The Thin Lens Equation
The thin lens equation relates object distance, image distance, and focal length for converging (convex) and diverging (concave) lenses.
Thin Lens Formula
1/f = 1/d_o + 1/d_i
f = focal length (positive: converging, negative: diverging)
d_o = object distance (positive: object in front)
d_i = image distance (positive: real image; negative: virtual)
Magnification: m = -d_i / d_o = h_i / h_oWorked Examples
Converging lens (f=+20cm), object at 60cm:
1/d_i = 1/20 - 1/60 = 3/60 - 1/60 = 2/60
d_i = 30cm (real image, same side as object exit)
m = -30/60 = -0.5 (inverted, half-size)
Object at 10cm (inside focal length):
1/d_i = 1/20 - 1/10 = -1/20
d_i = -20cm (virtual, upright image)Sign Convention Summary
- d_i > 0: real image (can project on screen)
- d_i < 0: virtual image (can't project, as in magnifying glass)
- m > 0: upright | m < 0: inverted
- |m| > 1: magnified | |m| < 1: diminished
Calculate lens properties: Free Lens Calculator
Thin Lens Quick-Reference Table
| Object distance (cm) | Focal length (cm) | Image distance (cm) | Magnification | Image type |
|---|---|---|---|---|
| ∞ (far away) | 10 | 10 | 0 | Real, inverted |
| 30 | 10 | 15 | −0.5 | Real, inverted |
| 20 | 10 | 20 | −1.0 | Real, inverted |
| 15 | 10 | 30 | −2.0 | Real, inverted |
| 5 | 10 | −10 | +2.0 | Virtual, upright |
How the Thin Lens Equation Works
The thin lens equation: 1/f = 1/d_o + 1/d_i, where f is focal length, d_o is object distance, and d_i is image distance (all in the same unit). Magnification m = −d_i/d_o. Positive d_i indicates a real image (on the opposite side from the object); negative d_i indicates a virtual image (on the same side as the object). Converging (convex) lenses have positive f; diverging (concave) lenses have negative f.
Camera lenses focus images by adjusting d_o through physical element movement. The human eye focuses by changing its lens curvature (and thus focal length) — a process called accommodation. Reading glasses use positive lenses to compensate for reduced accommodation in older eyes. Microscopes and telescopes combine multiple lenses to achieve high magnification.
Common Mistakes
- Sign convention errors: Use the real-is-positive convention consistently: d_o and d_i positive for real objects/images, negative for virtual. Inconsistency produces wrong image positions.
- Confusing focal length with optical power: Optical power P = 1/f in dioptres (f in metres). A +2D lens has f = 0.5 m. Eyeglass prescriptions use dioptres, not focal lengths in cm.
- Ignoring lens thickness: The thin lens approximation fails for thick lenses and multi-element systems. Camera lens design uses ray-tracing software, not the simple formula.
Frequently Asked Questions
A −2.5D lens is diverging (concave), with f = −0.4 m. It is prescribed for myopia (near-sightedness), where the eye focuses images in front of the retina. The diverging lens spreads light before it enters the eye, shifting the focal point back onto the retina. Positive prescriptions (convex lenses) correct hyperopia (far-sightedness).
A single lens suffers from aberrations — chromatic (different wavelengths focus at different distances), spherical (outer rays focus closer than paraxial rays), and others. Modern lenses use 6–20+ elements in groups, with different glass types and curvatures that cancel each other's aberrations, delivering sharp images across the frame at various apertures and focal lengths.
A magnifying glass is a converging lens used with the object inside its focal length (d_o < f). This produces a virtual, upright, magnified image at the near point (~25 cm for the average eye). The angular magnification M = 25/f (cm), so a 5 cm focal length lens gives 5× magnification. Reducing f increases magnification but reduces the field of view and working distance.