Reflection by spherical mirrors
Pole, centre and focus
Look into a shiny steel spoon. On the inside you see yourself upside down; on the back you see yourself upright and small. Both sides are curved mirrors.
In ray optics we draw light as rays: straight lines along which light travels. This works because the wavelength of light is tiny compared with mirrors and lenses.
Reflection obeys two laws. The angle of reflection equals the angle of incidence, and the incident ray, the reflected ray and the normal all lie in one plane.
A spherical mirror is a piece cut from a hollow sphere. If the inside is silvered and reflects, it is concave. If the outside reflects, it is convex.
The centre of the mirror’s surface is the pole, P. The centre of the sphere is the centre of curvature, C, and its radius is R. The line through P and C is the principal axis.
Now we need signs. In the Cartesian sign convention, every distance is measured from the pole. Distances in the direction of the incident light are positive; distances against it are negative.
Heights above the principal axis are positive, and heights below it are negative. We always draw the light coming from the left, so everything in front of the mirror has a negative distance.
Send in rays parallel to the axis and close to it. These are paraxial rays. After reflection from a concave mirror, they all meet at one point on the axis: the principal focus F.
For a convex mirror the reflected rays spread out, but they seem to come from a point F behind the mirror. The distance from P to F is the focal length f.
Since F and C of a concave mirror are in front, its f and R are negative. For a convex mirror both are positive, because F and C lie behind it.
Why f = R/2, and drawing ray diagrams
Why is the focus halfway between P and C? Follow one parallel ray that hits the mirror at M. The line CM is a radius, so it is the normal at M.
The ray makes an angle θ with this normal, and so does the reflected ray. By alternate angles, the angle MCP is also θ, and the reflected ray crosses the axis at F making an angle 2θ.
Drop a perpendicular MD to the axis. Then tan θ equals MD over CD, and tan 2θ equals MD over FD.
For paraxial rays θ is small, so tan θ is about θ and tan 2θ is about 2θ. Dividing, FD is half of CD. And since D is very close to P, FP is half of CP. So f equals R over 2.
A concave mirror of radius 30 centimetres therefore has a focal length of 15 centimetres; with signs, R is minus 30 and f is minus 15 centimetres.
To find an image, follow any two rays from the tip of the object. A ray parallel to the axis reflects through F. A ray through C hits the mirror head-on and comes straight back.
A ray through F reflects parallel to the axis. And a ray striking the pole reflects at an equal angle on the other side of the axis. Where the reflected rays meet is the image of the tip.
Place the object beyond C. The rays meet between F and C: a real, inverted, smaller image. Real means the light actually passes through it, so it can be caught on a screen.
Move the object between F and the pole. Now the reflected rays spread out. Extend them backwards, dashed: they meet behind the mirror in an upright, magnified, virtual image. That is a shaving or make-up mirror.
A convex mirror always gives an upright, smaller, virtual image, wherever the object is. It shows a wide view, so it is used as the rear-view mirror on scooters and cars.
Try it: ray diagrams for mirrors
- Drag the green object. The amber ray starts parallel and goes through F; the blue ray goes through C and comes back on itself.
- Start far away and move the object towards the mirror. Note where the image is real (in front) and where it becomes virtual (behind, dashed).
- Put the object exactly at C (u = −30 cm with f = 15 cm). The image is at C too, the same size, inverted.
- Switch to Convex. Whatever you do, the image stays upright, smaller and behind the mirror.
Signs first
A concave mirror has a radius of curvature of 30 cm. Using the Cartesian sign convention, what is its focal length?
- +15 cm
- −15 cm
- −30 cm
- +60 cm
Answer
B. f = R/2, so |f| = 30/2 = 15 cm. The focus of a concave mirror is in front of the mirror, against the direction of the incident light, so it is negative: f = −15 cm. (−30 cm is R itself; +15 cm would be a convex mirror.)
Object close to a concave mirror
An object is placed between the focus F and the pole P of a concave mirror. What kind of image is formed?
- Real, inverted and magnified, beyond C
- Real, inverted and smaller, between F and C
- Virtual, upright and magnified, behind the mirror
- No image: the reflected rays are parallel
Answer
C. Draw the parallel ray (reflects through F) and the ray along the line from F (reflects parallel). After reflection they spread apart and never meet in front. Extended backwards they meet behind the mirror: a virtual, upright, magnified image. This is how a shaving mirror works. (Parallel rays happen only when the object is exactly at F.)
Summary
- Focal length and radius:
f = R/2 - Concave mirror:
f < 0, R < 0 (F and C in front) - Convex mirror:
f > 0, R > 0 (F and C behind)
- Cartesian sign convention: measure from the pole; along the incident light is +, against it is −; heights up are +, down are −.
- Paraxial rays parallel to the axis meet at (concave) or seem to come from (convex) the principal focus F, halfway between P and C.
- Ray rules: parallel → through F; through F → parallel; through C → straight back; at P → equal angle on the other side.
- A real image is where the rays actually meet (on a screen); a virtual image is where they only seem to come from.
- A convex mirror always gives a virtual, upright, diminished image: a wide field of view for rear-view mirrors.