Two thin lenses of focal lengths f₁ and f₂ are kept in contact. Derive the focal length of the combination and write its power.

1. 1 · State the idea. The image formed by the first lens acts as the object for the second lens.
2. 2 · Draw the diagram. [Two thin convex lenses A and B in contact; object O; ray through A heading for I₁ (dotted); bent by B to meet the axis at I; label u, v₁ and v]
3. 3 · Lens A. 1/v₁ − 1/u = 1/f₁ ...(1)
4. 4 · Lens B. The lenses are thin and in contact, so I₁ is at v₁ from B: 1/v − 1/v₁ = 1/f₂ ...(2)
5. 5 · Add and compare.
   • 1/v − 1/u = 1/f₁ + 1/f₂
   • For one equivalent lens, 1/v − 1/u = 1/f
   • So 1/f = 1/f₁ + 1/f₂, and P = P₁ + P₂
6. 6 · Real life. Cameras, microscopes and telescopes combine lenses to get the power and sharpness they need.
7. 7 · Close in one line. For thin lenses in contact, powers add and magnifications multiply.

Scoring tip: State clearly that the lenses are thin and in contact. That is why the same v₁ appears in both equations.