The Mirror Formula & Magnification
One equation finds every mirror image; another tells you its size and whether it's upside down.
Lesson at a glanceShow the key points
- For paraxial rays, angles in the diagram give 1/u + 1/v = 2/R (18.1), valid for every case with signs.
- Parallel rays (u = ∞) meet at v = f, so f = R/2 (18.2): the focus is midway between pole and centre. 1/u + 1/v = 1/f (18.3).
- Two easy rays for an extended object: parallel to the axis → through F; toward C → straight back. (Also: through F → parallel; at the pole → mirror image angle.)
- Lateral magnification m = h₂/h₁ = −v/u (18.4); negative m means inverted.
With the sign convention in hand, a single short equation tells you where any spherical mirror forms its image, and a second one tells you how big it is and which way up.
Deriving the mirror formula
A point object O sits on the axis of a concave mirror. One ray goes along the axis, hits the pole P and comes straight back. Another ray OA hits the mirror at A and reflects through I, where the image forms. CA is the normal at A, so it bisects the angle OAI.
Call the angles that OA, CA and IA make with the axis α, β and γ, and let θ be the angle of incidence (= angle of reflection). The exterior angle of a triangle equals the sum of the opposite interior angles:
For a paraxial ray A is close to P, so the small angles are AP/PO, AP/PC and AP/PI:
With light from the left, O, I and C are all on the negative side: PO = −u, PI = −v, PC = −R. Substituting:
Focal length
A very distant object (u → ∞) sends parallel rays, which meet at the focus: v = f. Then 1/f = 2/R:
An object is 10 cm in front of a convex mirror of radius of curvature 30 cm. Where is its image?
Extended objects and magnification
For an object AB standing on the axis, find the image of the tip B with two convenient rays, then drop a perpendicular to the axis:
- a ray parallel to the axis reflects through F (or away from F for a convex mirror);
- a ray aimed at C hits the mirror normally and comes straight back;
- a ray through F reflects parallel to the axis;
- a ray striking the pole reflects symmetrically about the axis.
Comparing the right-angled triangles formed by the ray that strikes the pole gives the lateral magnification:
A 3.0 cm tall object stands at 1.25 f in front of a concave mirror (f is the magnitude of its focal length). Find the image’s height and orientation.
- 11/u + 1/v = 2/R = 1/f for every spherical mirror (with signs); f = R/2.
- 2Parallel → through F; toward C → straight back; through F → parallel; at P → symmetric.
- 3m = h₂/h₁ = −v/u; negative m is inverted.
Check yourself
Keep exploring
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