Lenses and optical instruments

Lenses refract light to form images. A converging (convex) lens brings parallel rays together at its principal focus; a diverging (concave) lens spreads them out. Lenses are in cameras, glasses, microscopes, projectors and your own eyes. The lens equation and magnification tell you exactly where an image will form and how big it will be.

Key facts

  • Converging lens: thicker in the middle; parallel rays meet at the principal focus, a distance f from the lens.
  • Diverging lens: thinner in the middle; it makes rays spread as if from a focus.
  • Magnification = image height ÷ object height = v ÷ u.
  • Lens equation (real is positive): 1/f = 1/u + 1/v; a virtual image has negative v.
  • Power P = 1 ÷ f (dioptres, f in metres); converging +, diverging −. Powers of thin lenses in contact add.
  • Object beyond 2f: real, inverted, smaller (camera). Between f and 2f: real, inverted, bigger (projector). Inside f: virtual, upright, bigger (magnifying glass).

Worked examples

Example 1

An object is 30 cm from a converging lens of focal length 10 cm. Where is the image, and what is the magnification?

  1. 1/v = 1/f − 1/u 1/v = 1/10 − 1/30 = 3/30 − 1/30 = 2/30
  2. So v = 15 cm
  3. Magnification m = v ÷ u = 15 ÷ 30 = 0.5

Answer: v = 15 cm; magnification 0.5

Example 2

A lens has a power of +4 D. What is its focal length, and what kind of lens is it?

  1. Invert f = 1 ÷ 4 = 0.25 m = 25 cm
  2. Positive power: converging.

Answer: 25 cm, converging

Example 3

A magnifying glass (f = 12 cm) is held 8 cm from a stamp. Find the image distance and magnification.

  1. 1/v = 1/12 − 1/8 = 2/24 − 3/24 = −1/24
  2. So v = −24 cm: a virtual image on the same side as the stamp
  3. Magnification 24 ÷ 8 = 3

Answer: v = −24 cm (virtual); magnification 3