Lenses

A lens refracts light at two curved surfaces. A convex (converging) lens has a positive focal length; a concave (diverging) lens a negative one. The same sign convention as for mirrors applies, measuring from the optical centre.

Lens formula: 1/v – 1/u = 1/f. Note the minus sign: it differs from the mirror formula. Magnification: m = v/u (no minus sign).

Example: convex lens, f = 20 cm, object at u = –30 cm. 1/v = 1/20 – 1/30 = 1/60, so v = +60 cm (real, on the other side). m = 60/(–30) = –2: inverted and twice as big.

A concave lens always gives a virtual, erect, smaller image of a real object.

Lensmaker's formula: 1/f = (n – 1)(1/R₁ – 1/R₂), where n is the lens's index relative to the surroundings. For a glass lens (n = 1.5) with R₁ = +20 cm and R₂ = –20 cm: 1/f = 0.5 × (2/20), so f = 20 cm.

In a liquid, n – 1 becomes (n_lens/n_liquid) – 1, which is smaller, so f gets longer. If the liquid has the same index as the glass, the lens bends no light at all and seems to vanish.

Power of a lens: P = 1/f, with f in metres. Unit: dioptre (D). f = 25 cm gives +4 D; a –2 D lens is concave with f = –50 cm. Opticians write prescriptions in dioptres.

Lenses in contact: powers simply add, P = P₁ + P₂, or 1/F = 1/f₁ + 1/f₂.

For lenses apart, work one at a time: the image made by the first lens is the object for the second.