Power and combination of lenses

Power of a lens

Narrated video. Transcript:

An optician’s prescription does not say “focal length 40 centimetres”. It says “minus 2.5”. That number is the power of the lens.

Send a ray parallel to the axis into a convex lens, at a height h above the optical centre. It bends and crosses the axis at the focus. The angle it turns through is δ.

From the triangle, tan δ = h/f. For small angles, δ is about h/f. So a lens with a shorter focal length bends the ray more: it is more powerful.

The power of a lens is defined as P = 1/f. When f is in metres, P is in dioptres, written D. One dioptre is the power of a lens of focal length one metre.

A convex lens of focal length 0.5 metre has power 1 over 0.5, which is +2 dioptres. A convex lens of 25 centimetres, 0.25 metre, has power +4 dioptres.

A converging lens has positive power. A diverging lens has negative power. So that prescription of −2.5 dioptres is a concave lens.

Its focal length is 1 over −2.5, which is −0.4 metre, or −40 centimetres.

Remember to convert to metres first. A 20 centimetre lens has power 1 over 0.2, which is 5 dioptres, not 1 over 20.

Lenses in combination

Narrated video. Transcript:

Now place two thin lenses A and B in contact, with focal lengths f₁ and f₂. An object O is in front of them.

Lens A alone would form an image I₁ at v₁: 1/v₁ − 1/u = 1/f₁. But before the light gets there, it passes through lens B.

For lens B, I₁ is the object. Since the lenses are thin and touching, its distance from B is also v₁. B forms the final image I at v: 1/v − 1/v₁ = 1/f₂.

Add the two equations. 1/v₁ cancels, leaving 1/v − 1/u = 1/f₁ + 1/f₂. The pair behaves like a single lens with 1/f = 1/f₁ + 1/f₂.

In terms of power, this is simply P = P₁ + P₂. For several lenses in contact, just add up the powers, with their signs.

Example: a +5 dioptre lens touches a −2 dioptre lens. Together P = +3 dioptres, a converging lens of focal length one-third of a metre, about 33 centimetres.

Put an object 60 centimetres in front. In metres, 1/v = P + 1/u = 3 − 1/0.6 = 3 − 1.67 = 1.33. So v = 0.75 metre, 75 centimetres behind the lenses.

If the lenses are apart, follow the light one lens at a time: the image by the first lens is the object for the second. The total magnification is m = m₁ × m₂.

Camera and microscope lenses are combinations like this. Combining lenses gives the power we need and also reduces the defects of a single lens.

Try it: combine two lenses

Interactive simulation. Things to try:

• With d = 0, compare the final image v₂ with the last readout, made by one lens of power P₁ + P₂. They agree.
• Now increase d. The lenses no longer act like P₁ + P₂; follow I₁ (from L₁) to the final image.
• Check that m = m₁ × m₂ in the readouts at any separation.
• Make P₂ = −P₁ with d = 0. The pair has zero power, like a flat sheet of glass.

Power from focal length

What is the power of a concave lens of focal length 25 cm?

1. +4 D
2. −4 D
3. −0.04 D
4. −25 D

Two lenses in contact

A convex lens of focal length 20 cm is placed in contact with a concave lens of focal length 40 cm. What is the focal length of the combination?

1. +40 cm (converging)
2. −40 cm (diverging)
3. +13.3 cm (converging)
4. +60 cm (converging)

Summary

• Power: P = 1/f (f in metres, P in dioptres D)
• Lenses in contact: 1/f = 1/f₁ + 1/f₂ + …, P = P₁ + P₂ + …
• Total magnification: m = m₁ m₂ m₃ …

• Power measures how strongly a lens bends light: shorter focal length, larger power.
• Converging (convex) lenses have positive power; diverging (concave) lenses have negative power.
• Thin lenses in contact act as one lens whose power is the sum of their powers.
• For separated lenses, the image formed by one lens is the object for the next.