ray from O leaves the mirror as if it came from the same point I, a distance d behind the mirror on the line through O at right angles to it. An eye looking into the mirror sees O there: I is the image of O. Repeat for every point of a larger object, and each point's image lies straight across from it, the same distance behind the mirror. The object and its image are symmetric about the mirror.
Four properties follow. The image is virtual: no light goes behind the mirror, so no light passes through I; the reflected rays only seem to come from there, and a screen held at I would stay dark. It is upright, because the top of the object has its image straight across from it, at the top. It is the same size as the object, because every point is copied straight across, so every distance between two points is copied exactly. And it is as far behind the mirror as the object is in front. What does look changed is left and right: raise your right hand, and the image raises the hand on the same side of the mirror, which, as the image faces you, is its left. In Chinese papers a plane mirror is 平面镜 and a virtual image is 虚像.
Because the image is as far behind the mirror as the object is in front, distances to the image come out doubled. A person 2 m in front of a mirror is 4 m from their image. Walk towards the mirror at 1 m/s, and the image comes towards the mirror at 1 m/s from the other side, so the gap between you closes at 2 m/s.
Turning a mirror
Keep a ray fixed and turn the mirror it falls on through an angle α. The normal is fixed to the mirror, so it turns through α as well, and the angle of incidence changes by α. The angle between the incident and reflected rays is 2i, so it changes by 2α. The incident ray has not moved, so all of that change belongs to the reflected ray: it turns through 2α, twice the angle the mirror turned. Turn the mirror the other way, and the reflected ray turns through 2α the other way.
The figure puts numbers in. At first the angle of incidence is 30°, and the reflected ray is 30° from the normal on the right. Turn the mirror through 15° so that its normal tilts 15° away from the incident ray. The incident ray is now 30° + 15° = 45° from the new normal, and the new reflected ray is 45° from it on the other side, which puts it 15° + 45° = 60° from the old normal. The reflected ray has turned through 60° − 30° = 30°, twice the 15°.
A note on curved mirrors
A curved mirror obeys the law of reflection at every point; only the direction of the normal changes from point to point. On a mirror that is part of a sphere, the normal at each point runs along the radius, through the sphere's centre. A concave mirror, curved inwards like the inside of a spoon, turns parallel rays close to its axis inwards so that they meet at one point, its focus. Light paths are reversible, so a small lamp placed at the focus sends out a parallel beam, which is how a torch or a car headlamp works. A convex mirror, curved outwards like the back of a spoon, spreads parallel rays apart, as if they came from a point behind it. Its image is virtual, upright and smaller than the object, and it shows a wider view than a flat mirror of the same size, which is why convex mirrors stand at blind corners on roads and in shops. No paper has asked about curved mirrors, and the syllabus line for optics names only the laws of reflection and refraction.
Your turn! · Practice questions
Things to remember
| Concept | What to know |
|---|---|
| Rays | In one uniform material light travels in straight lines, drawn as rays with arrows. |
| The normal | The line at right angles to the surface where the ray hits. Angles of incidence, reflection and refraction are all measured from it. |
| Angle to the surface | A ray at θ to the surface has i = 90° − θ. March 2026 Q28: 30° to the mirror gives r = 60°. |
| Law of reflection | Incident ray, reflected ray and normal in one plane; the reflected ray on the other side of the normal; r = i. The two rays also make equal angles with the surface. |
| Angle between the rays | i + r = 2i, or 180° − 2θ from the angle to the surface. December 2025 Q31: i = 30° gives 60°. |
| Glass and water | The reflected ray obeys r = i at any boundary, whatever the materials; only the refracted ray depends on them. April 2026 Q30: 45°, not the 30° of a refracted ray. |
| Reversibility | A ray sent back along the reflected ray returns along the incident ray (光路可逆). |
| Plane-mirror image | Virtual, upright, the same size, and as far behind the mirror as the object is in front; object and image are symmetric about the mirror. |
| Distances to the image | An object d from the mirror is 2d from its image. Walk towards the mirror at v and the gap to your image closes at 2v. |
| Turning a mirror | Turn the mirror through α with the ray fixed, and the reflected ray turns through 2α. |
| Curved mirrors | Concave: brings parallel rays to a focus. Convex: spreads them, giving a smaller, upright image and a wider view. Not asked so far. |
| Common slips | The angle to the surface taken as i; the angle of reflection given for the angle between the rays; a refraction angle given for the angle of reflection. All three papers have a wrong option at 30°. |