Mirror Equation — Physica

Equivalent forms

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

m = -\frac{v}{u}

A single reciprocal sum predicts every reflection — telescopes, shaving mirrors, and car headlights all bow to it.

Symbol	Name	SI unit	Dimension	Range
$f$	Focal length Distance from mirror to focal point (positive for concave, negative for convex)	m	L	-1 – 1
$u$	Object distance Distance from mirror to object (positive in front of mirror)	m	L	0.01 – 2
$v$	Image distanceoutput Distance from mirror to image (positive if real, negative if virtual)	m	L	-2 – 2

Unit systems

Symbol	SI	CGS	Imperial
$f$	m	cm	in
$u$	m	cm	in
$v$	m	cm	in

Where it holds

Valid for paraxial rays (small angles, near axis) on spherical mirrors. Breaks down for marginal rays (spherical aberration) and aspheric mirrors.

Dimensional analysis

1/f \to [L^{-1}]

1/v + 1/u \to [L^{-1}] + [L^{-1}] = L^{-1}

Both sides have units of inverse length.

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Discovery

Carl Friedrich Gauss · 1841

Gauss formalized paraxial mirror and lens imaging in his Dioptrische Untersuchungen, unifying centuries of geometric optics into a clean reciprocal relation.

Try this

Why does the rearview mirror say 'objects may be closer than they appear'?

A concave mirror has focal length f = 0.10 m. An object sits at u = 0.30 m. Find the image distance v.

Research status: stable

Real-world applications

Car side mirrors: convex mirrors give wide field of view at the cost of distance distortion.
Reflecting telescopes: large concave primaries focus distant starlight (Newton, Cassegrain).
Dental and shaving mirrors: concave at short distance creates magnified upright virtual image.
Solar furnaces: parabolic concave mirrors concentrate sunlight at focal point.

Common misconceptions

Image distance can be measured the same way regardless of real/virtual — sign convention is essential.
Convex mirrors form real images — they never do; convex always produces virtual, upright, diminished images.
Focal length equals radius — $f = R/2$ , not R.

Experimental verification

Verified by classroom experiments: shining parallel light onto concave mirrors confirms

f = R/2

, and varying object distance yields predicted image distances within 2% for spherical mirrors of small aperture.

Derivation

Consider a ray from object point O at distance u, reflecting off a spherical mirror of radius R at point P, and reaching image point I at distance v.

Using the geometry of similar triangles in the paraxial limit and the law of reflection, plus the relation

f = R/2

, one derives

1/v + 1/u = 2/R = 1/f

.

The proof generalizes from a single ray and holds for all paraxial rays.

Limiting cases

u \to \infty

⟶

v \to f

Parallel rays from infinity converge at the focal point.

u \to f

⟶

v \to \infty

Object at focal point produces parallel reflected rays — image at infinity.

u < f (concave)⟶ v < 0Object inside focal length forms a virtual, upright, magnified image.

What if…

What if the mirror is parabolic?

All parallel axial rays focus exactly at f with zero spherical aberration — the basis of high-quality reflecting telescopes.

What if

u = f

exactly?

Reflected rays emerge parallel; image is at infinity. This is how lighthouse beams are formed.

What if the object is virtual (u < 0)?

Occurs in optical systems where a converging beam is intercepted by a mirror; image can be real on the same side as the incoming light.

1

Object beyond focal point of concave mirror

Given ·

f:: 0.1
u:: 0.3

Find · v

Steps

Write $1/f = 1/v + 1/u$
Rearrange: $1/v = 1/f - 1/u$
Substitute: $1/v = 10 - 3.333 = 6.667\,\mathrm{m}^{-1}$
$v = 0.15\,\mathrm{m}$ ; image is real and inverted
Magnification $m = -v/u = -0.5$ (half size)

Result ·

1/v = 1/f - 1/u = 1/0.10 - 1/0.30 = 10 - 3.333 = 6.667

;

v = 0.15\,\mathrm{m}

(real, inverted)

2

Convex mirror (car side-view)

Given ·

f:: -0.2
u:: 1

Find · v

Steps

Convex mirror: f < 0
$1/v = 1/f - 1/u = -5 - 1 = -6\,\mathrm{m}^{-1}$
$v = -0.167\,\mathrm{m}$ (negative ⇒ virtual)
$m = -v/u = 0.167$ (upright, diminished)

Result ·

1/v = 1/(-0.20) - 1/1.0 = -5 - 1 = -6

;

v = -0.167\,\mathrm{m}

(virtual, behind mirror)

Thin Lens Equation→Lensmaker's Equation→Law of Reflection→