The Thin-Lens Equation

A lens bends light rays so they meet again at a point, forming an image. Where that image forms depends on how far the object sits from the lens and how strongly the lens bends light, a property called its focal length. The thin-lens equation ties these three distances together, so knowing any two lets you find the third.

The formula

1/f = 1/dₒ + 1/dᵢ

  • f — focal length (m): how strongly the lens bends light — where parallel rays converge
  • dₒ — object distance (m): how far the object is from the lens
  • dᵢ — image distance (m): how far the image forms from the lens

Worked example

A converging lens has a focal length of 10 cm. An object is placed 30 cm in front of it. How far from the lens does the image form?

  • f = 10 cm
  • dₒ = 30 cm
  1. 1/dᵢ = 1/f − 1/dₒ
  2. 1/dᵢ = 1/10 cm − 1/30 cm = 1/15 cm

dᵢ = 15 cm

Test yourself

Using the same 10 cm focal-length lens, if the object is moved to 20 cm away, how far from the lens does the image form?
  • 10 cm
  • Correct answer: 20 cm
  • 30 cm

Right! 1/dᵢ = 1/f − 1/dₒ = 1/10 cm − 1/20 cm = 1/20 cm, so dᵢ = 20 cm.

For the original example (dₒ = 30 cm, dᵢ = 15 cm), what is the magnification, and what does it tell you about the image?
  • Correct answer: m = −0.5 — inverted and half size
  • m = 0.5 — upright and half size
  • m = −2 — inverted and double size

Yes — m = −dᵢ/dₒ = −15 cm / 30 cm = −0.5, so the image is upside down and half the object's height.

Where you see this

A projector throwing a sharp picture is this equation on a wall: too close or too far and the image blurs, because for a given lens and screen distance only one object distance focuses. Your own eye runs it nonstop — its lens changes shape to keep the image of whatever you look at focused on your retina.

Common mistakes

The first slip is treating image distance as a property of the lens alone — it moves with the object: with a 10 cm focal length, an object at 30 cm images at 15 cm, but slide it to 20 cm and the image moves to 20 cm, per 1/dᵢ = 1/f − 1/dₒ. The second is reading magnification loosely: m = −dᵢ/dₒ = −15/30 = −0.5 carries two facts in one number — the minus says the image is inverted, the 0.5 says half-size.

How it connects

A lens is refraction harnessed: two curved surfaces each bending rays by Snell's law, shaped so all the bends converge — and where they converge is what this equation predicts. The mirror equation, next, turns out to be the identical algebra with reflection in refraction's seat, one of the module's pleasant surprises.

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