Reflection of Light in Curved Mirrors
Senior 2 · Physics
Reflection of Light in Curved Mirrors
Lesson 1 of 4
18 min
Look into the bowl of a shiny spoon and your face appears upside down; turn the spoon over and it is the right way up but tiny. The same two kinds of curved surface are at work in a moto's side mirror, the reflector behind a torch bulb, a dentist's small mirror, the round safety mirrors at blind corners on Kigali's hilly roads, and the giant mirrors of astronomical telescopes. To predict what each one does, you first need the vocabulary.
A spherical mirror is a mirror shaped like part of the surface of a hollow sphere.
| Term | Meaning |
|---|---|
| Pole (P) | The centre point of the reflecting surface of the mirror |
| Centre of curvature (C) | The centre of the sphere of which the mirror is a part |
| Radius of curvature (R) | The radius of that sphere: the distance from P to C |
| Principal axis | The straight line through P and C |
| Principal focus (F) | For a concave mirror, the point on the principal axis where rays parallel to the axis meet after reflection. For a convex mirror, the point behind the mirror from which such rays appear to come |
| Focal length (f) | The distance from the pole to the principal focus, PF |
| Aperture | The width of the mirror's reflecting surface |
A concave mirror has a real focus in front of it: light really passes through it, and you could catch sunlight there on a piece of paper. A convex mirror has a virtual focus behind it: light only seems to come from it.
For a mirror whose aperture is small compared with its radius of curvature, the principal focus lies halfway between the pole and the centre of curvature:
f = R / 2
Worked example. A concave mirror is cut from a sphere of radius 40 cm. Its focal length is f = 40 ÷ 2 = 20 cm. Parallel sunlight reflected from it comes to a bright point 20 cm in front of the pole.
A curved mirror obeys the same two laws of reflection as a flat mirror, at every point on its surface:
The only new idea is where the normal is. For a spherical mirror, the normal at any point is the radius drawn to that point, the line through the centre of curvature C. So a ray travelling along a radius (through C) hits the mirror at 90° and is reflected straight back along its own path.
The rule f = R/2 is only exact for rays close to the principal axis. Rays striking near the edge of a wide spherical mirror are reflected to a point slightly closer to the mirror, so the image is blurred. This is called spherical aberration. It is reduced by using a mirror with a small aperture, or removed by using a parabolic mirror, which brings all parallel rays to one exact focus. Car headlamps, satellite dishes and large telescopes use parabolic reflectors.
Tip
Memory hook: a concave mirror caves in and collects light (converges). A convex mirror bulges out and spreads light (diverges).
f = R/2, always check it. If a question gives R and asks for f (or the other way round), halve or double before anything else.
A convex mirror has a radius of curvature of 60 cm. What is its focal length?
Solution: f = R/2 = 60 ÷ 2 = 30 cm (a virtual focus, behind the mirror).
A concave mirror brings sunlight to a focus 12 cm from its pole. Find its radius of curvature.
Solution: R = 2f = 2 × 12 = 24 cm.
A ray strikes a concave mirror travelling along a line through its centre of curvature. What happens to it, and why?
Solution: It is reflected straight back along its own path. A line through C is a radius, so it is the normal to the mirror: the angle of incidence is 0°, so the angle of reflection is also 0°.
Why is the focus of a convex mirror called virtual?
Solution: Parallel rays reflected by a convex mirror spread apart. They never actually meet; they only appear to come from a point behind the mirror, where no light actually goes. A focus that light only appears to come from is virtual.
Remember
Sources and credits
