Spherical Mirrors: Concave & Convex

Spherical and near-spherical mirrors do the heavy lifting of modern optics. A reflecting telescope — from a backyard Newtonian to the segmented giants on mountaintops — gathers starlight on a large concave mirror, exactly Foucault's silvered glass scaled up. A car headlamp and a torch put the bulb at the focus of a concave reflector so the light leaves as a parallel beam; a satellite dish and a radio telescope do the same for radio waves, concentrating a faint parallel wavefront onto a receiver at the focus. Solar cookers and solar-thermal power towers are concave mirrors turning the parallel rays of the Sun into a blazing hot spot. On the diverging side, the passenger-side wing mirror and the convex mirrors bolted at blind corners in shops and car parks trade magnification for a wide field of view, shrinking a whole aisle or junction into one small upright image — the "objects in mirror are closer than they appear" warning is the price of that shrinking. Even a dentist's mirror is a small concave mirror, giving a magnified upright view of a tooth held inside its focus. Every one of them is $f = R/2$ and the three ray rules, built at scale.

Imagine a hollow glass sphere, silvered, and smashed; pick up one small curved shard and polish it. That shard is a spherical mirror. If you silver the outside so the hollowed face reflects, you have a concave mirror (it caves inward, like the inside of a spoon's bowl); silver the inside so the bulging face reflects and you have a convex mirror (it bulges outward, like the back of the spoon). Every spherical mirror carries a fixed skeleton of points, and naming them is half the battle: The centre of curvature $C$ is the centre of the sphere the shard was cut from. The radius of curvature $R$ is that sphere's radius — the distance from the mirror to $C$. The pole $P$ is the geometric centre of the mirror's own surface, the point you would touch at its middle. The straight line through $P$ and $C$ is the principal axis, the mirror's line of symmetry. And the principal focus $F$, the star of the whole chapter, is the point on the axis where rays arriving parallel to the axis converge after reflection (concave) or from which they appear to diverge (convex). The distance $PF$ is the focal length $f$.

The difference between concave and convex is entirely about which side of the mirror the interesting points fall on. For a concave mirror, $C$ and $F$ sit in front of the reflecting surface, out in the space where the light actually travels — which is why a concave mirror can gather real light to a real point and, for instance, set paper alight at its focus in sunlight. For a convex mirror, $C$ and $F$ sit behind the surface, where no light goes; the mirror spreads parallel light outward, and the focus is only the point the diverging rays appear to come from. That single geometric fact — front versus behind — is what the sign convention is about to encode as a plus or a minus.

A number like "the focal length is 20 cm" is meaningless for a curved mirror until you say on which side. The New Cartesian sign convention — the one used by CBSE, ICSE, JEE, NEET and AP Physics 2 — fixes this once and for all, and you must state it before you compute anything:

• The pole P is the origin, and all distances are measured from the pole along the principal axis.
• The incident light is taken to travel in the positive direction. Distances measured along the incident light (behind the mirror) are positive; distances measured against it (in front of the mirror, where a real object and the incoming light are) are negative.
• Heights measured upward, perpendicular to the axis, are positive; downward heights are negative.

Two consequences follow immediately and must become instinct. A real object sits in front of the mirror, on the incoming-light side, so its distance is measured against the incident direction and $u < 0$ — real objects always have negative object distance. And because a concave mirror's $F$ and $C$ lie in front of it, its focal length and radius are negative: $f < 0$, $R < 0$. A convex mirror's $F$ and $C$ lie behind it, so its focal length and radius are positive: $f > 0$, $R > 0$. Memorise the pair "concave negative, convex positive" — it is the hinge on which every mirror-formula answer in the next article turns.

where: $f$ is the focal length ($PF$) and $R$ the radius of curvature ($PC$), both carrying their sign. Because $F$ and $C$ are always on the same side of the pole, the sign takes care of itself: a concave mirror of $R = -30\ \mathrm{cm}$ has $f = -15\ \mathrm{cm}$; one of $R = -40\ \mathrm{cm}$ has $f = -20\ \mathrm{cm}$; a convex mirror of $R = +30\ \mathrm{cm}$ has $f = +15\ \mathrm{cm}$. The relation says the focus lies exactly halfway from the pole to the centre of curvature, and it is worth proving rather than swallowing.

To find where a spherical mirror puts an image, you trace rays from a point on the object and see where the reflected rays meet (real image) or appear to meet (virtual image). Three rays are especially easy to draw because you already know where each goes: The parallel ray — a ray from the object travelling parallel to the axis — reflects through the focus F (concave) or reflects so that it appears to come from F (convex). The focal ray — a ray heading through F (concave) or towards F (convex) — reflects parallel to the axis; it is the parallel ray run backwards. The radial ray — a ray heading through C (concave) or towards C (convex) — strikes the mirror along a radius, hits it head-on ($\theta_i = 0$), and reflects straight back on itself. Any two of these three pin down the image point where they cross; the third is a check. The whole of image-finding, for the rest of the chapter, is these three rules plus the sign convention.

A concave mirror is a shape-shifter: move the object along the axis and the image marches the other way, swelling, flipping and finally jumping behind the glass. The table below is the one every board expects you to know, anchored to a concave mirror of focal length $f = -20\ \mathrm{cm}$ (so $F$ is $20\ \mathrm{cm}$ in front and $C$ is at $R = -40\ \mathrm{cm}$, i.e. $40\ \mathrm{cm}$ in front). The next two articles compute every row with the mirror formula; here we read it off the rays.

Two features are worth burning in. First, a concave mirror gives a real, inverted image for every object beyond the focus, and only when the object comes inside the focus does the image flip to virtual, erect and magnified — that last case is the shaving or make-up mirror, where you hold your face closer than the focal length. Second, the image and object cross over at C: exactly at the centre of curvature the image is the same size, sitting right on top of C, inverted. A convex mirror, by contrast, is boringly reliable, and that is its virtue. For any real object, wherever you put it, a convex mirror forms an image that is virtual, erect and diminished, tucked between the pole and the focus behind the glass. It shrinks the whole world into a small upright picture and never makes a real image of a real object — which is exactly why it is the wide-angle mirror on every car and street corner.

Board level — focal length and the mirror's skeleton (CBSE and ICSE pattern)

A concave mirror has a radius of curvature of 30 cm, and a convex mirror has a radius of curvature of 30 cm. For each, find the focal length in the New Cartesian convention and state where the focus and centre of curvature lie relative to the mirror.

Step 1. Assign signs from the convention before touching a number. For the concave mirror, $C$ and $F$ are in front of the reflecting surface, on the incoming-light side, so distances there are negative: $R = -30\ \mathrm{cm}$. For the convex mirror they are behind, so $R = +30\ \mathrm{cm}$.

Step 2. Apply $f = R/2$. Concave: $f = -30/2 = \boxed{-15\ \mathrm{cm}}$. Convex: $f = +30/2 = \boxed{+15\ \mathrm{cm}}$. The magnitudes are equal; only the signs differ, and the signs are the physics.

Step 3. State the geometry, which is what the "and where" is asking for. The concave mirror's focus is $15\ \mathrm{cm}$ in front of the pole and its centre of curvature $30\ \mathrm{cm}$ in front. The convex mirror's focus is $15\ \mathrm{cm}$ behind the pole and its centre of curvature $30\ \mathrm{cm}$ behind — points that no light ever reaches, which is why the convex focus is virtual.

Competitive level — locate the image by rays, two positions (JEE and NEET pattern)

A concave mirror has focal length 20 cm (so F is 20 cm in front and C is 40 cm in front). Using only ray tracing, describe the image when the object is placed (a) at 60 cm from the mirror, and (b) at 30 cm from the mirror.

Step 1. For part (a), the object at $60\ \mathrm{cm}$ is beyond C ($C$ at $40\ \mathrm{cm}$). Draw the parallel ray from the object's tip: it reflects through F. Draw the radial ray through C: it returns on itself. The two reflected rays cross between F and C, below the axis.

Step 2. Read off the nature for (a). The rays actually cross in front of the mirror, so the image is real; it is below the axis, so inverted; and it is shorter than the object, so diminished. This is the general rule for any object beyond C, and it is how a shaving mirror held at arm's length shows a small upside-down room. (The mirror formula in the next article makes this the exact anchor $v = -30\ \mathrm{cm}$, $m = -0.5$.)

Step 3. For part (b), the object at $30\ \mathrm{cm}$ is between C and F. Draw the parallel ray (reflects through F) and the focal ray through F (reflects parallel to the axis). They cross beyond C, below the axis.

Step 4. Read off the nature for (b): the rays cross in front, so real; below the axis, so inverted; and taller than the object, so magnified. Moving the object from beyond C to inside C has swung the image from diminished to magnified, with the crossover exactly at C. (Formula anchor: $v = -60\ \mathrm{cm}$, $m = -2$.) Timing note: about two minutes for both. The speed trick is to memorise the six-row table so that "beyond C" instantly means "real, inverted, diminished, between F and C" without drawing — then the ray diagram is only a confirmation, not the whole labour.

Writing $f = R$ instead of $f = R/2$. The commonest slip on this topic across every board. The focus is halfway to the centre of curvature. A radius of $30\ \mathrm{cm}$ gives a focal length of $15\ \mathrm{cm}$, not $30\ \mathrm{cm}$.

Getting the sign of $f$ backwards. Concave means $f < 0$; convex means $f > 0$ — in the New Cartesian convention that CBSE, ICSE, JEE, NEET and AP Physics 2 all use. Some older ICSE and A-Level texts use a "real-is-positive" convention where a concave focal length is written positive; if your textbook does, name which convention you are in and stay in it, but the modern default and the one these boards mark to is New Cartesian.

Claiming a convex mirror makes a real image. It never does, for a real object — the image is always virtual, erect and diminished. A convex mirror cannot burn paper, cannot project onto a screen, and cannot magnify.

Board scope. Curved mirrors are on CBSE, ICSE, JEE Main, JEE Advanced, NEET, AP Physics 2 (Unit 13, Geometric Optics), IGCSE and A-Level. Note carefully: IB dropped geometric optics from its guide, so spherical mirrors are not on either IB SL or IB HL — if you are an IB candidate this is enrichment, not exam material. IGCSE treats spherical mirrors qualitatively (the ray rules and the image types), while JEE and NEET want the full quantitative treatment that the next two articles provide.

Texts. NCERT Science Class 10, Ch. 10 ("Light — Reflection and Refraction"), spherical mirrors and ray diagrams · NCERT Physics Class 12, Ch. 9 ("Ray Optics and Optical Instruments"), reflection by spherical mirrors · H. C. Verma, Concepts of Physics Vol. 1, "Geometrical Optics" · Cambridge A-Level 9702 and IGCSE 0625 on curved reflectors.

Provenance. Standard geometric optics. Recompute: $f = R/2$ for $R = -30 \to -15$, $R = -40 \to -20$, $R = +30 \to +15$ cm; the centre of curvature at $R = 2f$ (concave $f = -20 \to R = -40$); the paraxial focus of a parallel ray found by direct reflection off the sphere converging to $R/2$; the concave image-table anchors previewed in the worked example, $u = -60 \to v = -30$, $m = -0.5$ and $u = -30 \to v = -60$, $m = -2$; the convex case $f = +15$, $u = -20 \to v = +8.571$; and Foucault's $298000\ \mathrm{km/s}$ within one percent of the modern $299792\ \mathrm{km/s}$.

Prerequisites. Plane Mirrors and Image Formation — the flat limit of this chapter — and The Ray Model of Light and Laws of Reflection, whose single law is applied here point by point along a curved surface. Leads to. The Mirror Formula and Sign Convention, which turns this geometry into the equation $1/v + 1/u = 1/f$, and Magnification by Mirrors, which quantifies the size and orientation.

Go deeper (UG). The full treatment of curved-mirror imaging, including spherical aberration, the parabolic mirror and the wave-optics limits on resolution, is developed in Plane and Spherical Mirrors in the university optics track. It is enrichment only; every result on this page follows from the law of reflection and paraxial geometry, and no exam here will ask for the aberration theory.