Lens formula and power

For lenses, all distances are measured from the optical centre O, with the same sign convention as mirrors. A convex lens has a positive focal length; a concave lens a negative one. The object distance u is negative.

Lens formula: 1/v – 1/u = 1/f

Magnification: m = h′/h = v/u

Compare with mirrors: the lens formula has a minus where the mirror formula has a plus, and the lens magnification has no minus sign. Read m the same way: the sign tells real or virtual, the size tells larger or smaller.

The book's example: a 2.0 cm object, 15 cm from a convex lens of focal length 10 cm. u = –15 cm, f = +10 cm, so 1/v = 1/10 – 1/15 = 1/30 and v = +30 cm. m = v/u = 30/(–15) = –2, so h′ = –4.0 cm. The image is real, inverted and twice the size, on the other side of the lens. Before calculating, could you have guessed? The object is between F₁ and 2F₁, so the image must be beyond 2F₂, real and larger.

Power tells how strongly a lens bends light. A lens that bends light more brings it to a focus closer, so it has a shorter focal length:

P = 1/f (f in metres)

The unit is the dioptre (D): 1 D is the power of a lens of focal length 1 m.

Convex lenses have positive power; concave lenses negative. +2.0 D means f = +0.50 m (convex); –2.5 D means f = –0.40 m (concave). An optician's prescription for spectacles is written in dioptres in exactly this way.

When lenses are placed in contact, their powers simply add: +2.0 D and +0.25 D make +2.25 D. Opticians use this to test eyes with a set of lenses.