Convex and concave lenses

A drop of water as a lens

Put a drop of water on a printed page. The letters under the drop look larger. The drop has a curved surface, and it bends the light. A lens does the same, in a controlled way.

A lens is a piece of transparent material (glass or clear plastic) with two surfaces, of which at least one is curved. The light bends at each surface by refraction. A lens is the main part of a camera, a microscope, a telescope, a pair of spectacles and the eye itself.

Two kinds of lens

• A convex lens is thicker in the middle than at the edges. It converges a beam of parallel rays, so it is also called a converging lens.
• A concave lens is thinner in the middle than at the edges. It diverges a beam of parallel rays, so it is also called a diverging lens.

A good way to remember this: a convex lens bulges outward at the middle, like the shape of a seed of a lentil. (The word lens is the Latin word for a lentil.)

The terms of a lens

• The optical centre $O$ is the centre of the lens. A ray that goes through $O$ goes straight on, with no bending.
• The principal axis is the straight line through $O$ at right angles to the lens.
• A lens has two principal foci, $F_1$ and $F_2$, one on each side, at the same distance from $O$.
• The focal length $f$ is the distance $OF$.
• The point $2F$ on each side is at a distance $2f$ from the lens.

For a convex lens, a beam parallel to the principal axis meets at the focus $F_2$ on the other side of the lens. This focus is real. For a concave lens, the beam spreads out, and it seems to come from the focus $F_1$ on the side where the light comes from. This focus is virtual.

You should find that the convex lens sends every parallel ray through $F_2$. The concave lens spreads them, and the dashed lines meet at $F_1$. A short focal length bends the rays sharply; a long one bends them gently.

How to find the focal length of a convex lens

The Sun is so far away that its rays reach us almost parallel. Hold a convex lens so that the Sun shines on it, and hold a sheet of paper behind it. Move the paper to and fro until the spot of light is the smallest and the brightest. This spot is the focus. The distance from the lens to the paper is the focal length. (Take care, because the spot is very hot. Do not hold the lens in one place for long, because it can burn the paper.)

A worked example

A student finds the smallest spot of sunlight $16\ \text{cm}$ behind a convex lens. Where are $F_1$, $F_2$, $2F_1$ and $2F_2$?

1. The focal length. The Sun's rays are parallel, so the spot is at the focus: $f = 16\ \text{cm}$.
2. The foci. $F_1$ and $F_2$ are $16\ \text{cm}$ from the lens, one on each side.
3. The $2F$ points. They are $2 \times 16\ \text{cm} = 32\ \text{cm}$ from the lens, one on each side.

Three rays that we can always draw

To find an image, we follow rays from the top of an object. For a convex lens, three rays are easy to draw.

1. A ray parallel to the principal axis goes through the focus $F_2$ after the lens.
2. A ray through the optical centre $O$ goes straight on.
3. A ray through the focus $F_1$ comes out parallel to the principal axis.

For a concave lens, the rule is similar. A ray parallel to the axis spreads out, as if it came from $F_1$. A ray through $O$ goes straight on. A ray that is directed toward $F_2$ comes out parallel to the axis.

We need only two of the rays to find the image. The third is a check.

Practise with new numbers

Try it in Python

A student moved a sheet of paper behind a convex lens in sunlight, and measured the width of the bright spot. The program finds the focal length from the readings.

Remember these

• A convex lens is thick in the middle and converges light. A concave lens is thin in the middle and diverges light.
• A ray through the optical centre goes straight on.
• A lens has two foci, one on each side, at the same distance $f$ from the optical centre.
• Three rays: parallel to the axis, through $O$, and through $F_1$.

Check yourself

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

1. A lens makes the print on a page look smaller, wherever you hold it. Is it convex or concave?

2. A convex lens focuses the Sun's rays on a spot $20\ \text{cm}$ from the lens. What is its focal length?

3. A ray passes through the optical centre of a lens. How does it bend?

4. A lens has a focal length of $13\ \text{cm}$. How far apart are its two foci?