The thin lens equation
For a thin lens, the object distance, image distance and focal length are linked by:
1/f = 1/dₒ + 1/dᵢ
Solved for the image distance: dᵢ = f × dₒ ÷ (dₒ − f).
The magnification is m = −dᵢ ÷ dₒ, and the image height is m × object height.
Sign conventions
This calculator uses the common “real is positive” convention:
- f is positive for a converging (convex) lens, negative for a diverging (concave) lens.
- dₒ is positive for a real object in front of the lens.
- dᵢ is positive for a real image on the other side of the lens, negative for a virtual image on the object side.
- m negative means the image is inverted; positive means upright.
Worked example
An object 2 cm tall stands 30 cm in front of a converging lens with a 10 cm focal length.
- Image distance: 10 × 30 ÷ (30 − 10) = 15 cm, a real image behind the lens.
- Magnification: −15 ÷ 30 = −0.5, an inverted image half the size.
- Image height: −0.5 × 2 = −1 cm.
- Lens power: 1 ÷ 0.10 m = 10 diopters.
Where the image forms with a converging lens
| Object position | Image | Example |
|---|---|---|
| beyond 2f | real, inverted, smaller | camera |
| at 2f | real, inverted, same size | photocopier |
| between f and 2f | real, inverted, larger | projector |
| at f | no image (rays parallel) | searchlight |
| inside f | virtual, upright, larger | magnifying glass |
A diverging lens always gives a virtual, upright, smaller image, as in glasses for nearsightedness.
Lens power and eyeglasses
Lens power in diopters is 1 ÷ f, with f in meters. A prescription of −2.00 D is a diverging lens with a focal length of −50 cm. For thin lenses in contact, powers add, which makes diopters convenient for optometrists.
Limits
The thin lens equation assumes a lens much thinner than its focal length and rays close to the axis. Real camera lenses have several elements and aberrations, and thick lenses need their principal planes to apply the same equation.