Result

Mirror equation

(O2.1.1) O2.1 · Flat and Spherical Mirrors →

\frac{1}{d _{o}} + \frac{1}{d _{i}} = \frac{1}{f}, f = \frac{R}{2}

Statement

Relates object distance, image distance, and focal length for a spherical mirror; f is half the radius of curvature.

Symbols and units

$d_{o}, d_{i}$	object, image distance	m
$f$	focal length	m
$R$	radius of curvature	m

When it applies

Rays close to the mirror's axis (the paraxial approximation) on a spherical or flat mirror.

Don't use when

Assumes the paraxial approximation — rays close to the mirror's axis; rays far from the axis suffer spherical aberration and the equation becomes inaccurate.

Derivation

Follows from similar triangles formed by a ray diagram, relating object distance, image distance, and the mirror's radius of curvature.

\frac{1}{d _{o}} + \frac{1}{d _{i}} = \frac{1}{f}

Worked examples

The flat-mirror limit

As the radius of curvature R → ∞, so does f, and the mirror equation reduces to:

\frac{1}{d _{o}} + \frac{1}{d _{i}} = 0 \Rightarrow d_{i} = - d_{o}

The familiar result that a flat mirror's image sits as far behind the mirror as the object stands in front of it.

Watch out for

Sign convention (which side counts as positive for real versus virtual images) varies between sources — be consistent within a single problem, or a real image can come out looking virtual.

Connects to

Thin lens equation