1/f = 1/dₒ + 1/dᵢ – Lens & Mirror Practice
1/f = 1/dₒ + 1/dᵢ — Thin lens / spherical mirror equation
For a thin lens or spherical mirror, 1/f = 1/dₒ + 1/dᵢ. Equivalently f = dₒ dᵢ / (dₒ + dᵢ) when signs are positive as given.
Guide — how to use this formula
- 1. Keep distances in the same unit (cm or m).
- 2. f = dₒdᵢ/(dₒ+dᵢ); or invert after adding 1/dₒ + 1/dᵢ.
- 3. dᵢ = f dₒ/(dₒ − f) when dₒ > f.
- 4. Object at 2f → image at 2f for a converging lens.
Derivation
From similar triangles in a ray diagram:
- Lens / mirror equation:
\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} - Solve for focal length:
f = \frac{d_o d_i}{d_o + d_i}
Magnification |m| = dᵢ/dₒ (sign conventions vary by course).
Examples
Example: 30 cm & 60 cm
f = 20 cm.
Example: equal distances
dₒ = dᵢ → f = d/2.