Q.Derive the relation between $f$ and R for a spherical mirror.

Using the small-angle (paraxial) geometry of a ray reflecting near the pole of a spherical mirror, the relation $2/R=1/u+1/v$ follows, and setting the object at infinity ($v=f$) gives $f=R/2$.

Working

Consider a concave mirror with pole $P$, centre of curvature $C$ (radius $R=PC$), forming an image $I$ of an object $O$ on the principal axis, using a paraxial ray $OM$ striking the mirror at a point $M$ close to the pole and reflecting to $I$ (normal at $M$ is along $MC$).

Let $\alpha,\beta,\gamma$ be the (small) angles that $OM$, $CM$, $IM$ respectively make with the principal axis, and let $\theta$ be the common angle of incidence/reflection at $M$ (law of reflection).

Using the exterior-angle property of a triangle:

• In $\triangle OMC$: $\beta=\alpha+\theta\ \Rightarrow\ \theta=\beta-\alpha$
• In $\triangle CMI$: $\gamma=\beta+\theta\ \Rightarrow\ \theta=\gamma-\beta$

Equating the two expressions for $\theta$:

$\beta-\alpha=\gamma-\beta\ \Rightarrow\ 2\beta=\alpha+\gamma$

Since $M$ is close to the pole, for a ray at height $h$ above the axis, the small angles are (paraxial approximation, $\tan\theta\approx\theta$): …