m = −dᵢ/dₒ — Lens Magnification

Lens magnification — image size from object and image distances — with a worked camera-lens example.

m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}

Magnification equals image height over object height, or negative image distance over object distance.

What it means

Magnification tells you how the image compares to the object: $|m| > 1$ enlarges, $|m| < 1$ shrinks, and the negative sign means inverted (real image). The ratio of distances fixes the ratio of heights — similar triangles doing the optics.

Variables

Symbol	Name	Unit	Meaning
$m$	Magnification	dimensionless	Image size relative to object (negative = inverted)
$h_i$	Image height	m or cm	Image height
$h_o$	Object height	m or cm	Object height
$d_i$	Image distance	m or cm	Lens-to-image distance
$d_o$	Object distance	m or cm	Lens-to-object distance

Assumptions

Thin lens (paraxial approximation)
Distances measured from the lens center
Sign convention: real image distances positive

Worked example

Problem

An object is 30 cm from a converging lens of focal length 10 cm. Find the image distance and magnification.

Given:

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

Lens equation: $1/f = 1/d_o + 1/d_i$ → $1/d_i = 1/10 − 1/30 = 2/30$
So $d_i = 15\ \mathrm{cm}$
Magnification: $m = -d_i/d_o = -15/30 = -0.5$

m = −0.5 (inverted, half size)

The image forms 15 cm behind the lens, inverted and half the object's height — exactly what a camera does: shrink the world onto a small sensor.