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The lens equation

Waves, sound and light · level 7, lesson 3 of 6

A question to start

A camera lens has a focal length of 50 mm. To photograph a rock 2 m away, how far behind the lens must the sensor be?

The idea

For a thin lens of focal length \(f\), an object at distance \(u\) forms an image at distance \(v\), where

\[\frac{1}{f} = \frac{1}{u} + \frac{1}{v}.\]

(Here real distances are positive. A negative \(v\) means a virtual image, on the same side as the object.)

The magnification is the image height over the object height:

\[m = \frac{v}{u}.\]
Object distance \(u\) Image
Very far away at \(f\), tiny, upside down
Beyond \(2f\) between \(f\) and \(2f\), smaller, upside down
At \(2f\) at \(2f\), same size, upside down
Between \(f\) and \(2f\) beyond \(2f\), bigger, upside down
Closer than \(f\) virtual, upright, bigger

Worked examples

1. The camera. \(1/v = 1/0.050 - 1/2 = 20 - 0.5 = 19.5\), so \(v = 0.0513\) m, 51.3 mm. Magnification \(0.0513/2 = 0.026\): a 1 m rock makes a 26 mm image.

2. A projector. A slide 0.11 m from a lens of \(f = 0.10\) m: \(1/v = 10 - 9.09 = 0.91\), so \(v = 1.1\) m. Magnification \(1.1/0.11 = 10\).

3. A magnifier. An object 0.04 m from a lens of \(f = 0.05\) m: \(1/v = 20 - 25 = -5\), so \(v = -0.2\) m, virtual. Magnification \(0.2/0.04 = 5\).

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Try it

  1. A lens has \(f = 0.20\) m. An object is 0.60 m away. Find \(v\) and the magnification.
  2. Where must an object be placed to give a same-size image with a lens of \(f = 0.15\) m?
  3. A camera with \(f = 0.10\) m photographs a distant mountain. Where is the image?
  4. A lens of \(f = 0.10\) m makes a virtual image 0.30 m from the lens. How far away is the object?
Answers
  1. \(1/v = 5 - 1.667 = 3.333\): \(v = 0.30\) m; \(m = 0.30/0.60 = 0.5\).
  2. At \(2f = 0.30\) m.
  3. At the focal point, 0.10 m behind the lens.
  4. \(1/u = 1/0.10 - 1/(-0.30) = 10 + 3.33 = 13.33\): \(u = 0.075\) m.

Common mistake

Adding the distances instead of their reciprocals. The equation adds \(1/u\) and \(1/v\). Work out \(1/v\) first, then turn it upside down.

Where it leads