S4.10.6 Chapter 6

Lenses

Held far from a window, a magnifying glass puts a small, upside down picture of the window on a sheet of paper. Why is the picture upside down?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Rays from the top of the window cross after the lens and reach the bottom of the picture.
You picked A. Read on. At the end, the chapter comes back to your guess.
B The lens turns the picture over, as you would turn a coin.
You picked B. Read on. At the end, the chapter comes back to your guess.
C The lens bends every ray down.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Draw ray diagrams for a converging lens
Explain how lenses correct short sight and long sight

A lens is a clear piece of glass or plastic with curved faces. It bends light by refraction at each face. A converging lens is thicker in the middle and brings parallel rays together. A diverging lens is thinner in the middle and spreads them apart.

The principal focus, F, of a converging lens is the point where rays parallel to the axis meet after the lens. The axis is the line through the middle of the lens at right angles to its faces. The distance from the lens to F is the focal length, f.

Parallel rays meet at the principal focus. A fatter lens bends them more, so its focus is nearer.

A lens works both ways, so it has a focus on each side, at the same distance f. A fatter, more curved lens has a shorter focal length. A real lens bends light at both faces. To keep the picture clear, we draw one bend, at one line through the middle.

To find the image of an object, draw an arrow for the object. Then trace three rays from its tip. These three rules are enough.

Rule one: a ray that travels parallel to the axis bends and goes through the focus on the far side. Rule two: a ray that goes through the centre of the lens carries straight on. Rule three: a ray that passes through the focus on the near side leaves the lens parallel to the axis.

Where the rays meet is the image of the tip. If the rays really cross there, the image is a real image. You can catch it on a screen. It is upside down. The picture of the window is a real image.

An object beyond 2F gives a real, upside down, smaller image between F and 2F.

An object at 2F gives a real, upside down image of the same size, also at 2F.

An object between F and 2F gives a real, upside down, larger image beyond 2F.

Move the object closer, to a point inside F. The rays now leave the lens spreading out. They never cross on the far side. Look backwards along the rays, and they seem to come from one point on the object's side.

An object inside F gives an upright, larger image on the same side as the object. This is a magnifying glass.

This image is a virtual image. Light does not really pass through it, so it cannot be caught on a screen. Your eye sees it because the rays seem to come from it. A virtual image from a converging lens is upright and larger. This is how a magnifying glass works.

The image from a converging lens. A real image is upside down. A virtual image is upright. An object at F gives no image: the rays leave parallel.

Diagrams give the shape of the answer. A rule gives the numbers. For a thin lens, 1/f = 1/u + 1/v. Here u is the distance from the object to the lens and v is the distance from the lens to the image. All are in the same unit.

This book counts distances to a real image as positive, and a virtual image has a negative v. Other books use other sign rules and write the formula another way. The physics is the same. To find v, rearrange: 1/v = 1/f − 1/u.

The magnification is the height of the image divided by the height of the object. It also equals v ÷ u when you ignore the sign. A magnification of 2 means the image is twice as tall as the object. It has no unit.

Worked example

An object is 4 cm from a converging lens of focal length 3 cm. Find the image distance and the magnification.

For a real image, 1/v = 1/f − 1/u. Here f = 3 cm and u = 4 cm.
1/v = 1/3 − 1/4 = 4/12 − 3/12 = 1/12.
So v = 12 cm. The image is real and 12 cm from the lens.
The magnification is v ÷ u = 12 ÷ 4 = 3.

Answer: 12 cm and a magnification of 3

Any object between F and 2F gives a larger real image, as in the worked example.

Now the diverging lens. It spreads parallel rays so that they seem to come from a focus on the object's side. The three rays bend the other way. A ray parallel to the axis leaves as if from the near focus. A ray through the centre goes straight on. A ray aimed at the focus on the far side leaves parallel to the axis.

A diverging lens always gives an upright, smaller, virtual image between the lens and the object.

Now turn to the eye. The cornea and the lens of the eye act together as a converging lens. They make a real, upside down image of what you look at on the retina at the back of the eye. The brain turns the picture the right way up.

A muscle changes the shape of the eye lens. To look at something close, the lens is made fatter, so its focal length is shorter. To look at something far, it is made thinner. This focusing is why a clear eye sees both near and far.

In short sight, the eye bends light too much, or the eyeball is too long. Rays from a far object cross in front of the retina, and the image on the retina is blurred. Near things are clear. A diverging lens in front of the eye spreads the rays a little and moves the image back onto the retina.

Short sight without glasses. The rays from a far object meet in front of the retina, so the picture on the retina is blurred.

Short sight corrected with a diverging lens. The rays now meet on the retina.

In long sight, the eye bends light too little, or the eyeball is too short. Rays from a near object would meet behind the retina, so the image on the retina is blurred. Far things are clear. A converging lens in front of the eye helps to bend the rays and moves the image forward onto the retina.

Long sight corrected with a converging lens. The image is back on the retina. The object is drawn near, as a book would be.

Opticians give the strength of a lens as its power: P = 1 ÷ f, with f in metres. The unit is the dioptre, D. A converging lens has a positive power and a diverging lens a negative one. A short focal length is a high power.

A short sighted person sees clearly only as far as a distance called the far point. A diverging lens in front of the eye should make a faraway object seem to be at the far point. So its focal length equals the far point distance, with the sign of a diverging lens.

Worked example

A short sighted person sees clearly only up to 0.5 m. What power of lens do they need?

The lens must make faraway things look as if they are 0.5 m away. So it is a diverging lens with focal length 0.5 m.
The size of the power is 1 ÷ f = 1 ÷ 0.5 = 2 D.
A diverging lens has a negative power, so the lens is −2 D.

Answer: The lens is −2 D

Try it

Make a picture with a lens

You need: a lens from an old pair of reading glasses, or a magnifying glass, a sheet of white paper, a ruler, a room with a window or a lamp across the room.

Be safe: Never look at the Sun, and never point a lens at the Sun. A lens focuses sunlight into a hot spot that can burn skin, paper and eyes. Choose a window or a lamp, and keep the Sun out of the view. Do not look into a bright lamp through the lens.

Hold the paper up as a screen, with its back to the window. Hold the lens between the window and the paper.
Move the lens towards and away from the paper until a sharp small picture of the window appears on the paper.
Measure the distance from the lens to the paper. This is about the focal length f, because the window is so far away.
Work out the power: P = 1 ÷ f, with f in metres.

What you will see: The picture on the paper is upside down and smaller than the window. It is a real image. The window is far away, so the image forms very near the focus.

If you cannot try it, this is what happens.

A wrong idea many people have

A converging lens always makes an upside down image.

Why it feels right: The first picture we see, a window on a screen, is upside down. Most pictures of lenses show an upside down image.

The test that breaks it: Hold a magnifying glass close to some print. The print looks larger and the right way up. That image is virtual, and the object is inside the focal length.

What is true: A real image from a converging lens is upside down. When the object is inside F, the image is virtual, upright and larger.

A wrong idea many people have

A short sighted person needs a lens that magnifies, so a converging lens.

Why it feels right: We call a magnifying glass a lens that helps us see, and a short sighted person sees things badly.

The test that breaks it: In short sight, the rays from a far object cross in front of the retina. A converging lens would bring them together even sooner, and make it worse.

What is true: Short sight needs a diverging lens, to spread the rays. Long sight needs a converging lens, to bring them together sooner.

A wrong idea many people have

A lens has one focus.

Why it feels right: Pictures of parallel rays arriving from one side show only one point where the rays meet.

The test that breaks it: Turn the lens round and send the rays from the other side. They meet at a point the same distance from the lens on the other side.

What is true: A lens has a focus on each side, at the same distance f. Rule three uses the focus on the object's side.

Key words

principal focus: the point where rays that arrive parallel to the axis meet after a converging lens.
focal length: the distance from the lens to its principal focus. Its symbol is f.
real image: an image where rays really cross. It can be caught on a screen, and it is upside down.
virtual image: an image where the rays only seem to cross. It cannot be caught on a screen.
magnification: the height of the image divided by the height of the object.
short sight: a fault in which far things are blurred. A diverging lens corrects it.
power: the strength of a lens, P = 1 ÷ f with f in metres.
dioptre: the unit of the power of a lens, written D.

Back to your guess

The answer is A: Rays from the top of the window cross after the lens and reach the bottom of the picture.

Rays from the top of the window go through the lens, bend, and cross. After crossing they are below the axis, so they land at the bottom of the picture. Rays from the bottom land at the top. So the picture is upside down.

Keep it

Copy this into your exercise book.

Three rays: parallel through the far focus, through the centre straight, through the near focus out parallel. 1/f = 1/u + 1/v. Short sight: diverging. Long sight: converging. P = 1 ÷ f.

Draw: A converging lens with its axis and F marks, an object arrow, and the three rays meeting at an upside down image.