Lenses

Thin lens formula

For a thin lens with the Cartesian sign convention: 1/v – 1/u = 1/f Magnification m = h′/h = v/u. A convex (converging) lens has positive f; a concave (diverging) lens has negative f.

Lens maker's formula

1/f = (n₂₁ – 1)(1/R₁ – 1/R₂) where n₂₁ is the refractive index of the lens material relative to the surrounding medium and R₁, R₂ are the radii of the first and second surfaces (with signs). For an equiconvex lens with R₁ = +20 cm, R₂ = –20 cm and n = 1.5: 1/f = 0.5 × (1/20 + 1/20), so f = 20 cm.

If the lens is placed in a liquid, n₂₁ becomes smaller, so f increases. A glass lens (n = 1.5) of focal length 20 cm in air has focal length 80 cm in water (n = 4/3), because (1.5 – 1) ÷ (1.5/(4/3) – 1) = 4. If the liquid has the same index as the lens, the lens behaves like a plane sheet.

Power

Power P = 1/f, with f in metres; unit dioptre (D). A converging lens has positive power and a diverging lens negative power. A lens of f = –25 cm has P = –4 D.

Combination of lenses

For thin lenses in contact: 1/f = 1/f₁ + 1/f₂, so P = P₁ + P₂. For lenses one after the other, the total magnification is m = m₁ × m₂. Lenses of +5 D and –3 D in contact give +2 D, so f = 50 cm.

Worked example

A convex lens of focal length 20 cm forms an image of an object at u = –30 cm. 1/v = 1/f + 1/u = 1/20 – 1/30 = 1/60, so v = +60 cm (real, on the other side). m = v/u = –2: inverted and twice the size.

Key points

• 1/v – 1/u = 1/f; m = v/u.
• 1/f = (n – 1)(1/R₁ – 1/R₂).
• P = 1/f (metres), in dioptres.
• Lenses in contact: powers add.
• A lens in a denser liquid has a longer focal length.

Common mistakes

• Using f in centimetres when calculating power.
• Forgetting that R₂ is negative for a biconvex lens.