Near point, far point and accommodation

Looking at a far tree, and then at a book

Look out of a window at a far tree. Then look at the page of a book at once. You do not have to do anything, but the eye does a lot. It must put a sharp image on the retina in both cases, and the distance from the lens to the retina does not change. How is this possible?

The lens changes its shape

The distance between the lens and the retina is fixed, about $2.5\ \text{cm}$. For a sharp image, the focal length of the lens must change with the distance of the object. The eye does this by changing the shape of its lens.

• Far object: the ciliary muscles relax. The lens becomes thinner. Its focal length is longer, and its power is smaller.
• Near object: the ciliary muscles contract. The lens becomes thicker. Its focal length is shorter, and its power is larger.

The ability of the eye to change the focal length of its lens is called accommodation.

You should find that a far object needs about $40\ \text{D}$. Bring it close, and you must raise the power to keep the spot small. Below about $25\ \text{cm}$ the eye would need more than a young lens can give.

How much does the power change?

For a very far object, the image distance is $2.5\ \text{cm}$, and the lens needs a focal length of $2.5\ \text{cm}$. Its power is $\dfrac{1}{0.025\ \text{m}} = 40\ \text{D}$.

For a nearer object, the lens needs extra power, $\dfrac{1}{u}$ with $u$ in metres. The closer the object, the more the eye must add.

Near point and far point

• The near point is the nearest place at which the eye can still see an object clearly. For a young adult with a normal eye, it is about $25\ \text{cm}$ from the eye. We also call it the least distance of distinct vision. When an object is closer than the near point, the lens cannot become thick enough, and the image is blurred.
• The far point is the farthest place at which the eye can see an object clearly. For a normal eye, the far point is at infinity.

So a normal eye sees clearly from $25\ \text{cm}$ to infinity.

Why do we hold a book at about 25 cm?

At $25\ \text{cm}$, the eye can see a small object clearly without a strain. At a shorter distance, the ciliary muscles must work hard, and the eye is tired. For this reason, a book should not be held very close to the eyes.

A worked example

A child holds a picture $30\ \text{cm}$ from her eye. The distance from her lens to her retina is $2.5\ \text{cm}$. What focal length must her eye lens have to see the picture clearly?

Given: $v = +2.5\ \text{cm}$ (the image is on the retina) and $u = -30\ \text{cm}$.

Step 1. We want $f$, so $\dfrac{1}{f} = \dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{2.5\ \text{cm}} + \dfrac{1}{30\ \text{cm}} = \dfrac{12 + 1}{30\ \text{cm}} = \dfrac{13}{30\ \text{cm}}$.

Step 2. $f = \dfrac{30\ \text{cm}}{13} = 2.31\ \text{cm}$, a power of $43.3\ \text{D}$.

The focal length is shorter than the $2.5\ \text{cm}$ for a far object, as expected for a near object.

Practise with new numbers

Try it in Python

The program finds the focal length and the power that the lens of the eye needs for objects at different distances.

Remember these

• The distance from the lens to the retina is fixed. The eye changes the focal length of its lens, and this is accommodation.
• Far object: thin lens, long focal length. Near object: thick lens, short focal length.
• For a normal eye, the near point is about $25\ \text{cm}$ and the far point is at infinity.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What happens to the ciliary muscles when you look at a near object?

2. A pencil is brought closer and closer to your eye. What do your ciliary muscles do, and when does the pencil blur?

3. The retina is $2.5\ \text{cm}$ behind the lens. By how much does the power change between looking at a star and at a page $50\ \text{cm}$ away?

4. Where is the far point of a normal eye?