Why can you read the board but not the book, or the book but not the board? Fix the eye with the right lens

How to use this simulation

1. Start with Myopia and the object at 400 cm. The rays meet in front of the retina and the readout says the image is blurred. The far point is 80 cm, so anything beyond that cannot be focused.
2. Tick "Wear the corrective lens". A concave lens appears in front of the eye. Its power reads 1.25 D (negative) and the rays now meet on the retina.
3. Drag the far point from 80 cm down to 30 cm and back. A worse eye (a nearer far point) needs a stronger concave lens. Watch the power change.
4. Switch to Hypermetropia. Drop the object to 25 cm, the usual reading distance. The rays would meet behind the retina. Wear the lens: a convex lens of 3.00 D pulls them forward.
5. Move the near point slider. The further the near point, the stronger the convex lens you need.
6. Try Presbyopia, put on the reading lens, then slide the object out to 600 cm. The reading lens now blurs the distance. That is why people with reading glasses take them off to look across the room.

How a healthy eye focuses

Think of the eye as a camera. Light goes through the cornea and the lens and lands on the retina, a screen of light-sensitive cells at the back. For you to see a sharp picture, the rays from each point of an object must meet at one point on the retina, not in front of it and not behind it.

School physics simplifies the eye to a single converging lens 2.5 cm from the retina. By the lens formula, a lens that brings parallel rays to a focus 2.5 cm away has a power of 1 ÷ 0.025 m = 40 D. (A real eye is nearer 60 D, because the cornea does most of the bending, but the textbook picture gives the right shape of every answer.)

The retina cannot move, so when the object comes closer the eye must change the lens. The ciliary muscles squeeze the lens, making it fatter and stronger. This is accommodation. To read at 25 cm, the lens needs a power of 44 D, which is 4 D more than at rest. Watch the lens in the animation: it gets fatter as it works harder.

Myopia: the eye that is too strong

A myopic (short-sighted) person sees near objects clearly but distant ones blurred. The cause is an eyeball that is too long, or a lens and cornea that bend light too much. Either way the relaxed eye is too powerful and brings parallel rays together in front of the retina. By the time the light reaches the retina it has crossed over and spread out again into a blurred disc.

In the animation the myopic eye has a far point of 80 cm. Its relaxed lens works out at 41.25 D, which is 1.25 D too strong. An object at 400 cm is beyond the far point, so the image forms in front of the retina; the focus error reads 1.00 D.

The cure is to weaken the system with a concave (diverging) lens. The lens should make a distant object look as if it sits at the far point. Using the lens formula with u = −∞ and the image at the far point v = −80 cm, the focal length comes out equal to the far point, with a negative sign: f = −80.0 cm. In dioptres, P = 1/f = −1.25 D.

With the lens on, the focus error drops to 0.00 D and the image falls on the retina. Notice also the uncorrected near point: this eye can only focus down to 19.0 cm, closer than the usual 25 cm. That is why short-sighted children read with the book almost touching the nose.

Hypermetropia: the eye that is too weak

A hypermetropic (long-sighted) person sees distant objects well but nearby ones are blurred, so the book has to be held far away. The eyeball is too short, or the lens is too flat. Even at full squeeze the eye cannot add enough power for a close object, and the rays would meet behind the retina if the retina were not in the way.

Take the near point at 100 cm. The strongest lens this eye can make is 41.00 D. A book at 25 cm needs more than that, so the focus error is −3.00 D: the image sits behind the retina, at 2.70 cm from the lens instead of 2.5 cm.

The cure is a convex (converging) lens, which adds the missing power. We want an object held at the comfortable 25 cm to look as if it were at the eye's real near point of 100 cm, with a virtual image there. The lens formula gives 1/f = 1/v − 1/u = −1/100 + 1/25 per cm, and the answer is f = 33.3 cm, P = +3.00 D.

Switch the lens on and the focus error becomes 0.00 D. The rays meet on the retina.

Presbyopia: the lens that stiffens with age

Presbyopia looks like long sight, and the lens that cures it is the same convex lens, but the cause is different. The eye's lens loses its springiness as years go by, usually from the forties, so the muscles can no longer make it fat. The near point drifts outward. A young eye can read at 25 cm; an older one may need the page at arm's length.

Long sight is a fault of shape, present from childhood; presbyopia is a fault of ageing, and it happens to everyone. People who already have myopia can get it too, and then the glasses need two powers: a concave top for the distance and a convex bottom for reading. That is a bifocal lens.

With a near point of 80 cm the reading lens is +2.75 D, a focal length of 36.4 cm. Look at what happens when the same wearer looks across the room through those reading glasses. Put the object at 600 cm with the lens on: the image falls in front of the retina and the focus error is 2.58 D. Reading glasses are the wrong tool for distance.

Dioptres and prescriptions

Opticians do not quote focal lengths; they quote power. The power of a lens is P = 1/f, with f in metres, and the unit is the dioptre (D). A convex lens has positive power and a concave lens has negative power. A lens of f = 25 cm has P = 4 D; a lens with f = −50 cm has P = −2 D.

The big advantage is that powers add. Two thin lenses held together have P = P₁ + P₂, so you can combine a +3 D and a −1 D lens into a +2 D lens without any fractions. Your prescription of −2.5 D tells the optician your far point: f = 1 ÷ 2.5 = 40 cm, so the world beyond 40 cm is a blur without glasses.

One more simplification: glasses sit about a centimetre or two from the eye, and school problems ignore that gap. For strong prescriptions the real-life difference matters, which is why contact lenses carry slightly different numbers.

Solved problems

Five questions in the style that turns up in class tests and admission papers. The numbers come from the same functions the animation runs.

Mistakes that cost marks

Check this list before the test. Each one has cost someone a mark.

• Mixing up the lenses. Myopia needs a concave lens, hypermetropia a convex one. Remember that a concave lens spreads the rays, so it weakens an eye that is too strong.
• Saying the myopic eye brings the image behind the retina. It is in front of the retina. Hypermetropia is behind.
• Using the near point as the focal length of the hypermetropia lens. The focal length comes from the lens formula with u = −25 cm and v = −(near point).
• Forgetting the sign of power. Concave is negative. Writing P = 2 D for a myopic eye loses the mark.
• Using centimetres in P = 1/f. The formula needs metres.
• Treating presbyopia as the same as long sight. The lens is the same, but long sight is a fault of shape and presbyopia comes from ageing.
• Drawing the correction lens after the eye. The spectacle lens sits in front, and its job is to make the object look as if it were within the eye's range.

Other problems, and the companion page

Astigmatism is a different fault: the cornea is not round, so rays in one direction focus differently from rays in another. It needs a cylindrical lens, not a spherical one. A cataract is a clouding of the lens, and no spectacle lens cures it; surgery replaces the lens.

If you want to see how the eye is built, how the retina turns light into signals and how the muscles squeeze the lens, read the biology companion page, Human eye (human-eye-biology). The physics page you are on now takes that structure as given and asks only where the rays go. For the lens formula itself, the lens ray diagram page is a good warm-up.

Frequently asked questions