Spherical mirrors and their terms

The back of a spoon

Look at your face in the inside of a steel spoon. You see it upside down and small. Turn the spoon over and look at the outside. Now your face is upright and small. The two sides of the spoon are two kinds of spherical mirror.

A spherical mirror is a part of the surface of a hollow sphere, with one side polished so that it reflects light.

• If the inside of the sphere reflects, the mirror is a concave mirror. It curves inward, like a cave.
• If the outside of the sphere reflects, the mirror is a convex mirror. It bulges outward.

The terms of a spherical mirror

These words describe every spherical mirror.

• The pole $P$ is the centre of the mirror surface.
• The centre of curvature $C$ is the centre of the hollow sphere of which the mirror is a part. For a concave mirror, $C$ is in front of the mirror. For a convex mirror, $C$ is behind it.
• The radius of curvature $R$ is the radius of that sphere. It is the distance $PC$.
• The principal axis is the straight line through $P$ and $C$.
• The principal focus $F$ is the point where rays that are parallel to the principal axis meet after reflection (or seem to come from, for a convex mirror).
• The focal length $f$ is the distance $PF$.

Where does the focus come from?

Send a beam of rays parallel to the principal axis toward a concave mirror. After reflection, all the rays pass through one point on the axis. This is the principal focus, and it is a real point. A concave mirror is also called a converging mirror.

Send the same beam toward a convex mirror. After reflection, the rays spread out. They do not meet, but they seem to come from one point behind the mirror. This is the principal focus of a convex mirror, and it is a virtual point. A convex mirror is also called a diverging mirror.

The focal length is half the radius

For a mirror with a small opening (a small piece of the sphere), the focal length is half of the radius of curvature:

So $F$ is exactly half way between $P$ and $C$. If a concave mirror has $R = 40\ \text{cm}$, then $f = 20\ \text{cm}$, and $F$ is $20\ \text{cm}$ from the pole.

You should find that a narrow beam meets at $F$, half way to $C$. A bigger mirror bends the rays less, so $F$ moves away. Spread the beam wide on a small mirror, and the outer rays cross before $F$: the focus is no longer one sharp point.

A worked example

A shaving mirror has a radius of curvature of $56\ \text{cm}$. Where is its focus?

1. The rule. $f = \dfrac{R}{2}$.
2. Put in the number. $f = \dfrac{56\ \text{cm}}{2} = 28\ \text{cm}$.
3. So the focus is $28\ \text{cm}$ in front of the pole, half way to $C$.

Three rays that we can always draw

To find where an image forms, we follow rays from the top of the object. Three rays are easy to draw, because we know how each is reflected.

1. A ray parallel to the principal axis is reflected through the focus $F$ (for a convex mirror: it seems to come from $F$).
2. A ray through the focus $F$ (for a convex mirror: directed toward $F$) is reflected parallel to the principal axis.
3. A ray through the centre of curvature $C$ falls on the mirror at a right angle, so it is reflected back along the same path.

We need only two of these rays to find the image, because two lines meet at one point. The third is a check.

Practise with new numbers

Try it in Python

The program follows rays at different heights to a concave mirror, and finds where each one crosses the axis.

Remember these

• A concave mirror reflects from the inside of a sphere, and a convex mirror from the outside.
• $P$ is the pole, $C$ is the centre of curvature, $F$ is the focus. $PC = R$ and $PF = f$.
• The focal length is half the radius of curvature: $f = R/2$.
• Three rays for a ray diagram: parallel to the axis, through $F$, and through $C$.

Check yourself

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

1. The radius of curvature of a concave mirror is $34\ \text{cm}$. What is its focal length?

2. Which mirror has a real focus?

3. A ray goes through the centre of curvature and falls on a mirror. How is it reflected?

4. Is the centre of curvature of a convex mirror in front of it or behind it?