Using a ray diagram, derive the mirror equation for a concave mirror. Also write the expression for its linear magnification.

1. 1 · State the rule. Use the New Cartesian sign convention: distances are measured from the pole P, positive along the incident light and negative against it.
2. 2 · Draw the ray diagram. [Concave mirror; object AB beyond C; ray parallel to the axis reflects through F; ray to P reflects at an equal angle; real, inverted image A′B′ between F and C; label P, F, C, M, u and v]
3. 3 · Write two pairs of similar triangles.
   • Triangles A′B′P and ABP: B′A′/BA = B′P/BP
   • Triangles A′B′F and MPF: B′A′/BA = B′F/FP (as PM = BA)
4. 4 · Equate and put in the signs.
   • (B′P − FP)/FP = B′P/BP
   • B′P = −v, FP = −f, BP = −u
   • (v − f)/f = v/u
5. 5 · Simplify to the result. Divide by v: 1/f − 1/v = 1/u, so 1/v + 1/u = 1/f.
6. 6 · Write the magnification. From the first pair: −h′/h = (−v)/(−u), so m = h′/h = −v/u.
7. 7 · Close in one line. The same equation works for convex and plane mirrors when the signs are put in correctly.

Scoring tip: Mark the signs on your diagram before you substitute. Examiners look for a clear statement of the sign convention.