Reflection of parallel beams

A parallel beam is a bundle of light rays travelling in exactly the same direction, side by side — sunlight reaching Earth is the standard example, since the Sun is so far away that its rays arrive essentially parallel. What a mirror does to such a bundle depends entirely on the shape of its reflecting surface.

• A plane (flat) mirror has the same flat surface everywhere, so every ray strikes it and bounces off in the same way relative to the surface. The whole bundle changes direction together but stays parallel — it neither spreads out nor comes together.
• A concave mirror curves inward, like the inside of a bowl. Rays striking near the edges are tilted more than rays striking near the centre, and this extra tilt bends every ray inward, toward the mirror's axis, so the whole bundle converges, meeting at (roughly) one point called the focus.
• A convex mirror curves outward, like the back of a spoon. Here the extra tilt at the edges bends rays outward instead, so the whole bundle diverges, spreading apart as if the rays were coming from a point behind the mirror.

Every point on a mirror obeys the same underlying law of reflection: the reflected ray leaves at the same angle to the local surface normal (the line perpendicular to the surface at that point) as the incoming ray made with it.

where: $\theta_i$ is the angle of incidence and $\theta_r$ is the angle of reflection, both measured from the normal at the exact point the ray strikes. On a flat mirror the normal points the same way everywhere, so parallel rays all get deflected by the same amount and stay parallel. On a curved mirror the normal tilts by a different amount at every point, so parallel rays get bent by different amounts — inward for a concave surface, outward for a convex one — which is the entire reason curvature turns a parallel beam into a converging or diverging one.

There is no focal-length formula to derive at this level — the reasoning behind converging and diverging beams rests entirely on how a curved surface changes the direction of its normal from point to point, and it is worth walking through carefully because it is the single idea behind everything else in this article and the one exam-writers probe hardest.

Step 1. Picture a parallel beam as several rays, side by side, all pointing in the same direction — say, straight down onto a mirror lying flat below them. At every point where a ray strikes any mirror, the law of reflection applies locally: the reflected ray makes the same angle with the surface normal at that point as the incoming ray did.

Step 2. On a plane mirror, the surface is flat everywhere, so the normal points in exactly the same direction at every point on the mirror. Since every incoming ray meets an identically oriented normal, every ray is deflected by exactly the same amount. The rays that started parallel are still parallel after reflecting — the whole bundle has simply been turned around, like a block moving as one piece.

Step 3. On a concave mirror, the surface curves inward like a bowl, so the normal tilts more and more as you move away from the centre of the mirror toward its edges — near the edges, the normal points noticeably inward, toward the mirror's central axis. A ray striking near the edge therefore gets reflected at a steeper inward angle than a ray striking near the centre. The result is that rays which entered perfectly parallel leave the mirror tilted toward each other by different amounts depending on where they struck, and they cross and meet at (approximately) a single point in front of the mirror — the beam has converged.

Step 4. On a convex mirror, the curve bulges outward, so the normal at the edges tilts the opposite way, outward, away from the axis. Rays striking near the edge are reflected outward more strongly than rays near the centre, so the whole bundle spreads apart after reflecting instead of narrowing — the beam has diverged, and if you trace the reflected rays backward, they appear to spread out from a single point behind the mirror. The whole story, then, is that curvature makes the normal direction vary across the mirror's surface, and that varying normal is what turns a parallel beam into a converging or a diverging one; a flat surface has no such variation, so it leaves the beam exactly as parallel as it found it.

Question: A student sets up three identical mirrors side by side — one plane, one concave, one convex — all facing the Sun on a clear day, and holds a sheet of white paper at the same fixed distance in front of each. Predict what she will see on the paper in each case, and explain which single mirror could realistically be used (very carefully, with adult supervision) to make the paper begin to smoke.

Step 1. In front of the plane mirror, the reflected sunlight lands on the paper as a bright patch roughly the same size as the mirror itself, because the parallel rays of sunlight remain parallel after reflecting — the patch is simply the beam redirected, not concentrated or spread.

Step 2. In front of the concave mirror, the reflected rays are converging toward a focus. If the paper happens to sit close to where that focus falls, the sunlight lands as a small, intensely bright spot, because all the energy that struck the whole curved mirror is now being funnelled into a tiny area.

Step 3. In front of the convex mirror, the reflected rays are diverging, so the patch of light on the paper is larger than the mirror itself and noticeably dimmer, because the same amount of energy that struck the mirror is now spread over a wider area.

Step 4. Concentrating a fixed amount of sunlight energy into a smaller area raises the intensity (energy delivered per unit area) at that spot; spreading it over a larger area lowers the intensity. Only the concave mirror shrinks the illuminated area below what fell on the mirror, so only the concave mirror can raise the intensity enough to heat the paper to smoking point. The concave mirror, positioned so the paper sits at its bright focus point, is the one that can make the paper smoke — the plane mirror only redirects the light, and the convex mirror spreads it out and makes it weaker, never stronger.

Competitive stretch

If the same three mirrors were instead used to widen a narrow torch beam so it could light up a whole room evenly, the convex mirror would now be the useful choice, since its diverging reflection is exactly what is needed to spread a beam out — the same physics that makes it unsuitable for concentrating sunlight makes it ideal for wide-angle illumination, which is why convex mirrors are used in the wide security mirrors at shop corners and in vehicle side mirrors.

• Mixing up converge and diverge — converge means the rays are pulled together (concave mirror); diverge means the rays are pushed apart (convex mirror). Say the word out loud with the mirror's shape in mind to check it makes sense.
• Assuming any curved mirror concentrates light — only a concave (inward-curving) mirror converges parallel rays to a point; a convex (outward-curving) mirror always spreads them out, never concentrates them.
• Thinking a plane mirror does nothing to a parallel beam — it does change the beam's direction as a whole, it simply does not make the rays converge or diverge, because its normal does not vary across the surface.
• Forgetting the safety rule when discussing a concave mirror focusing sunlight — never look directly at the Sun, at a concave mirror pointed at the Sun, or at the bright concentrated spot it produces.

Ask the tutor: Want to see how this same converging idea explains why a concave mirror also makes a magnified image when you hold your face close to it?