Spherical mirrors and conjugate points

A spherical mirror is a spherical cap with either the inner or outer surface reflecting. If the concavity faces the incident light the mirror is concave, otherwise it is convex. Every spherical mirror has three characteristic points: the vertex $V$, the centre of the cap; the centre of curvature $C$, the centre of the sphere; and the focus $F$, halfway between $V$ and $C$. The focal length is therefore

$f = \frac{R}{2}$

where $R$ is the radius of the sphere. The focus is the point where, after reflection, the rays arriving parallel to the optical axis converge.

To find graphically where the image of a point forms, three privileged rays are used, of which it is enough to draw two.

Principle — Ray construction (concave mirror)

• A ray parallel to the optical axis is reflected passing through the focus $F$.
• A ray passing through the focus $F$ is reflected parallel to the axis.
• A ray passing through the centre $C$ is reflected back on itself.

The intersection of (at least) two of these rays gives the position of the image.

The graphical construction translates into an exact algebraic relation. If $p$ is the object–mirror distance and $q$ the image–mirror distance, the conjugate points equation holds:

Key formula

$\ev{\dfrac{1}{p} + \dfrac{1}{q} = \dfrac{1}{f}}$ $I = -\dfrac{q}{p}$ Concave: $f > 0$; convex: $f < 0$. $q > 0$: real image; $q < 0$: virtual image.

The magnification $I = -q/p$ gives the size of the image relative to the object and its orientation: if $I$ is negative the image is inverted, if positive it is upright. The sign of $q$ instead distinguishes a real image — which forms in front of the mirror and can be captured on a screen — from a virtual one, which appears to come from behind the mirror.

Collegamenti

Argomenti: Ottica Concetti: Lenti e specchi · Riflessione

Esercizi collegati: Problema — Raggi per oggetto oltre 2f (lente) · Problema — Raggi per specchio concavo con p minore di f · Esercizio svolto — Specchio concavo