Thin lenses and the lens maker’s formula

The lens maker’s formula

Narrated video. Transcript:

A thin lens is a piece of glass with two spherical surfaces, so close together that we can ignore its thickness. Light refracts twice: once going in, once coming out.

Call the radius of the first surface R₁ and of the second R₂. The glass has index n₂ and the surroundings n₁. An object sits at distance u on the left.

The first surface alone would form an image I₁ at distance v₁ inside the glass. From the single-surface result: n₂/v₁ − n₁/u = (n₂ − n₁)/R₁.

The light never reaches I₁, because it meets the second surface first. For that surface I₁ acts as the object, and the light goes from glass back into the surroundings: n₁/v − n₂/v₁ = (n₁ − n₂)/R₂.

Add the two equations. The n₂/v₁ terms cancel, leaving n₁/v − n₁/u = (n₂ − n₁)(1/R₁ − 1/R₂).

For an object at infinity, the image forms at the focus, v = f. That gives the lens maker’s formula: 1/f = (n₂₁ − 1)(1/R₁ − 1/R₂), where n₂₁ = n₂/n₁.

Example: a biconvex lens of glass, n = 1.5, has both radii 20 centimetres. With signs, R₁ = +20 and R₂ = −20. So 1/f = 0.5 × (1/20 + 1/20) = 0.5 × 1/10. f = +20 centimetres.

Now put the same lens in water, n = 1.33. n₂₁ becomes 1.5 ÷ 1.33 = 1.128, so n₂₁ − 1 is only 0.128. The focal length grows to 20 × 0.5 ÷ 0.128, about 78 centimetres.

The lens bends light much less in water, because the glass and the water have nearly the same refractive index. If they were equal, the lens would vanish optically.

The thin lens formula

Narrated video. Transcript:

Put the lens maker’s formula back into our sum and divide by n₁. We get the thin lens formula: 1/v − 1/u = 1/f. Note the minus sign; the mirror equation had a plus.

A convex lens has f positive: it is converging. A concave lens has f negative: it is diverging. Each lens has two foci, F₁ and F₂, one on each side, at equal distances from the optical centre O.

Three easy rays locate an image. A ray parallel to the axis passes through F₂, or seems to come from F₁ for a concave lens. A ray through the optical centre goes straight on.

And a ray through F₁, or heading towards F₂ for a concave lens, comes out parallel to the axis.

Example: a 2 centimetre object is 25 centimetres from a convex lens of focal length 15 centimetres. 1/v = 1/f + 1/u = 1/15 − 1/25 = 2/75. So v = +37.5 centimetres.

Positive v: the image is on the far side, real. The magnification of a lens is m = v/u = 37.5 ÷ −25 = −1.5. The image is inverted and 3 centimetres tall.

Move the object inside the focus, and the rays leave the lens spreading out. The image is virtual, upright and magnified, on the same side as the object. That is a magnifying glass.

A concave lens, f = −15 centimetres, with the object at −30 centimetres: 1/v = −1/15 − 1/30 = −3/30. v = −10 centimetres, and m = −10 ÷ −30 = +1/3.

A concave lens always gives a virtual, upright, smaller image, on the same side as the object, just like a convex mirror.

Try it: images through a lens

Interactive simulation. Things to try:

• This is the example: f = +15 cm, u = −25 cm. Check v = +37.5 cm and m = −1.5.
• Drag the object to 2F₁ (u = −30 cm). Where is the image, and how big?
• Move the object inside F₁. The rays diverge and the image becomes virtual and upright.
• Switch to Concave. Now the image is always virtual, upright and smaller.

Try it: make your own lens

Interactive simulation. Things to try:

• Biconvex, n = 1.5, R = 20 cm gives f = +20 cm, as in the video.
• Change the shape to plano-convex. One flat face halves the power: f doubles to 40 cm.
• Tick “Lens in water”. The focal length jumps to about 78 cm.
• Make n smaller than 1.33 with the lens in water. The convex lens now diverges!

Convex lens image

An object is 30 cm from a convex lens of focal length 20 cm. Find the image distance and the magnification.

1. v = +60 cm, m = −2
2. v = +12 cm, m = −0.4
3. v = −60 cm, m = +2
4. v = +60 cm, m = +2

A lens under water

A biconvex glass lens (n = 1.5) has a focal length of 20 cm in air. What is its focal length in water (n = 1.33)?

1. About 15 cm
2. About 27 cm
3. About 78 cm
4. About −78 cm

Summary

• Lens maker’s formula: 1/f = (n₂₁ − 1)(1/R₁ − 1/R₂)
• Thin lens formula: 1/v − 1/u = 1/f
• Magnification: m = h′/h = v/u

• A thin lens = two refracting surfaces; the image formed by the first is the object for the second.
• Convex (converging) lens: f > 0. Concave (diverging) lens: f < 0.
• For a lens, v > 0 means a real image on the far side; v < 0 means a virtual image on the object’s side.
• Ray rules: parallel → through F₂; through O → undeviated; through F₁ → parallel.
• In a denser surrounding medium a lens is weaker, because n₂₁ − 1 is smaller.