Derivation of Mirror Formula | Convex Classes Jaipur

The mirror formula relates the object distance (u), image distance (v), and focal length (f) of a spherical mirror:

$$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$$

This formula is valid for both concave and convex mirrors when the New Cartesian Sign Convention is applied.

Sign Convention Recap

• Distances are measured from the pole (P) of the mirror.
• Distances measured along the direction of incident light are taken as positive.
• Distances measured against the direction of incident light are taken as negative.
• Heights above the principal axis are positive; below are negative.

Geometrical Setup

Consider a concave mirror:

• C = Centre of curvature
• F = Focus
• P = Pole
• Object AB is placed at distance u from P.
• Image A′B′ is formed at distance v from P.

Draw two rays:

1. Ray parallel to the principal axis → passes through focus (F).
2. Ray through centre of curvature (C) → reflects back along the same path.

These rays intersect at A′, forming the image.

Step‑by‑Step Derivation

Step 1: Similar Triangles

From the ray diagram:

• Triangle ABP (object side) and triangle A′B′P (image side) are similar.
• Triangle A′B′F and triangle ABF are also similar.

Step 2: Ratio Relations

From similarity:

$$\frac{AB}{A\text{′}B\text{′}}=\frac{PB}{PB\text{′}}$$

and

$$\frac{AB}{A\text{′}B\text{′}}=\frac{FB}{FB\text{′}}$$

Step 3: Express Distances

• PB = u (object distance)
• PB′ = v (image distance)
• PF = f (focal length)

Using geometry:

$$\frac{PB}{PB\text{′}}=\frac{u}{v}$$

and

$$\frac{FB}{FB\text{′}}=\frac{u-f}{f}$$

Step 4: Equating Ratios

Since both ratios equal $\frac{AB}{A\text{′}B\text{′}}$:

$$\frac{u}{v}=\frac{u-f}{f}$$

Step 5: Simplify

Cross‑multiply:

$$u\cdot f=v(u-f)$$

$$uf=uv-vf$$

$$uv-uf=vf$$

Divide throughout by $uvf$:

$$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$$

Final Mirror Formula

$$\boxed{{\displaystyle \frac{1}{f}=\frac{1}{v}+\frac{1}{u}}}$$

This is the mirror formula, valid for both concave and convex mirrors.

Applications

• Concave mirrors: telescopes, headlights, shaving mirrors.
• Convex mirrors: vehicle side mirrors for wider field of view.
• Used in optical instruments to calculate image position without ray diagrams.

Conclusion

The mirror formula is derived using geometry and sign conventions, making it a universal relation for spherical mirrors. At Convex Classes Jaipur, we provide step‑by‑step derivations, diagrams, and solved examples so students can master concepts and score high in exams.