Introduction & Reflection by Spherical Mirrors

Light as Rays, Mirrors as Geometry

Light is an electromagnetic wave (Chapter 8) — but when it travels around objects much larger than its wavelength (a few hundred nanometres), it behaves as if moving in straight lines: rays. Ray optics is the geometry of light.

The laws of reflection hold at every point of any surface, plane or curved:

1. Angle of reflection = angle of incidence (both measured from the normal).
2. Incident ray, reflected ray and normal lie in one plane.

For a spherical mirror, the normal at any point runs along the radius — the line from the centre of curvature C to the point. The geometric centre of the mirror is its pole P; the line joining P and C is the principal axis.

The Cartesian Sign Convention (Learn It Once, Use It Everywhere)

Every formula in this chapter assumes the Cartesian sign convention:

1. All distances are measured from the pole of the mirror (or the optical centre of a lens).
2. Distances along the direction of incident light are positive; distances against it are negative.
3. Heights above the principal axis are positive; below, negative.

With this one convention, a single mirror formula and a single lens formula handle every case — concave or convex, real or virtual.

Immediate consequences (with light incident from the left):

• Object distance u is negative (object sits left of the pole).
• Concave mirror: f and R negative; convex mirror: f and R positive.
• A real image (left of a mirror) has v negative; a virtual image (behind the mirror) has v positive.

[NEET Important] Most lost marks in ray optics are sign errors. Write u, v, f, R with signs before substituting — every time.

The Principal Focus and f = R/2

Send a paraxial beam (rays close to the pole, making small angles with the axis) parallel to the principal axis:

• Concave mirror: reflected rays converge to a point F on the axis.
• Convex mirror: reflected rays appear to diverge from a point F behind the mirror.

F is the principal focus; the pole-to-focus distance is the focal length f. (A parallel beam at a small angle to the axis converges in the focal plane through F.)

Why $f = R/2$: consider a parallel ray hitting the mirror at M. The radius CM is the normal, so the ray reflects making equal angles $\theta$ with CM. Geometry gives $\angle$MCP $= \theta$ and $\angle$MFP $= 2\theta$; for small angles, $FD = MD/2\theta$ and $CD = MD/\theta$, so $FD = CD/2$ — for paraxial rays D approaches P, hence

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

Key Point: $f = R/2$ holds for both mirror types (with signs: concave $R<0 \Rightarrow f<0$; convex $R>0 \Rightarrow f>0$) and only for paraxial rays — wide beams suffer blurring (spherical aberration, the reason for parabolic mirrors in telescopes).

Solved Examples

Example 1: Focal lengths with signs [NEET Numerical]

Write the focal lengths (with signs) of (a) a concave mirror of R = 20 cm, (b) a convex mirror of R = 32 cm.

Solution:

1. $f = R/2$ in magnitude; signs from the convention.
2. (a) Concave: $f = -20/2 = -10$ cm.
3. (b) Convex: $f = +32/2 = +16$ cm.

Example 2: The side-view mirror's focus [NEET Numerical]

A car's side-view (convex) mirror has R = 2 m. Where is its focus, and can sunlight be concentrated by it?

Solution:

1. $f = +R/2 = +1$ m — the focus lies 1 m behind the mirror.
2. Reflected sunlight only appears to diverge from this virtual focus; it is never actually concentrated.
3. A concave mirror (real focus) is what burns paper in sunlight.

Example 3: Normal along the radius

Why is the normal to a spherical mirror at any point taken along the radius through that point?

Solution:

1. The normal must be perpendicular to the tangent plane of the surface at the point of incidence.
2. For a sphere, every radius is perpendicular to the surface where it meets it.
3. So the line from the centre of curvature C to the incidence point is the normal — which is why C anchors all mirror geometry.

Example 4: Paraxial means… [JEE pattern]

State the paraxial approximation and why mirror formulas need it.

Solution:

1. Rays incident close to the pole, making small angles with the principal axis (so $\tan\theta \approx \theta$).
2. Only such rays reflect through a single sharp focus; marginal (wide) rays cross the axis at slightly different points.
3. All formulas of this chapter — $f = R/2$, the mirror and lens equations — are paraxial results.

Example 5: Sign-convention drill [NEET Numerical]

An object stands 25 cm in front of a concave mirror of focal length 10 cm. Write u and f with correct signs (do not solve yet).

Solution:

1. Light travels from the object towards the mirror; distances against the incident light are negative.
2. $u = -25$ cm (object on the incident side).
3. $f = -10$ cm (concave focus on the incident side). The actual image hunt happens in Section 2.

Example 6: Focal plane in action [JEE Numerical]

A parallel beam strikes a concave mirror ($f = -20$ cm) at a small angle of 0.1 rad to the principal axis. Where does it converge?

Solution:

1. A tilted parallel beam converges in the focal plane — the plane through F perpendicular to the axis.
2. Distance from the axis: $y = |f|\theta = 20 \times 0.1 = 2$ cm.
3. Answer: 20 cm in front of the mirror, 2 cm off the axis.

Example 7: Concave vs convex at a glance

List the sign pattern (u, f, R) for the two mirror types with a real object.

Solution:

1. Always: $u < 0$ for a real object (it sits against the incident direction).
2. Concave: $R < 0$, $f < 0$ — real focus in front.
3. Convex: $R > 0$, $f > 0$ — virtual focus behind. Memorise the pattern; it removes half the errors of this chapter.

Example 8: Why curved surfaces still obey plane laws

Justify using the plane-mirror laws of reflection for spherical mirrors.

Solution:

1. The laws of reflection are local — they hold at each point of any reflecting surface.
2. At each point, the curved mirror behaves like the tiny tangent plane there, with the normal along the radius.
3. The curvature only changes the normal's direction from point to point — which is exactly what focuses the beam.

Example 9: R from f [NEET Numerical]

A shaving (concave) mirror must have its focus 25 cm from the pole. What radius of curvature should the maker grind?

Solution:

1. $|R| = 2|f| = 50$ cm.
2. With signs: $f = -25$ cm, $R = -50$ cm.
3. The centre of curvature sits 50 cm in front of the mirror — twice as far as the focus, always.

Example 10: One convention, all cases

What does NCERT mean by 'a single formula handles all cases'?

Solution:

1. With the Cartesian convention, $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$ (Section 2) works for concave AND convex mirrors, real AND virtual images.
2. You never memorise case-wise rules; the signs carry the geometry.
3. The same economy repeats for lenses with $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$.