Lens Equation — 1/f = 1/do + 1/di

The lens equation 1/f = 1/do + 1/di explained: focal length, object and image distances, sign conventions, and a worked example.

\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

The reciprocals of image distance and object distance add up to the reciprocal of the focal length.

What it means

The lens equation links three distances: where the object is, where the image lands, and the lens's focal length. It says a lens of fixed power trades object distance against image distance — move the object closer and the image pushes farther out. The sign convention carries the physics: a positive image distance means a real image you can project on a screen; a negative one means a virtual image on the same side as the object, like a magnifying glass view. Camera autofocus is this equation running in reverse, shifting the lens until the sensor sits exactly at di.

Variables

Symbol	Name	Unit	Meaning
$f$	Focal length	m or cm	Focal length of the lens (positive for converging)
$do$	Object distance	m or cm	Distance from object to lens
$di$	Image distance	m or cm	Distance from lens to image (positive = real image)

Assumptions

Thin lens (thickness negligible vs focal length)
Paraxial rays — rays close to the optical axis
Real-is-positive sign convention

Derivation

Applying Snell's law at each spherical surface of a thin lens and using the small-angle approximation gives the refraction at a single surface; combining two surfaces and taking the thin-lens limit yields $1/f = 1/d_o + 1/d_i$ .

Worked example

Problem

An object sits 30 cm in front of a converging lens of focal length 10 cm. Where does the image form?

Given:

do = 30 cm
f = 10 cm (converging lens)

Write the lens equation: $1/f = 1/d_o + 1/d_i$
Rearrange: $1/d_i = 1/f - 1/d_o = 1/10 - 1/30$
Common denominator: $1/d_i = 3/30 - 1/30 = 2/30 = 1/15$
Invert: $d_i = 15\ \mathrm{cm}$

di = 15 cm (real image, on the opposite side of the lens)

The image forms 15 cm past the lens — closer than the object, as expected for an object outside the focal point of a converging lens. The magnification m = −di/do = −0.5, so the image is real, inverted and half-sized.