Reflection by Spherical Mirrors

Geometry of a Spherical Mirror

A spherical mirror is part of a reflecting sphere. The geometric centre of the surface is its pole (P), the centre of the sphere of which the mirror is a part is the centre of curvature (C), and the radius of that sphere is the radius of curvature (R). The line joining P and C is the principal axis. A mirror whose reflecting surface curves inwards is concave; one that curves outwards is convex.

For a curved surface the normal at the point of incidence is taken along the radius — the line joining C to that point. The laws of reflection (equal angles, coplanar rays) still apply, because a spherical mirror behaves as a large number of tiny plane mirrors. Throughout NEET problems we restrict attention to small-aperture mirrors and to paraxial rays, those incident close to the pole and making small angles with the axis.

Cartesian Sign Convention

Before any formula can be used safely, distances must be signed consistently. NCERT §9.2.1 and NIOS §20.1.2 both adopt the Cartesian sign convention: all distances are measured from the pole, the object is placed on the left so incident light travels left to right, distances measured along the incident light (to the right of P) are positive, and those opposite to it (to the left of P) are negative. Heights above the principal axis are positive; heights below are negative.

The consequence for focal lengths is fixed and worth memorising: because the focus of a concave mirror lies in front of it (the negative side, same side as the object), $f$ and $R$ are negative for a concave mirror; because the focus of a convex mirror lies behind it (the positive side), $f$ and $R$ are positive for a convex mirror.

Focal Length: f = R/2

Consider a ray parallel to the principal axis striking a concave mirror at M, with C its centre of curvature (Fig. 3). The radius CM is the normal, so the reflected ray crosses the axis at the focus F. Drop MD perpendicular to the axis. By the geometry, $\angle MCP = \theta$ (angle of incidence at M) and the reflected ray makes $\angle MFP = 2\theta$ with the axis.

Then $\tan\theta = \dfrac{MD}{CD}$ and $\tan 2\theta = \dfrac{MD}{FD}$. For paraxial rays $\theta$ is small, so $\tan\theta \approx \theta$ and $\tan 2\theta \approx 2\theta$, giving

$$\frac{MD}{FD} = 2\,\frac{MD}{CD} \quad\Rightarrow\quad FD = \frac{CD}{2}.$$

For paraxial rays the foot D lies very close to the pole P, so $FD = f$ and $CD = R$. Therefore

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

The focal length of a spherical mirror is half its radius of curvature, with the sign carried from the convention above. This relation depends only on geometry, not on the medium, which is why a mirror's focal length does not change when it is immersed in water.

Ray Diagrams and Image Formation

To locate an image, trace any two of four convenient rays from an object point. Their intersection (or the intersection of their backward extensions) marks the image point. An image is real if the reflected rays actually converge; it is virtual if they only appear to diverge from a point when produced backwards.

Drawing three rays does not imply only three rays leave a point; infinitely many do, in all directions. Point $A'$ is the image of $A$ precisely because every ray from $A$ that strikes the mirror passes through $A'$ after reflection. This is why covering part of a mirror reduces brightness but never removes part of the image.

The Mirror Equation

Take a real, inverted image $A'B'$ of an object $AB$ formed by a concave mirror (Fig. 4). The right triangles $A'B'F$ and $MPF$ are similar, and (since $\angle APB = \angle A'PB'$) so are $A'B'P$ and $ABP$. Equating ratios of magnitudes gives

$$\frac{B'P - FP}{FP} = \frac{B'P}{BP}.$$

Now apply the sign convention. Light travels from object to mirror, the positive direction; the object, image and focus are all reached by moving opposite to the incident light, so $B'P = -v$, $FP = -f$, $BP = -u$. Substituting and simplifying:

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

This is the mirror equation. Although derived for a real image in a concave mirror, with proper signs it holds for every case — concave or convex, real or virtual image.

Magnification

The linear (transverse) magnification is the ratio of image height $h'$ to object height $h$, with heights signed by the convention. From the similar triangles $A'B'P$ and $ABP$, applying the signs gives

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

The sign of $m$ carries physical meaning. A negative $m$ marks a real, inverted image (object and image on the same negative side, but $h'$ opposite in sign to $h$). A positive $m$ marks a virtual, erect image. The magnitude tells whether the image is enlarged ($|m|>1$), the same size ($|m|=1$) or diminished ($|m|<1$). A convex mirror always gives a virtual, erect, diminished image, so $0 < m < 1$ for it — the reason convex mirrors are used as wide-field side-view mirrors.

Worked Examples