Module 9 · geometrical optics
For this whole module, light is a straight line that bends at boundaries. That is a lie, and it is a superbly useful one: it gets you mirrors, lenses, cameras, spectacles, rainbows and the eye, and it does not break down until Module 12. Almost everything here is what you can work out with a ruler — until Bench 4, where colour forces you to admit that light has a wavelength.
Light bounces off a smooth surface with the angle of reflection equal to the angle of incidence, both measured from the normal — the line perpendicular to the surface. That is the entire law, and everything about mirrors follows from it plus one habit of your brain: your eye assumes light travelled in a straight line to reach it. It has no way to know about the bounce, so it places the object along the straight-line direction the ray arrived from — which is behind the mirror, where nothing is.
You are 170 cm tall. What is the shortest mirror on which you can see yourself from head to toe?
And does it depend on how far back you stand? Think before you press.
Two things worth noticing about the image itself. It is virtual — no light actually arrives at the place you seem to see yourself, so you could not catch it on a screen held there. And it is exactly as far behind the mirror as you are in front, which is why the room behind you in a mirror looks twice as deep as it is.
Everyone says so, and it is wrong — or at least badly described. A mirror does not swap left and right; it swaps front and back. Point at the mirror: your image points back at you, along the same line. Your right hand stays on the same side of the room. The only direction reversed is the one running straight into the glass — so your image is you with front and back swapped, and nothing else. We misread that as a left-right swap for one reason: to compare yourself with your image you imagine turning yourself around to face the same way, and turning around does swap your own left and right. So we blame the mirror for a swap we performed ourselves — and left and right are the only pair we could confuse this way, because our two sides look so nearly alike. Which is why the writing on an ambulance bonnet is drawn with the letters mirrored, so that a driver looking in the rear-view mirror reads it the right way round.
Light goes slower in glass or water than in vacuum — about 200,000 km/s in glass instead of 300,000. That single fact is what the refractive index records: n = c/v, where c is light's speed in vacuum and v its speed inside the material. So glass is about 1.5, water 1.33, diamond 2.42. And that slowing is why a ray arriving at an angle bends.
Going from glass into air, you raise the angle of incidence past about 42°. What happens to the ray?
The critical angle is where sin θ₂ would have to equal 1: sin θc = n₂/n₁. For glass to air it is 42°, for water to air 49°, and for diamond a mere 24° — which is most of why a diamond sparkles; the rest is dispersion, coming in Bench 4. Light entering a well-cut diamond bounces around inside, unable to escape from most faces, until it finds one of the few directions it can leave from, and it leaves there brilliantly.
| Where you have seen it | What is happening |
|---|---|
| A straw in a glass of water looks snapped | rays from the submerged part bend away from the normal on leaving the water; your brain, still assuming straight lines, places that part higher and to one side. |
| A swimming pool looks shallower than it is | same reason, applied to the bottom. It is about three-quarters of its true depth — the ratio 1/1.33. |
| Optical fibre | light fired down a glass thread hits the wall at a shallow angle every time, exceeds the critical angle, and cannot get out. It bounces its way across an ocean with almost no loss. Almost every message you have ever sent has travelled part of its journey like this. |
| A road mirage on a hot day | the air just above the hot road surface is warmer, and warm air is thinner, so its n is very slightly smaller. Picture that air as a stack of thin layers, each with a slightly smaller n than the one above it. The ray refracts a little at every layer boundary, always away from the normal, and hundreds of small bends add up to one smooth curve. Light from the sky is curved up into your eye, so you see sky where the road should be. Not water reflecting: refraction through air whose n changes gradually. |
| The Sun is already below the horizon when you watch it set | the atmosphere bends its light around the curve of the Earth by about half a degree — roughly the Sun's own width. Every sunset you have ever seen was over before you saw it. |
Put a coin in an empty opaque mug and back away until the rim just hides it. Hold still and have someone slowly pour water in. The coin rises into view — the rays from it now bend as they leave the water, and reach your eye over the rim. Then look up through the water surface from underneath, in a pool or a big bowl: everything above the water is squashed into one bright circle overhead, about 97° wide — which is twice the 49° critical angle, once on each side. Outside that circle you cannot see out at all: those directions lie past the critical angle, so the underside of the surface acts as a mirror and shows you the pool floor instead. Fish live inside that circle. It is called Snell's window.
A lens is a piece of glass with curved surfaces, shaped so that the rays leaving one point are each bent by just the right amount to pass through one other point. There are two kinds. In a converging lens the middle is thicker than the edge. Near the edge the surface is steeply tilted, so a ray arriving there meets that surface far from its normal and is bent a lot; near the middle the surface is almost flat to the ray and it is barely bent at all. Rays that were spreading apart are therefore brought together. In a diverging lens the middle is thinner than the edge and the same reasoning runs the other way: rays that were parallel are made to spread apart, as though they had come from a point. Bench 3 lets you switch between the two.
Two words first. The axis is the straight line through the centre of the lens at right angles to it — the line you would look along. The focal point is the place where the lens collects all the rays that arrive parallel to that axis: point the lens at something very far away and its light gathers at this one spot. There is one focal point on each side, the same distance out, and that distance is the focal length, f. A more strongly curved lens bends light more, so its f is shorter.
Now the rules — two let you find any image with a ruler, and a third is a useful check.
Where those rays meet is where the image is. Where you put the object decides which kind of image you get. Put it further from the lens than the focal point and the rays are brought back together on the far side: the image is real — you can catch it on a screen, and it is upside down. Put it closer than the focal point and the rays are still spreading when they leave the lens, so they never cross and no real image forms anywhere; your eye traces them back and sees a virtual, upright, enlarged image on the object's own side. That is a magnifying glass.
The eye is a lens with a fixed image distance — the retina is where it is and cannot move. So to focus on things at different distances, the eye changes its focal length instead, by squeezing the lens fatter with a ring of muscle. That is called accommodation, and it is why reading for hours is tiring: you are holding a muscle contracted.
In 1666 Newton let a beam of sunlight through a prism and spread it into colours. That much was known. What he did next was the actual experiment: he passed one single colour through a second prism and showed that it did not split further, and then recombined the whole spread back into white. The colours were not being manufactured by the glass. They were in the sunlight all along, and the prism merely separated them.
It works because n is not a single number — it depends a little on the colour of the light. Colour, for light, means wavelength: the distance from one crest of the wave to the next (this is the one idea from Module 7 you will need). Violet has the shortest wavelength of the visible colours, about 400 nanometres; red has the longest, about 700. From here on, λ (lambda) is the symbol for wavelength. Violet light travels a touch slower in glass than red does, so it bends a touch more, and after the prism's two surfaces the gap is wide enough to see. Splitting light by colour this way is called dispersion — the word the bench's second slider uses.
A raindrop is a sphere. Sunlight entering one bends on the way in, bounces off the inside of the back surface, and bends again on the way out. That bounce is not total internal reflection: the ray meets the back well inside water's 49° critical angle, so most of the light passes straight out and is lost, and only a small share reflects. That is why a rainbow is faint and needs strong sun behind you. Now trace that path for rays entering the drop at every possible height. They do not come out evenly spread: they crowd together, and almost all of them leave within a degree or two of the same angle. Stand with the Sun behind you, look 42° away from the direction of your own shadow, and that crowded light is what arrives. Red and violet crowd at angles about 2° apart, because their refractive indices are slightly different — so every drop sends red to your eye from slightly higher in the sky than violet, and the arc has red on the outside.
Air molecules are far smaller than λ, the wavelength of the light, and such small scatterers throw light sideways with an efficiency that goes as 1/λ⁴ — an extremely steep dependence. Blue light, with roughly two-thirds the wavelength of red, is scattered about five times as strongly. So blue sunlight is kicked out of the direct beam and bounced around the sky, and when you look away from the Sun, that scattered blue is what reaches you.
At sunset the light travels a much longer slanting path through the atmosphere. By the time it arrives, nearly all the blue has been scattered away somewhere else, and what is left in the direct beam is red and orange. The blue missing from a sunset is blue that has been thrown sideways out of the beam — and sideways-thrown blue is exactly what makes the daytime sky. One process, seen from two directions.
Two reasons, and both are needed. The Sun emits less violet than blue to begin with, and — the bigger effect — your eye's three colour receptors respond weakly to violet and strongly to blue. The sky genuinely does scatter violet more than blue; you simply are not equipped to notice. A camera sensor with a different response curve photographs a slightly different sky, which is worth remembering the next time a photograph disagrees with your memory of a colour.
1. You stand 2 m from a plane mirror. How far away does your image appear to be from you?
2. A ray of light passes from air into a glass block along the normal — dead perpendicular to the surface. Inside the glass it:
3. You use a converging lens as a magnifying glass, holding it close to a stamp. The image you see is: