Reflection by Spherical Mirrors

A spherical mirror is a portion of a reflecting sphere. The geometry of reflection at every point obeys the same law (angle of incidence equals angle of reflection), but because the normal direction changes from point to point across the surface, parallel rays converge (concave) or diverge (convex) after reflection.

Concept

Key terms:

• Pole (P): the geometric centre of the mirror.
• Centre of curvature (C): centre of the sphere from which the mirror is a part.
• Radius of curvature (R): $|PC|$.
• Principal axis: line through $P$ and $C$.
• Focus (F): point on the axis where paraxial rays parallel to the axis converge (concave) or appear to diverge from (convex).
• Focal length (f): $|PF|$.

Cartesian sign convention. Place the origin at the pole $P$ with the incident light travelling along $+x$. All distances measured against the direction of incident light are negative; heights below the axis are negative.

Consequences:

• For a concave mirror, $f< 0$ and $R< 0$.
• For a convex mirror, $f>0$ and $R>0$.
• A real object placed in front has $u< 0$.

Derivation

Relation $f = R/2$. Consider a paraxial ray $AB$ parallel to the axis striking a concave mirror at $B$. The normal at $B$ passes through $C$. Let $\angle ABC = \theta$. By the law of reflection the reflected ray makes the same angle with the normal, so $\angle FBC = \theta$ where $F$ is where the reflected ray meets the axis.

Triangle $FBC$ is isoceles ($\angle FBC = \angle FCB = \theta$), so $FB = FC$. In the paraxial limit $FB \approx FP$, and since $FC = PC - PF = R - f$,

$f = R - f \quad\Rightarrow\quad f = \frac{R}{2}$

Mirror formula. With the same paraxial geometry, let an object be at distance $|u|$ from $P$ on the axis and its image form at distance $|v|$. From similar triangles (object/image with the principal ray and the chief ray through $F$),

$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

Magnification.

$m = \frac{h'}{h} = -\frac{v}{u}$

A negative $m$ means an inverted image; $|m|>1$ means magnified.

Worked Example

A 2 cm object is placed 15 cm in front of a concave mirror of focal length 10 cm. Locate the image and find its height.

Sign convention: $u = -15$ cm, $f = -10$ cm.

$\frac{1}{v} = \frac{1}{f} - \frac{1}{u} = \frac{1}{-10} - \frac{1}{-15} = -\frac{1}{10} + \frac{1}{15} = -\frac{1}{30}$

So $v = -30$ cm: a real image 30 cm in front of the mirror.

$m = -\frac{v}{u} = -\frac{-30}{-15} = -2$

Image height $= -2 \times 2 = -4$ cm (inverted, magnified).

Common Confusions

• Forgetting that $u$ is negative for real objects. Writing $u = +15$ cm gives the wrong sign for $v$.
• Treating $f$ as positive for a concave mirror. Always assign the sign first, then substitute.
• Confusing magnification sign: negative magnification means inverted (real image, for a single mirror); positive means erect (virtual image).

Key Takeaways

• $f = R/2$ in the paraxial approximation.
• Cartesian sign convention: distances measured opposite to incident light are negative.
• Mirror formula $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$ holds for both mirror types with signs included.
• Magnification $m = -v/u$.