Images in curved mirrors

Naming the geometry. Every spherical mirror is described using five landmarks. The centre of curvature ($C$) is the centre of the sphere the mirror is sliced from; the radius of curvature ($R$) is the distance from $C$ to the mirror. The pole ($P$) is the geometric centre of the mirror surface itself, and the principal axis is the straight line through $P$ and $C$, extended in both directions. The focus ($F$) is the point on that axis where rays travelling parallel to the axis meet after reflection (concave), or the point they appear to come from after reflection (convex); the focal length ($f$) is the distance from $P$ to $F$.

The one formula this topic needs. The syllabus states, without asking you to prove it, that the focus always sits exactly halfway between the pole and the centre of curvature:

where: $f$ is the focal length and $R$ is the radius of curvature, both measured in the same length unit (usually centimetres). This single relation is enough to answer almost every numerical question this topic sets — given either $R$ or $f$, you get the other by doubling or halving.

Four rays you can trust. Ray diagrams do not need every ray leaving an object — just two of these four standard rays, chosen for how simply they reflect: A ray travelling parallel to the principal axis reflects through $F$ (concave) or reflects so that it appears to come from $F$ (convex). A ray travelling towards $F$ reflects parallel to the principal axis — the exact reverse of the first rule. A ray travelling towards $C$ strikes the mirror along the normal (since every radius is perpendicular to the sphere's surface), so it reflects straight back along its own path. A ray striking the pole $P$ reflects with equal angles on either side of the principal axis, by the plain law of reflection.

How the image changes with object position (concave mirror). With the object beyond $C$, the image is real, inverted and diminished, forming between $F$ and $C$. With the object exactly at $C$, the image is real, inverted and the same size, also at $C$. With the object between $C$ and $F$, the image is real, inverted and enlarged, forming beyond $C$. With the object at $F$, the reflected rays leave parallel and never meet, so the image forms at infinity. With the object between $F$ and $P$, the reflected rays only appear to meet behind the mirror, giving a virtual, upright, enlarged image — this is the shaving-mirror and dentist's-mirror case. Concave mirrors are chosen wherever a magnified or a strongly directed beam is wanted: shaving and make-up mirrors, torches, headlight reflectors and solar cookers.

Convex mirrors are the simple case. Because the surface always curves away from the object, reflected rays always diverge — they can never actually cross in front of the mirror, only appear to meet behind it. So a convex mirror gives exactly one kind of image, for every object position without exception: virtual, upright, and diminished. That constant diverging behaviour also means a convex mirror gathers light from a wider spread of directions than a flat mirror of the same size, so it shows a wider field of view, even though everything in it looks smaller and farther away. That trade — less detail for more coverage — is exactly what a driver or a shopkeeper wants, which is why convex mirrors are used as vehicle rear-view mirrors and shop security mirrors.

Both behaviours trace back to one idea: which way the mirror bends relative to the incoming light. A concave surface curves towards the source, so a bundle of parallel rays gets tipped inward and physically crosses at a real point in front of the mirror — that crossing is where a screen would catch the image, which is why concave-mirror images are often real. A convex surface curves away, so the same bundle gets tipped outward; the rays never cross in front of the mirror, only their backward extensions meet behind it. A point where light only appears to come from, with no light actually passing through, is by definition a virtual image — so a convex mirror can never form anything else. This also explains why a concave mirror's image keeps changing while a convex mirror's never does. As the object slides towards a concave mirror it eventually crosses $F$, and that crossing flips the reflected rays from converging (real image) to diverging (virtual image). A convex mirror has no equivalent crossing: its surface diverges rays for every object position, so there is only ever one row in the image table to remember, not five.

A concave mirror has a radius of curvature of $30\text{ cm}$. Consider two situations with this same mirror: first, an object placed exactly at $C$; second, the same object moved further out, beyond $C$. Find the focal length, and state how the image changes between the two cases.

Step 1. Find the focal length using $f = R/2$. With $R = 30\text{ cm}$, $f = 30/2 = 15\text{ cm}$. This value of $f$ does not change as the object moves — it depends only on the mirror's curvature, never on where the object sits.

Step 2. Locate and describe the image with the object at $C$ ($30\text{ cm}$ from the pole). A ray aimed at $C$ strikes the mirror along the normal and reflects straight back on itself, so it gives no new information about the image position other than confirming it lies on the axis at $C$ itself. Using the parallel-ray and $F$-ray rules together instead: rays from the object's tip meet again exactly at $C$. The image is real, inverted, and the same size as the object.

Step 3. Now move the object beyond $C$, say to $40\text{ cm}$ from the pole, and describe the image qualitatively. This is the "beyond $C$" case from the core idea: the image is still real and inverted, but now diminished, and it forms between $F$ ($15\text{ cm}$) and $C$ ($30\text{ cm}$) — closer to the mirror than the object.

Step 4. Compare the two. Moving the object farther from $C$ moves the image closer to $F$ and shrinks it; moving the object towards $F$ (from $C$ back down towards $15\text{ cm}$) does the opposite — the image grows and moves farther out, beyond $C$. Answer: $f = 15\text{ cm}$ in both cases; at $C$ the image is real, inverted, same-size, at $C$; beyond $C$ the image is real, inverted, diminished, and lies between $F$ and $C$.

Competitive stretch

The same mirror ($f = 15\text{ cm}$) is now compared with a convex mirror of the same focal length. An object is placed $40\text{ cm}$ from each mirror. Without any calculation beyond what you already know, state one respect in which the two images agree and one in which they differ. They agree that both images are still tied to the same fixed $f = 15\text{ cm}$, since $f$ is a property of the mirror alone, not of the object's position, for either mirror. They differ completely in nature: the concave mirror at $40\text{ cm}$ (beyond $C = 30\text{ cm}$) gives a real, inverted, diminished image, while the convex mirror — whatever the object distance — always gives a virtual, upright, diminished image. The one thing both share is diminished size; everything else about how the image forms is opposite.

• Confusing $C$ with $F$: since $f = R/2$, the focus is always the midpoint of $P$ and $C$, never at the same place as $C$.
• Assuming a convex mirror can ever produce a real or inverted image — it cannot, at any object distance, because its curved surface always diverges reflected rays outward.
• Treating a concave mirror's image as having one fixed description — its nature depends entirely on where the object is placed relative to $F$ and $C$.
• Trying to derive $f = R/2$ from mirror geometry — this syllabus gives the relation directly; the geometric proof is not expected at this level.

Ask the tutor: Want to trace ray diagrams for a few more object positions, or check your own sketch of a concave or convex mirror image? Ask the AI tutor to walk through images in curved mirrors with you step by step.