E. Physics: Light and Geometric Optics · Light and Geometric OpticsThe Ray Model and the Laws of Reflection
E2.2 E2.1 A1.4 A1.5 A1.6 A1.8About 70 minutes
Learning goals- I can draw rays and normals and measure angles of incidence and reflection.
- I can state and apply the two laws of reflection.
- I can explain the difference between specular and diffuse reflection.
- I can carry out a safe investigation of the law of reflection and explain sources of error.
Words to know
ray
A straight line with an arrow showing the direction light travels.
normal
A line perpendicular (at 90°) to a surface at the point where a ray strikes it.
incident ray
The ray that travels towards a surface.
reflected ray
The ray that bounces off a surface.
angle of incidence (θi)
The angle between the incident ray and the normal.
angle of reflection (θr)
The angle between the reflected ray and the normal.
specular reflection
Reflection from a smooth surface; parallel rays stay parallel and an image can form.
diffuse reflection
Reflection from a rough surface; parallel rays scatter in many directions.
Why can you see yourself in a spoon but not in a sweater?
Light from a lamp hits your sweater and your spoon. Both reflect light, so you can see both. But only the shiny spoon shows you a picture of your face. The difference comes down to how each surface reflects, and that is controlled by two simple laws.
The ray model
In a uniform material, light travels in straight lines. (You can see this in the sharp edges of shadows, and in a beam of sunlight through dusty air.) So we can draw light as rays: straight lines with arrows that show the direction the light travels. A bundle of rays is a beam.
When a ray hits a surface, we draw a normal: a dashed line perpendicular (at 90°) to the surface at the point where the ray hits. All angles in optics are measured from the normal, never from the surface.
Reflection vocabulary
- Incident ray: the ray coming in towards the surface.
- Point of incidence: where the incident ray hits the surface.
- Reflected ray: the ray bouncing off the surface.
- Angle of incidence (θi): the angle between the incident ray and the normal.
- Angle of reflection (θr): the angle between the reflected ray and the normal.
- Plane mirror: a flat, smooth, shiny reflecting surface.
The laws of reflection
- The angle of incidence equals the angle of reflection: θi = θr.
- The incident ray, the reflected ray and the normal all lie in the same plane. (If you drew them on a flat sheet of paper, all three would fit on the sheet.)
Worked exampleExample 1: Watch out for the angle to the surface
A ray of light strikes a plane mirror. The angle between the ray and the mirror surface is 35°. Find the angle of incidence, the angle of reflection, and the angle between the incident and reflected rays.
- Angles are measured from the normal, which is at 90° to the surface.
- θi = 90° − 35° = 55°.
- By the law of reflection, θr = θi = 55°.
- The angle between the incident and reflected rays is θi + θr = 55° + 55° = 110°.
θi = 55°, θr = 55°, and the rays are 110° apart. A very common mistake is to answer 35°.
Specular and diffuse reflection
Every reflection obeys the laws of reflection. What changes is the surface.
- Specular (regular) reflection happens on a smooth surface such as a mirror, still water or polished metal. Parallel incident rays all meet the surface with the same normal, so they leave parallel. The reflected light keeps its pattern, so you see a clear image.
- Diffuse reflection happens on a rough surface such as paper, cloth or a painted wall. Under a microscope the surface is bumpy, so the normal points in a different direction at every spot. Parallel incident rays leave in many directions. You can see the object from anywhere in the room, but you see no image.
Diffuse reflection is why you can read this page from any angle. It also explains a driving hazard: a wet road at night becomes smooth, so headlight beams reflect specularly away from the driver and into oncoming drivers' eyes, and painted lines become hard to see.
Investigating the law of reflection
Here is how a Grade 10 class tests the law of reflection, with sample data.
Question: How does the angle of reflection compare with the angle of incidence for a plane mirror?
Hypothesis: If the angle of incidence increases, then the angle of reflection will increase by the same amount, because light reflects symmetrically about the normal.
Variables: independent: angle of incidence; dependent: angle of reflection; controlled: same mirror, same light source, mirror kept on the same line.
Method (summary): Draw a line for the mirror and a normal on paper. Draw incident lines at 15°, 30°, 45°, 60° and 75° from the normal. Stand the mirror on the mirror line, shine a narrow beam along each incident line, mark the reflected beam with two dots, then join the dots and measure θr with a protractor.
| θi (°) | 15 | 30 | 45 | 60 | 75 |
|---|
| θr measured (°) | 15 | 31 | 44 | 60 | 76 |
| Difference (°) | 0 | +1 | −1 | 0 | +1 |
Analysis and conclusion: In every trial, θr matched θi within ±1°. The small differences are random (some positive, some negative), so they are best explained by measurement uncertainty, not by a real difference. The evidence supports the hypothesis: θr = θi.
Two mirrors at 90°
When a ray reflects off two mirrors that meet at a right angle, it comes back parallel to the direction it came from, no matter what angle it came in at. This corner reflector idea is used in bicycle reflectors and road signs, and in three dimensions (three mirrors at right angles, like the inside corner of a box) it sends light straight back to its source.
Worked exampleExample 2: Following a ray through two mirrors
Two plane mirrors meet at 90°. A ray hits mirror 1 with an angle of incidence of 30°. Find the angle of incidence at mirror 2 and show that the outgoing ray is parallel to the incoming ray.
- At mirror 1: θr = θi = 30°. The reflected ray makes 90° − 30° = 60° with the surface of mirror 1.
- The reflected ray, mirror 1 and mirror 2 form a triangle. Its angles add to 180°: 60° + 90° + x = 180°, so the ray makes x = 30° with the surface of mirror 2.
- So the angle of incidence at mirror 2 is 90° − 30° = 60°, and θr = 60°.
- The ray is turned by 180° − 2(30°) = 120° at mirror 1 and by 180° − 2(60°) = 60° at mirror 2. Total turn: 120° + 60° = 180°.
θi at mirror 2 = 60°. A total turn of 180° means the outgoing ray travels in exactly the opposite direction, parallel to the incoming ray.
Drawing reflection diagrams
- Draw the normal first, as a dashed line at 90° to the surface.
- Measure angles from the normal with a protractor.
- Put arrows on rays to show direction.
- Label θi and θr, and give your diagram a title.
- For curved surfaces, the normal is still perpendicular to the surface at the point of incidence (you will use this in the curved-mirror lessons).
Just for funWhy is a mirror such a good friend?
It always reflects before it speaks, and it never takes your side... it just shows you both.
Hands-on activityComb-and-flashlight ray box
You need: a flashlight, a plastic comb (or a card with a narrow slit), a small flat mirror, modelling clay or a book to stand the mirror up, a protractor, a ruler, white paper and a pencil.
- Draw a straight line across the paper for the mirror. Draw a normal at 90° in the middle of it, and mark the point of incidence.
- Draw incident lines at 15°, 30°, 45°, 60° and 75° from the normal.
- Stand the mirror upright on the mirror line. In a dim room, shine the flashlight through one gap of the comb so a thin beam travels along the first incident line.
- Put two dots on the reflected beam, remove the mirror, join the dots to the point of incidence, and measure θr. Repeat for each angle.
- Record a table like the sample above. Calculate the difference θr − θi for each trial and write a conclusion that mentions measurement uncertainty.
- Rub a little petroleum jelly or soap on a second, old mirror (or use aluminium foil, then crumple and flatten it) and observe what happens to the beam: compare specular and diffuse reflection.
What to look for: θr within 1° to 2° of θi every time; the crumpled foil spreads the beam in many directions.
Safety (grown-up note): Handle mirrors carefully; tape any sharp edges. If a laser pointer is used instead of a flashlight, a grown-up must supervise: use only a low-power pointer, keep the beam on the table, and never point it at anyone's eyes or at an aircraft (that is illegal in Canada).
Your turn
1State the two laws of reflection.
- θi = θr. 2. The incident ray, reflected ray and normal lie in the same plane.
2What is a normal, and why do we need it?
A line drawn perpendicular (90°) to a surface at the point of incidence. All angles of incidence and reflection are measured from the normal.
3A ray hits a mirror with an angle of incidence of 25°. What is the angle of reflection?
4A ray makes an angle of 70° with a mirror's surface. Find θi and θr.
θi = 90° − 70° = 20°; θr = 20°.
5Explain why you can see a clear image in a calm lake but not on a lawn.
Calm water is smooth, giving specular reflection that keeps the rays in order and forms an image. Grass is rough, giving diffuse reflection that scatters rays in all directions.
6Does a rough surface obey the law of reflection? Explain.
Yes. Each ray still has θi = θr at its own point; the normals point in different directions because the surface is bumpy, so the reflected rays scatter.
More practice: warm-up
1What do we call the ray that bounces off a surface?
2A ray travels along the normal and hits a mirror. What is its angle of incidence?
0°. It reflects straight back along the normal.
3Which kind of reflection lets you read a book from any angle?
5From what line are angles of incidence measured?
More practice: on your own
1The angle between an incident ray and its reflected ray is 84°. Find the angle of incidence.
θi + θr = 84° and θi = θr, so θi = 42°.
2A student measures θr = 47° when θi = 45°. Give two possible sources of error.
For example: the beam is wide, so it is hard to mark its centre; the mirror was not exactly on the line (the reflecting surface is behind the glass); the protractor was misaligned or misread.
3Explain why wet roads at night are harder to see on than dry roads.
Water makes the road smooth, so headlight light undergoes specular reflection away from the driver, instead of diffuse reflection back towards the driver. The road and lines look dark, and oncoming drivers get glare.
4A ray hits a horizontal mirror at θi = 35° coming from the left. Describe (or sketch) the reflected ray.
It leaves to the right, at 35° from the normal (55° above the mirror surface), in the same plane as the incident ray and the normal.
5Why is the mirror's "line" in your diagram usually drawn at the back of the glass in a careful investigation?
In an ordinary mirror the reflecting silver layer is at the back of the glass. Lining up the back surface with the drawn line gives more accurate angles.
Challenge
1Two plane mirrors meet at 90°. A ray hits the first mirror with θi = 50°. Find θi at the second mirror and the total angle the ray is turned through.
After mirror 1, the ray makes 90° − 50° = 40° with mirror 1's surface. Triangle: 40° + 90° + x = 180°, so x = 50° with mirror 2's surface, and θi = 90° − 50° = 40° at mirror 2. Turn at mirror 1: 180° − 100° = 80°; at mirror 2: 180° − 80° = 100°. Total 180°: the ray returns parallel to its incoming path.
2A mirror is rotated by 10° while the incident ray stays fixed. Show that the reflected ray turns by 20°.
Rotating the mirror by 10° rotates its normal by 10°, so θi changes by 10°. Since θr = θi, θr also changes by 10° on the other side of the normal, which has itself moved 10°. Altogether the reflected ray turns by 10° + 10° = 20° (twice the mirror's rotation).
3Explain why a smooth white wall and a mirror can both reflect almost all the light that hits them, yet only the mirror forms an image.
Both reflect most of the light, but the white wall is rough on a tiny scale, so it reflects diffusely: rays from one point of an object leave in all directions and no longer come from a single image point. The mirror reflects specularly, so the rays from each object point stay organized and appear to come from one image point.
Quiz
1. The angle of incidence is measured between the incident ray and:
2. A ray strikes a mirror at an angle of incidence of 32°. The angle of reflection is:
3. A ray makes an angle of 20° with a mirror's surface. Its angle of reflection is:
4. Which surface produces diffuse reflection?
5. The second law of reflection says that the incident ray, reflected ray and normal:
6. Two mirrors meet at 90°. A ray reflects off both. The outgoing ray is:
Big idea: Expectation E2.2 asks students to investigate the laws of reflection through inquiry. The key skills are drawing the normal, measuring angles from the normal, and recognizing that small differences in data come from measurement uncertainty.
Common difficulties:
- Measuring from the surface. This is the most common error. If your teen gets 35° where the answer is 55°, ask: "Where is your normal?"
- Thinking diffuse reflection breaks the law. Every ray obeys θi = θr; rough surfaces just have normals pointing every which way.
- Sloppy diagrams. Encourage a sharp pencil, a ruler, a protractor, arrows on every ray and labels.
How to help: Do the comb-and-flashlight activity together; it takes about 20 minutes. If you use a laser pointer, supervise closely: low power only, never towards eyes, never out a window.