notesonly.in

**Worked Example** *Problem:* A double‑convex glass lens has a refractive index n = 1.5. The front surface has a radius of curvature R1 = +12 cm and the back surface has a radius of curvature R2 = –12 cm. The lens is thin. Find the focal length f. *Step 1 – Identify the formula.* We use the thin‑lens Lensmaker’s formula: 1/f = (n – 1)(1/R1 – 1/R2). *Step 2 – Plug in the numbers.* n – 1 = 1.5 – 1 = 0.5. 1/R1 = 1/12 cm⁻¹. 1/R2 = 1/(–12 cm) = –1/12 cm⁻¹. So 1/R1 – 1/R2 = 1/12 – (–1/12) = 2/12 = 1/6 cm⁻¹. *Step 3 – Calculate 1/f.* 1/f = 0.5 × (1/6) = 0.5 × 0.1667 = 0.0833 cm⁻¹. *Step 4 – Find f.* f = 1 / 0.0833 ≈ 12 cm. *Answer:* The focal length of the lens is about 12 cm. This means light rays that pass through the lens will converge at a point 12 cm behind the lens. *Check:* Since both surfaces bulge outwards, the lens is converging and should have a positive focal length, which it does.