Light — Reflection and Refraction

Why This Matters

Look into the back of a steel spoon. You look upright, but tiny. Dip a pencil into a glass of water. It looks broken at the water’s surface. A coin at the bottom of a bucket looks closer to the top than it really is. Your car’s side mirror even says “objects are closer than they appear.”

All of these happen because light does just two things. It bounces off surfaces. This is called reflection. And it bends when it moves from one material into another. This is called refraction. The best part is that both follow exact rules. The rules are so exact that you can calculate where an image will form, how big it will be, and whether it is upright or upside-down. You only need two simple formulas.

This chapter is the science behind torches, shaving mirrors, magnifying glasses, cameras, telescopes, and the spectacles that millions of people wear. Once you get the sign rules right, the rest is just careful arithmetic.

The Big Idea

Before the new ideas, here is the one fact from earlier classes that everything in this chapter is built on.

Light travels in straight lines until it hits a surface. At a mirror it reflects — it bounces back, with the angle of incidence equal to the angle of reflection. When it passes into a new material it refracts — it bends, because its speed changes. Curved mirrors and lenses use these two rules to bring rays together or spread them apart. This forms images, and we find where those images are using the mirror formula and the lens formula.

Two ideas run through the whole chapter:

1. Reflection bends light back. Refraction bends light as it passes through. Refraction happens only because light moves at different speeds in different materials.
2. To do the maths without getting confused about “is this plus or minus?”, we use one strict set of rules — the New Cartesian Sign Convention. Learn it once, and every numerical becomes a simple, step-by-step job.

Let’s Break It Down

Reflection and the two laws

The rules of reflection lean on two words you have met before — “perpendicular” and “the normal”. Let’s get them crystal clear first.

When light hits a shiny, polished surface it bounces off. This bouncing follows two laws of reflection:

1. The angle of incidence equals the angle of reflection. (The angle of incidence is the angle of the incoming ray. The angle of reflection is the angle of the bounced-off ray. We measure both from the normal — the line drawn straight out from the surface, at 90°, at the exact spot where the ray hits.)
2. The incoming ray, the bounced-off ray, and the normal all lie flat on the same plane (the same flat sheet).

But why should the two angles come out exactly equal? There is a lovely reason: light always takes the quickest path. Say light has to leave a point A, touch the mirror, and reach your eye at B. Of all the bent paths from A to the mirror to B, the shortest one is the only one light actually takes. Here is the trick to find it. Mark a point B′ the same distance behind the mirror as B is in front of it (its mirror image). The straight line from A to B′ is the shortest possible distance — and it crosses the mirror at exactly one point. The real path A → that point → B is the same length as A → B′, because B and B′ are mirror-images. And at that crossing point, simple geometry of the two mirror-image triangles forces the incoming angle to equal the outgoing angle. So “shortest path” and “i = r” are the same rule, just described two ways.

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A plane mirror is just a flat mirror. The image you see in it is:

• virtual (you cannot catch it on a screen — more on this soon),
• erect (the right way up),
• the same size as you,
• as far behind the mirror as you are in front of it, and
• laterally inverted — left and right are swapped.

Lateral inversion is why “AMBULANCE” is written back-to-front on the front of an ambulance. When the driver ahead sees it in their mirror, the letters flip and read correctly.

Spherical mirrors — the vocabulary

A spherical mirror is a small piece cut out of a shiny ball (a sphere). There are two kinds:

• Concave mirror — the shiny surface curves inward, like the inside of a spoon. It brings light rays together. We say it converges light.
• Convex mirror — the shiny surface curves outward, like the back of a spoon. It spreads light rays apart. We say it diverges light.

Figure 9.2 below puts both mirrors side by side and labels the five key points you’ll use in every mirror problem.

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Learn these five terms — every problem uses them:

• Pole (P): the centre point of the mirror’s surface.
• Centre of curvature (C): the centre of the ball (sphere) that the mirror was cut from.
• Radius of curvature (R): the distance from P to C.
• Principal focus (F): the point where rays that come in parallel to the axis meet after reflecting (concave mirror), or the point they seem to come from (convex mirror).
• Focal length (f): the distance from P to F.

The one relationship you must remember is f = R/2. The focus sits exactly halfway between the pole and the centre of curvature.

This is not a random rule — it falls straight out of the law of reflection, and you can see why with one ray. Take a ray coming in parallel to the axis and hitting the mirror at a point M. The normal there is the radius MC (it points to the centre of curvature C). The incoming ray and the radius make some angle θ; by i = r the reflected ray leaves at the same angle θ on the other side of the normal. Now look at the triangle MCF, where F is the focus. Its angle at C is θ (between the radius and the axis), and its angle at F is also θ (the incoming ray was parallel to the axis, so these are equal “alternate” angles). Two equal angles means the triangle is isosceles, so the two sides facing them are equal: FM = FC. For a ray close to the axis, M sits almost above the pole P, so FM is practically the same as FP. That gives FP = FC — meaning F is the exact midpoint of P and C. Half of R is f. That is the whole reason.

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Why a curved mirror reflects the way it does

Here is a question that confuses many students. How do you know which way a ray bounces off a curved mirror? The good news is simple. A curved mirror follows the exact same law as a flat one — angle of incidence = angle of reflection. You just need to find the normal at the spot where the ray hits.

For a spherical mirror, the normal is very easy to find. It is the line drawn from the hit point straight back to the centre of curvature C (this line is a radius of the sphere). A radius always meets a sphere at a right angle (90°). So that radius is the normal at that point. Once you have the normal, you reflect the incoming ray so that i = r (angle in equals angle out). That fixes the direction of the bounced ray. No guessing needed.

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Try this for several parallel rays. The further a ray is from the axis, the more its normal tilts. So each ray reflects a little more steeply, and they all cross at the same point — F. That is why a concave mirror brings light together. A convex mirror is the opposite. Its centre of curvature is behind the surface, so the normals point outward and the reflected rays spread apart. They look like they come from F behind the mirror. So whenever you are not sure which way a ray goes off a curved mirror, follow this rule: first draw the normal towards C, then apply i = r.

Real and virtual images — what’s the difference?

The words real and virtual appear in every row of the table below. So let’s be clear about what they mean. After a mirror or lens bends light, the rays do one of two things. They either actually meet at a point, or they only look like they come from a point.

• A real image forms where the bounced (or bent) rays actually cross each other. Real light truly reaches that spot. So you can catch a real image on a screen (like a paper screen or a cinema screen). From a single mirror or lens, a real image always comes out upside-down (inverted).
• A virtual image forms where the rays do not meet. The rays spread apart, and only their backward extensions (drawn as dashed lines in the diagrams) seem to meet — behind the mirror, or on the same side as the object for a lens. No real light reaches that spot. So you cannot catch a virtual image on a screen. You can only see it by looking into the mirror or lens. A virtual image is always the right way up (erect).

Here is the one-line test. Hold a paper screen where you think the image is. If a sharp picture lands on it, the image is real. If nothing lands on it, the image is virtual.

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Real images around you:

• The picture a cinema projector throws on the screen. (The film inside is loaded upside-down on purpose. That is because the real image it makes is inverted, so it lands the right way up on the screen.)
• The image your camera forms on its sensor, and the image your eye forms on your retina. Both are real and inverted. (Your brain quietly flips it the right way up, so you never notice.)

Virtual images around you:

• Your reflection in a flat bathroom mirror. It looks like “you” standing the same distance behind the glass. But there is only a wall back there. No light reaches that point, and you could never catch it on a screen. So it is virtual (and upright).
• The enlarged word you see under a magnifying glass, and the wide view in a convex rear-view mirror or shop mirror. Both are virtual and upright.

Here is a link to the maths coming up. A negative magnification (m) means a real, inverted image. A positive m means a virtual, erect image. That one sign tells you the whole story.

Images in a concave mirror

A concave mirror is the interesting one. Its image changes completely depending on where you place the object. Here is the full picture:

Object position	Image position	Size	Nature
At infinity	At F	Point-sized	Real, inverted
Beyond C	Between F and C	Diminished	Real, inverted
At C	At C	Same size	Real, inverted
Between C and F	Beyond C	Enlarged	Real, inverted
At F	At infinity	Highly enlarged	Real, inverted
Between P and F	Behind the mirror	Enlarged	Virtual, erect

Notice the pattern. As the object moves closer to the mirror, the image moves further away and gets bigger. This continues until the object crosses the focus. After that, the image flips to virtual, erect, and enlarged. (This is the shaving-mirror or make-up-mirror position.)

To find the image of a nearby object, you only need to draw two rays from the top of the object:

1. One ray drawn parallel to the axis, which reflects back through F.
2. One ray drawn to the pole P, which reflects back at the same angle on the other side of the axis.

The point where these two reflected rays cross is the top of the image.

One case is special and does not follow this “two rays from the top” method. That is a very distant object. Its light reaches the mirror as a parallel beam, instead of as rays spreading out from a nearby top (see Figure 9.6 below).

Object at infinity → a point image at F

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Why parallel rays from an object at infinity? Every point on an object sends out light in all directions. But the further away the object is, the less those rays have spread out by the time they reach the mirror. From something very far away — the Sun, a distant hilltop, a star — the rays from one point have travelled so far that they arrive almost perfectly parallel. (This is the same reason we treat sunlight as parallel beams.) So for an “object at infinity”, we do not draw two rays from a visible top. Instead we draw a bundle of parallel rays, and the concave mirror brings them all together at one point at F.

Real-life use: this is how a reflecting telescope and a solar cooker / solar furnace work. A large concave mirror gathers the almost-parallel rays from a faraway source and concentrates them at the focus.

Object beyond C → image between F and C

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Real-life use: this is the small, upside-down version of yourself you see when you stand well back from a concave shaving mirror. The same kind of small, real image is what a reflecting telescope’s main mirror forms of a far-off scene, before the eyepiece magnifies it.

Object at C → image at C (same size)

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Real-life use: since the object and its image sit together at C, this is a neat trick used in the lab to measure a concave mirror’s radius of curvature. Slide a screen until the sharp image lands right next to the object. That distance is R (and so f = R/2).

Object between C and F → image beyond C

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Real-life use: a real and magnified image is useful in a solar concentrator (the dish in a solar cooker forms a large, intense, hot image of the Sun) and for the big bright spot a floodlight reflector throws onto a distant surface.

Object at F → image at infinity

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So is there an image, or not? When the reflected rays come out exactly parallel, they never actually cross. They would only “meet” infinitely far away. That is all “image at infinity” means: no real image forms that you could catch on a screen at any normal distance. This is the in-between case. It sits exactly between a real image (object beyond F) and a virtual image (object inside F). The useful part comes from doing the reverse trip: a bright bulb placed at F sends light out as that parallel beam. And if you look into such a mirror, your relaxed eye focuses the parallel rays onto your retina, so the source looks like it is infinitely far away.

Real-life use: run this case backwards. Put a bulb exactly at the focus, and the mirror sends every ray out as one strong parallel beam that barely spreads out with distance. That is exactly how torches, car headlights, searchlights, and lighthouse reflectors throw a beam that reaches far away.

Object between P and F → image behind the mirror

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Real-life use: you want a right-way-up and bigger view of your own face from close up. So this is the shaving / make-up mirror position. The same upright-and-magnified trick lets a dentist’s mirror show an enlarged view of a tooth.

Images in a convex mirror

A convex mirror is simpler. No matter where you put the object, the image is always virtual, erect, and diminished (smaller). It sits between P and F, behind the mirror. This wide, shrunk-down view is exactly why a convex mirror is used as a vehicle’s rear-view mirror. It shows a large area, even though everything looks smaller (which is why things seem “closer than they appear”).

Object at infinity → a point image at F (behind)

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Object at a finite distance → image between P and F (behind)

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Real-life use: because the image is always small and upright, a convex mirror fits a very wide area into a small piece of glass. This makes it perfect for vehicle rear-view / side mirrors (you can see several lanes at once), shop anti-theft mirrors, and mirrors placed at blind corners on roads and in corridors. The trade-off is that everything looks smaller, and so it looks further away. That is why the side mirror warns “objects are closer than they appear.”

Now that you’ve seen both mirrors in action, here’s a quick check to see if you can pick the right one for the job.

The New Cartesian Sign Convention

Before doing any calculation, set up the rules for signs. Put the pole at the origin (the zero point), and lay the principal axis along the x-axis:

• The object always sits on the left. So light travels from left to right.
• Distances measured against the incoming light (to the left) are negative. Distances measured along the light (to the right) are positive.
• Heights above the axis are positive. Heights below the axis are negative.

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Here are the results you will use again and again. The object distance u is always negative (the object is on the left). A concave mirror’s focal length is negative (its F is in front). A convex mirror’s focal length is positive (its F is behind). For lenses: a convex lens has positive f, and a concave lens has negative f.

Mirror formula and magnification

The mirror formula links the three distances:

1/v + 1/u = 1/f

Here u = object distance, v = image distance, and f = focal length. Magnification tells you how big the image is compared to the object:

m = h′/h = −v/u

Here h is the object’s height and h′ is the image’s height. A negative m means the image is real and inverted. A positive m means the image is virtual and erect.

Let’s put both formulas to work on a typical numerical, watching how the signs do all the heavy lifting.

Refraction — why light bends

Now let’s look at the bending. When light passes from one see-through material into another at an angle, it changes direction. Why? Because its speed changes. Light is fastest in a vacuum (3 × 10⁸ m/s, which is 300 million metres per second). It slows down inside glass or water.

A rarer medium is one where light travels faster (like air). A denser medium is one where light travels slower (like glass).

• Going from a rarer medium into a denser one (air → glass), light slows down and bends towards the normal.
• Going from a denser medium into a rarer one (glass → air), light speeds up and bends away from the normal.

But why does a change in speed make light turn?

Slowing down by itself does not make something turn. A car braking in a straight line just goes slower; it does not move sideways. Light bends only when it crosses the boundary at an angle. The reason is surprisingly simple, once you stop picturing a single thin ray. Instead, picture a wide front moving forward together — like a row of soldiers marching in step, shoulder to shoulder.

Now march that row at an angle, off a hard road and onto soft mud. (Here the mud is the slower, denser medium.) They do not all reach the mud at the same moment. The soldiers at one end reach the mud first and slow down, while the others are still marching fast on the road. One end is now dragging, and the other end is racing ahead. So the whole row swings round — it changes direction, turning towards the mud. Later, when they step back onto firm road, the first ones to reach it speed up again, and the row swings back the other way.

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Light’s wavefront does exactly this. Think of the front as the line of soldiers. The edge that enters the denser medium first slows down first, so the front pivots. It turns towards the normal when going into a slower medium, and away from the normal when coming back out. The fronts even crowd closer together in the slower medium (this is a shorter wavelength), just like the bunched-up soldiers.

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It even explains a special case. If light hits the surface head-on (straight along the normal), the whole front slows down at the same moment. Nothing drags behind, so the ray goes straight through with no bend at all. This is exactly what you see when you look straight down into a pond.

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This is exactly why a pencil in water looks bent, why a pond looks shallower than it really is, and why a coin seems to “rise” when you pour water over it. Light from underwater bends as it leaves the surface. So the object appears to be in a slightly shifted place.

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Snell’s law and refractive index

How much the light bends follows the laws of refraction:

1. The incoming ray, the bent (refracted) ray, and the normal all lie on the same plane.
2. Snell’s law: for a given pair of materials, sin i / sin r is always the same fixed number. (Here i is the angle of incidence and r is the angle of refraction.) That fixed number is called the refractive index (n).

The refractive index compares the speed of light in two places. The (absolute) refractive index of a material is:

n = (speed of light in vacuum) / (speed of light in the material) = c/v

So water’s n = 1.33 means light travels 1.33 times faster in a vacuum than in water. A higher refractive index means the material is optically denser. That means light slows down more in it, and so it bends more.

Why a diamond’s huge n = 2.42 makes it sparkle

Diamond’s refractive index is 2.42. This is one of the highest of any everyday material, and that single number is why a diamond flashes in a way that plain glass never can. Two things work together:

• Light gets trapped inside it. When light inside a dense material tries to escape out into the air, there is a limit. Past a certain steep angle — called the critical angle — the light cannot get out at all. Instead it reflects completely back inside. (This is called total internal reflection. You will study it properly in Chapter 10.) The bigger the refractive index, the smaller this critical angle. For diamond it is only about 24°, compared to about 42° for glass. So almost any ray that enters a diamond hits a back face too steeply to escape. It bounces around inside, and finally comes back out the top, towards your eye. A diamond is cut at carefully chosen angles, exactly so that this trapped light is sent straight back up. That bright, white sparkle is called the diamond’s brilliance.
• It splits white light into colours. A high refractive index also bends different colours by different amounts. (Violet slows down and bends the most; red bends the least.) So each time light bends through the diamond, white light spreads out into a tiny rainbow. These coloured flashes are what jewellers call a diamond’s fire.

Glass (n is about 1.5) does both of these much more weakly. Its critical angle is larger, so more light just leaks out instead of bouncing back. That is why cut glass sparkles a little, but never as much as a real diamond.

Try a quick one to tie together bending direction and the speed formula n = c/v.

Lenses — converging and diverging

A lens is a piece of see-through material with at least one curved surface. There are two types:

• Convex (converging) lens — thicker in the middle. It brings parallel rays together at the focus. (A magnifying glass is a convex lens.)
• Concave (diverging) lens — thinner in the middle. It spreads parallel rays apart, so they appear to come from the focus.

A convex lens has two foci (F₁ and F₂, one on each side). It also has a point at its centre called the optical centre (O). A ray that passes through O goes straight on without bending.

Why does the ray through O escape bending, when every other ray gets bent? Zoom right in on the middle of the lens. The front and back surfaces are curved overall, but at the very centre they are almost flat and parallel to each other — just like the two faces of a thin glass slab. And you already know what a parallel-sided slab does (see Figure 9.18 earlier): the ray bends towards the normal going in, then bends away from the normal by the same amount coming out. So it leaves parallel to how it came in — only shifted sideways a little. Because the lens is so thin, that sideways shift is almost zero. So the ray through O looks like it sails straight through without bending at all. That is the only ray you can trust to stay perfectly straight.

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Why a lens bends light the way it does

A lens works by refraction, and it uses the same rule you saw with the glass slab. Entering the glass, light slows down and bends towards the normal. Leaving the glass, light speeds up and bends away from the normal. The normal at any point on a lens surface is just the line drawn at 90° to the surface there. On a curved surface, this normal tilts from point to point. That is what lets the lens bend different rays by different amounts.

The easiest way to see the result is to picture the lens as a stack of prisms. The top half of a convex lens is shaped like a prism with its thick base towards the axis. The bottom half is like a prism with its thick base towards the axis from below. A prism always bends light towards its thicker base. So the top half turns rays downward and the bottom half turns them upward — both towards the axis — so all the rays meet at F. A concave lens is the opposite. Its prisms have their bases pointing outward, so it spreads light apart.

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So, just like with the mirror, you never have to guess. Find the normal to the surface, apply the bending rule (towards the normal going in, away from it coming out), and you will see that a convex lens always nudges rays towards the axis.

Images in a convex lens

Draw two rays from the top of the object:

1. One ray parallel to the axis, which bends to pass through F₂.
2. One ray through the optical centre O, which goes straight on.

The point where these two rays cross is the top of the image.

Object at infinity → a point image at F₂

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Real-life use: the objective lens of a refracting telescope (and of binoculars) collects the almost-parallel light from a distant star, planet, or hill. It brings that light to a sharp real image at its focus, which the eyepiece then magnifies. The same focusing of parallel rays lets a burning glass or a solar concentrator gather the Sun’s rays onto one tiny spot — hot enough to burn paper or boil water. Because the source is so far away, the image is basically just a point. That is the price of squeezing all that light into one place.

Object beyond 2F₁ → image between F₂ and 2F₂

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Real-life use: the camera — and the lens in your phone and even in your own eye (Chapter 10). The scene is far bigger than the sensor, the film, or the retina. So you need a real image (one that actually lands on the sensor) that is shrunk to fit. The exact image distance changes a little as the subject moves nearer or further away. So the camera re-focuses by sliding its lens. Your eye does the same job, but by changing the shape of its lens.

Object at 2F₁ → image at 2F₂ (same size)

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Real-life use: a photocopier or scanner set to 1:1, where the copy must come out exactly the same size as the original. This special position is also the standard lab setup for measuring a convex lens’s focal length. Slide the object and the screen until the image is real, inverted, and the same size. The object distance you then measure is 2f, so f is half of that.

Object between F₁ and 2F₁ → image beyond 2F₂

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Real-life use: the slide / film projector, the cinema projector, the overhead projector, and a photographic enlarger. All of these need a real image (so it can be caught on a distant screen or a sheet of paper) that is bigger than the small original. Because the image is inverted, slides and film are loaded upside-down on purpose, so the picture lands the right way up on the screen. The closer the slide moves towards F₁, the larger and further away the projected image becomes. That is why you move the projector back to fill a bigger screen.

Object at F₁ → image at infinity

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Real-life use: run this case backwards. Put a bright source exactly at F₁, and the rays leave the lens as a parallel beam that barely spreads out however far it travels. This is how a searchlight, spotlight, or lighthouse beam made with a lens works. It is also how a collimator / condenser sends a steady parallel beam into spectrometers and other optical instruments. It is the lens version of putting a bulb at a concave mirror’s focus.

Object within F₁ → image on the same side (magnifying glass)

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Real-life use: the magnifying glass and the jeweller’s loupe. Hold the lens close so the object sits just inside F, and you see an upright, enlarged image that you can look straight at. Nothing is projected here. The magnified image is virtual and on the same side as the object, so it can only be viewed, never caught on a screen. This same “object just inside the focus” trick is how the eyepiece of a microscope or telescope enlarges the real image made by the first lens, and how a clip-on macro lens for a phone camera works.

Images in a concave lens

A concave (diverging) lens is the simplest of all. No matter where you put the object, the image is always virtual, erect, and diminished (smaller). It sits between the lens and F₁, on the same side as the object. The image barely changes as the object moves. It only shrinks a little and creeps closer to the lens. You build it with the same two rays: the parallel ray spreads out as if it came from F₁, and the central ray goes straight on. The image is where their backward extensions cross.

Object at infinity → a point image at F₁

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Object beyond 2F₁ → image between O and F₁

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Object at 2F₁ → image between O and F₁

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Object between F₁ and 2F₁ → image between O and F₁

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Object within F₁ → image very close to the lens

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Real-life uses — the same at every object position. A concave lens always makes the same kind of image — virtual, erect, and diminished — no matter where the object is. So all its uses depend on that one property, not on the object distance:

• Spectacles for short sight (myopia). A short-sighted eye bends light too strongly and focuses distant objects in front of the retina. A concave lens spreads the incoming rays apart a little first. Then the relaxed eye can focus them correctly on the retina. (You will meet this again in Chapter 10.)
• Door peephole (the “door viewer”). Its wide, small, always-upright image fits a whole hallway into a tiny hole. So you can see a broad view of who is outside, without opening the door.
• Eyepiece of a Galilean telescope (and old opera glasses). A diverging eyepiece gives an upright view — unlike the upside-down view of a simple convex eyepiece. An upright view is what you want for looking at distant things on Earth.
• Spreading a beam in optical systems. A concave lens is used to widen a narrow beam (a “beam expander” in lasers and projectors). It is also used as a correcting part inside good camera and binocular lenses, to keep the final image sharp.

In short, use a concave lens whenever you want a small, upright, wide view, or simply to spread light apart.

Lens formula, magnification and power

The lens formula looks like the mirror formula, but with a minus sign:

1/v − 1/u = 1/f

And the magnification for a lens is m = h′/h = v/u. (Note: there is no minus sign here, unlike with mirrors. Be careful not to mix them up.)

Here’s the lens formula in action — notice the minus sign and the plain m = v/u, so you don’t slip into the mirror version.

Finally, the power of a lens tells you how strongly it bends light:

P = 1/f (with f in metres), measured in dioptres (D).

A convex lens has positive power. A concave lens has negative power. For example, a +2.0 D lens is a convex lens with f = +0.50 m. When you place lenses together, their powers simply add up: P = P₁ + P₂ + … This is exactly how opticians stack test lenses on top of each other to find your eyeglass prescription.

Common Mistakes

Quick Check

Practice Problems

Summary

• Light reflects off mirrors (angle of incidence = angle of reflection). It refracts (bends) when it passes between materials, because its speed changes.
• Concave mirrors bring light together (converge); convex mirrors spread it apart (diverge). For both, the focal length is f = R/2.
• A concave mirror’s image changes with the object’s position (real and inverted in most positions; virtual, erect, and enlarged when the object is inside F). A convex mirror’s image is always virtual, erect, and diminished.
• Mirror formula: 1/v + 1/u = 1/f, and magnification m = −v/u. Always use the New Cartesian Sign Convention (object distance is negative; concave f is negative, convex f is positive).
• Light bends towards the normal when entering a denser medium, and away from it when leaving. Refractive index n = c/v. A higher n means optically denser, which means it bends light more.
• Convex lenses converge (f is positive); concave lenses diverge (f is negative). Lens formula: 1/v − 1/u = 1/f, and magnification m = v/u.
• Power P = 1/f (with f in metres), measured in dioptres. Convex power is positive, concave power is negative. The powers of lenses placed together simply add up.

What’s Next

You now understand how a single mirror or lens bends light to form an image. But the most amazing optical instrument you own is your eye — a living convex lens that focuses all by itself. In Chapter 10: The Human Eye and the Colourful World, you will see how the eye forms images, why some people need the very spectacle lenses you just studied (to fix short sight and long sight), and how refraction makes the sky blue, the sunset red, and splits white light into a rainbow through a prism.