Spherical Mirrors & the Sign Convention
The vocabulary of curved mirrors, and a sign rule that makes one formula fit every case.
Lesson at a glanceShow the key points
- A spherical mirror is part of a hollow sphere: reflecting on the inside is concave, on the outside convex.
- Pole P (middle of the mirror), centre of curvature C, radius R, principal axis through P and C.
- Paraxial rays parallel to the axis converge at (concave) or seem to diverge from (convex) the focus F; focal plane and focal length PF.
- Real image: reflected rays actually meet. Virtual image: they only seem to come from it. A virtual object: incident rays converge toward a point behind the mirror.
- Coordinate sign convention: pole at origin, x-axis along the principal axis, positive along the incident light; u, v, R, f are x-coordinates; heights up are positive.
Look into a shiny spoon: one side shows you upside down, the other right way up but tiny. Curved mirrors can magnify, shrink and flip images, and a few words and one sign rule are enough to describe them all.
The parts of a spherical mirror
A spherical mirror is a piece cut from a hollow sphere, usually glass silvered on one side. If light reflects from the hollow (inner) side it is a concave mirror; from the bulging (outer) side, a convex mirror.
- Centre of curvature C: the centre of the sphere. Its radius is the radius of curvature R.
- Pole P: the middle point of the mirror’s surface.
- Principal axis: the line through P and C.
Focus
Send a narrow beam parallel to the principal axis onto a concave mirror, close to the pole. The reflected rays all cross the axis at (nearly) one point: the focus F. On a convex mirror the reflected rays spread out, but they seem to come from one point behind the mirror, its focus.
The distance PF is the focal length; the plane through F perpendicular to the axis is the focal plane.
Rays far from the axis miss the focus slightly, which matters for big mirrors (you’ll meet this again under aberrations).
Tracing an image
To find the image of a point, follow two rays from it to the mirror. At each point of incidence the normal is the line to C; reflect each ray by the laws of reflection.
- If the reflected rays actually meet, they form a real image (it can be caught on a screen).
- If they diverge, they seem to come from a virtual image.
Real and virtual objects
Usually rays diverge from a point object in front of the mirror: a real object. But a lens could send converging rays toward the mirror; the point where they would have met, behind the mirror, then acts as a virtual object. In short: where the incident rays meet (or would meet) is the object; where the reflected rays meet (or seem to) is the image.
The sign convention
Mirror formulas involve the object distance, image distance, focal length and radius. To make one formula work in every situation we use a coordinate sign convention:
- The pole is the origin and the principal axis is the x-axis.
- The positive x-direction is the direction of the incident light (usually left to right).
- u, v, R and f are the x-coordinates of the object, image, centre of curvature and focus: positive if on the positive side of P, negative otherwise.
- Heights above the axis are positive, below negative.
| Situation (light from the left) | u | v | R, f |
|---|---|---|---|
| Concave mirror, real image | − | − | − |
| Concave mirror, virtual image (object inside F) | − | + | − |
| Convex mirror, real object | − | + | + |
- 1Concave (reflects on the inside) and convex (outside) mirrors; pole P, centre C, radius R, principal axis.
- 2Paraxial rays parallel to the axis meet at (or seem to come from) the focus F; PF is the focal length.
- 3Real image: rays meet. Virtual image: rays only seem to come from it. Converging incident rays define a virtual object.
- 4Coordinate convention: pole at origin, + along incident light, u, v, R, f are x-coordinates; up is positive.
Check yourself
Keep exploring
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