Light · Model
Reflection: mirrors and scattering
A mirror and a sheet of paper send back about the same amount of light and obey exactly the same rule. Only one of them shows you your face, and the reason is the surface rather than the rule.
Start here
Two white surfaces. One shows your face.
A mirror and a sheet of white paper both send back almost all the light that falls on them. Hold either up to a window and the room brightens. Only one of them shows you your own face.
What is different about what the light does at the two surfaces?
Both surfaces send back most of the light, and both obey the law of reflection at every single point. The difference is the surface itself. A mirror is smooth, so parallel rays stay parallel and the arrangement of the light survives — and the arrangement is the image. Paper is rough, so each ray leaves at its own angle and the arrangement is gone.
Every ray that meets a surface obeys the same rule. Draw a line at right angles to the surface where the ray lands — that line is called the normal — and measure both angles from it. Then the angle of reflection equals the angle of incidence, always, on every surface, however rough. A ray is always drawn with an arrow on it, because which way the light is going is part of the answer.
What changes between a mirror and a sheet of paper is not the rule but the surface. A mirror is smooth on the scale of light, so the normal points the same way everywhere and a set of parallel rays comes off still parallel: the pattern survives, and the pattern is the image. That is specular reflection. Paper is rough on that scale, so every tiny facet has its own normal pointing a different way. Each ray still reflects correctly, and the set of them leaves in all directions with the pattern destroyed. That is diffuse scattering, and it is why you can see the paper from anywhere in the room and cannot see yourself in it.
Some of the light is neither reflected nor scattered but absorbed, its energy taken up by the surface. A matt black card is rough like paper and absorbs most of what lands on it, so very little leaves in any direction at all.
Stand a mirror up and look at yourself. The image is the same size as you and the same way up, and it sits as far behind the glass as you are in front of it. Nothing is there: no light comes from behind the mirror. Your eye follows the reflected rays back along the straight lines they arrived on, and the image is where those lines meet. That is what virtual means. The one axis a plane mirror does reverse is the one running towards it and away from it — near and far. Writing looks backwards because you had to turn the page round to face the glass, and turning it is what swapped its left and right.
At the bench · a ray box, a protractor and four surfaces
One rule. Four different results.
Change a control to begin
A single narrow ray from a ray box lands on a surface, with the normal drawn in at the point where it lands. Set the angle it comes in at, and set what it lands on.
Commit first. A ray hits a mirror at 30° to the normal. The mirror is swapped for a sheet of white paper and the ray comes in at exactly the same 30°. What happens to the angle of reflection of that one ray?
—
What it lands on
Angle of incidence
—
measured from the normal
Angle of reflection
—
measured from the normal
Roughly how much leaves again
—
the rest is absorbed
How the rays leave
—
Writing it down · the shape of this relationship
Angle of reflection = angle of incidence
Both measured from the normal
Two sides that always balance. Nothing is being added up here and nothing is being shared out, so there is nothing to cover: whatever one side reads, the other reads too.
i · angle of incidence, from the normal · °
r · angle of reflection, from the normal · °
Worked example · one step at a time
A ray strikes a plane mirror at 20° to the mirror surface. What is the angle of reflection?
Step 0 of 5
Convert
20° stays 20° · 90° stays 90°
Both angles are in degrees and the rule compares them directly, so there is nothing to convert.
Formula
r = i, both measured from the normal
The rule is an equality, so the two angles are always the same number.
Insert
i = 90° − 20°
The 20° given is to the mirror surface, and the normal is at right angles to that surface.
Fine-tune
90 − 20 = 70, so i = 70°
Degrees taken from degrees leave degrees.
Answer
r = 70°
Seventy degrees from the normal, which is 20° from the mirror on the other side.
Your turn · the same five steps
Your ray arrives at 40° from the normal.
Write each line out yourself — starting by deciding whether anything needs converting. Then check your working and tick the lines you had.
The five lines · tick what you had
Convert
the angle is already in degrees
The protractor reads in degrees and the rule compares degrees with degrees, so there is nothing to convert.
Formula
r = i, both from the normal
An equality: whatever the incoming angle is, the reflected one matches it.
Insert
i = 40°
Read from the protractor on the bench, measured from the dashed normal.
Fine-tune
r = 40°, so the ray leaves at 50° to the surface
The normal is at right angles to the surface, so the two angles add to 90°.
Answer
r = 40°
On the far side of the normal from the incoming ray, and the same on any of the four surfaces.
The five lines above give 40°, which is the angle the bench above is drawing.
Key fact
The angle of reflection equals the angle of incidence, both measured from the normal — the line at right angles to the surface. On a smooth surface parallel rays stay parallel and the pattern survives as an image, which is specular reflection. On a rough surface every facet has its own normal, so the rays leave in all directions and the pattern is lost, which is diffuse scattering. Some light is absorbed at every surface.
Think again
“Rough surfaces break the law of reflection.”
Not one ray disobeys it. Zoom in far enough on a sheet of paper and it is a landscape of fibres, each facet flat and each with its own normal pointing wherever that facet happens to face. A ray landing on one of them reflects at exactly the angle it arrived at, measured from that facet’s normal — and because the facets point every which way, the rays that started off parallel finish scattered. The law holds perfectly. What is lost is the arrangement, and the arrangement was the image.
“Angles in reflection are measured from the mirror.”
They are measured from the normal, the line drawn at right angles to the surface, and it is a convention worth being fussy about because it is the one that keeps working. A ray at 20° to the mirror is at 70° to the normal, and if you quote 20° as the angle of incidence every later answer is wrong by the same amount. Measuring from the normal also survives being taken to a curved mirror, where there is no single surface to measure from, and to refraction in the next lesson, where the two materials meet at one point.
Mastery ladder
Not started yet.
Rungs 3 and 4 you mark yourself.
Rung 1 · Calculate
A ray strikes a plane mirror at 25° to the mirror surface. What is the angle of reflection?
Rung 2 · The one that catches people
You can see a sheet of white paper from anywhere in the room but cannot see your face in it. Which statement is right?
Rung 3 · Explain
Explain why a mirror shows an image and a sheet of white paper does not, using the words normal, parallel and scattering.
Rung 4 · Take it somewhere new
A wet road at night is dangerous to drive on partly because it reflects headlights very differently from a dry one. Explain what changes when the road is wet, and why oncoming drivers are dazzled by long streaks of light.
Key note
The angle of reflection equals the angle of incidence, both measured from the normal — the line at right angles to the surface at the point where the ray lands. A smooth surface keeps parallel rays parallel, so the arrangement of the light survives and an image forms: specular reflection. A rough surface gives every facet its own normal, so the rays leave in every direction and the arrangement is lost: diffuse scattering. At every surface some light is absorbed instead.
Going further
Almost everything you can see is being seen by diffuse scattering. Only a handful of objects — mirrors, still water, polished metal, glass at a glancing angle — reflect specularly, and those are precisely the ones that show you something other than themselves. A room lit by a single lamp is visible in every corner because every rough surface in it is scattering light in all directions at once, which is also why the shadows are soft.
Smooth is a comparison with the wavelength of light, not with your finger. Visible light has a wavelength of a few ten-thousandths of a millimetre, so a surface has to be flat to well within that to act as a mirror. Radio waves have wavelengths measured in metres, and a wire mesh with centimetre holes is a perfect mirror to them — which is why a satellite dish can be a grid rather than a solid sheet, and why the door of a microwave oven has a metal grid you can see straight through.
Before this lesson
Connects to
At GCSE this becomes
- Ray diagrams for plane and curved mirrors, virtual images, specular and diffuse reflection at a boundary, and the relationship between surface roughness and wavelength.
Where to next
Ask Mr Badmus AI
Got an angle and a surface, and want to know where the ray goes?
The bench is a teaching model. Angles are drawn to scale from the normal and the law of reflection is applied exactly. The percentages of light leaving again are round teaching figures for a typical surface of each kind and depend on the colour of the light and the angle it arrives at. The scattered fans are drawn as five rays spread over a fixed angle so the spread can be seen; a real rough surface sends light out over the whole half-space above it, and the number and spacing of the drawn rays carry no information. The surface profiles are drawn far rougher than any of these materials really are, since paper is rough only on the scale of the wavelength of light.
Lesson content © MrBadmusAI.