Spherical mirrors: key terms, ray rules, images and uses
f = R/2. A concave mirror gives real, inverted images unless the object is inside F. A convex mirror always gives virtual, erect, smaller images.
Words for spherical mirrors
| Term | What it means |
|---|---|
| Spherical mirror | A mirror whose reflecting surface is part of a hollow sphere |
| Concave mirror | Reflection takes place at the caved-in (hollow) side; the outer side is painted. Like the inside of a steel spoon |
| Convex mirror | Reflection takes place at the bulging side; the inner side is painted. Like the back of a steel spoon |
| Pole (P) | The centre point of the reflecting surface |
| Centre of curvature (C) | The centre of the sphere that the mirror is a part of. It is in front of a concave mirror and behind a convex mirror. It is not on the mirror itself |
| Radius of curvature (R) | The radius of that sphere, that is, the distance PC |
| Principal axis | The imaginary line through P and C, extended on both sides |
| Principal focus (F) | Concave: the point on the axis where rays parallel and close to the axis meet after reflection. Convex: the point from which they seem to come after reflection |
| Focal length (f) | The distance PF. For mirrors with a small aperture, R = 2f |
| Aperture | The width (diameter) of the reflecting surface |
Why f is half of R
- 1Draw a concave mirror with pole P, centre of curvature C and focus F on the axis.
- 2A ray IM, parallel to the axis, strikes the mirror at M. CM is a radius, so it is the normal at M.
- 3The reflected ray MF passes through F. By the law of reflection, ∠IMC = ∠CMF (∠i = ∠r).
- 4IM is parallel to the axis, so ∠IMC = ∠MCF (alternate angles). So ∠MCF = ∠CMF.
- 5Two angles of triangle CMF are equal, so the sides opposite them are equal: MF = CF.
- 6For a mirror of small aperture, M is very close to P, so MF ≈ PF. Then PF = CF.
- 7★So PC = PF + CF = 2PF, that is, R = 2f. The focus lies midway between P and C.
★ key step
Four ray rules for mirrors (any two rays are enough)
- Ray to the pole: a ray striking P at a slant reflects at the same angle on the other side of the axis.
- Parallel ray: a ray parallel to the axis passes through F after a concave mirror. After a convex mirror it goes outwards as if it came from F.
- Ray through C: a ray through C (or aimed at C, for a convex mirror) falls along the normal and retraces its path.
- Ray through F: a ray through F (or aimed at F, for a convex mirror) comes back parallel to the axis.
- Where two reflected rays actually meet, the image is real. Where they only seem to meet when drawn back as dotted lines, it is virtual.
Images by concave and convex mirrors
| Object at | Image at | Size | Nature |
|---|---|---|---|
| Concave: infinity | F | Highly diminished | Real, inverted |
| Concave: beyond C (2F) | Between F and C | Smaller | Real, inverted |
| Concave: at C (2F) | C | Same size | Real, inverted |
| Concave: between C and F | Beyond C | Larger | Real, inverted |
| Concave: at F | Infinity | Highly enlarged | Real, inverted |
| Concave: between P and F | Behind the mirror | Larger | Virtual, erect |
| Convex: anywhere in front | Between P and F, behind the mirror | Always smaller | Virtual, erect |
How to draw the ray diagrams for concave and convex mirrors
Start every concave mirror diagram the same way. Draw the axis and the mirror on the right, with P on the mirror. Mark F and C on the left, with C twice as far from P as F. Draw the object AB with A at the top, and two rays from A. For a convex mirror, draw it bulging towards the object, with F and C behind it. Every line behind a mirror is dotted.
- 1Object at infinity. Draw two rays parallel to the principal axis coming from the left. After reflection, draw both passing through F. They meet at F.Image at F: point-sized, real, inverted.
- 2Object beyond C. Stand AB to the left of C. From A, draw a ray parallel to the axis and reflect it through F. From A, draw a ray through F to the mirror and reflect it parallel to the axis. The rays cross below the axis between F and C: mark A′ and draw A′B′ down to the axis.Image between F and C: smaller, real, inverted.
- 3Object at C. Stand AB on C. Use the same two rays. They cross below the axis exactly at C. Draw A′B′ pointing down, as long as AB.Image at C: same size, real, inverted.
- 4Object between C and F. Stand AB between C and F. The ray parallel to the axis reflects through F. The ray from A through F reflects parallel to the axis. They cross below the axis beyond C. Draw a long A′B′ pointing down.Image beyond C: enlarged, real, inverted.
- 5Object at F. Stand AB on F. The ray parallel to the axis reflects through F. Draw a second ray from A along the line from C. It hits the mirror along the normal and returns on itself. The two reflected rays are parallel, so they meet only at infinity.Image at infinity: highly enlarged, real, inverted.
- 6Object between P and F. Stand AB between F and the mirror. The ray parallel to the axis reflects through F. A second ray from A in line with C returns on itself. The reflected rays spread apart. Extend both behind the mirror as dotted lines: they meet at A′. Draw A′B′ upright, larger and dotted.Image behind the mirror: enlarged, virtual, erect.
- 7Convex mirror, object at infinity. Draw two rays parallel to the axis hitting the mirror. Reflect them so they spread outwards. Extend each backwards as a dotted line: they meet at F behind the mirror.Image at F: point-sized, virtual, erect.
- 8Convex mirror, object anywhere in front. From A, draw a ray parallel to the axis. Reflect it outwards and dot it back to F. From A, draw a ray aimed at C: it returns on itself; dot its line to C. The two dotted lines meet at A′ between P and F. Draw A′B′ upright, small and dotted.Image between P and F: diminished, virtual, erect.
Uses of mirrors
| Mirror | Use | Why it works |
|---|---|---|
| Plane | Looking glass, periscope, kaleidoscope, seeing round corners | Gives an erect, same-size image and turns light through a set angle |
| Concave | Torches, search-lights and vehicle headlights | A bulb at F gives a strong parallel beam |
| Concave | Shaving and make-up mirrors | A face between P and F gives an erect, enlarged image |
| Concave | Dentist's mirror | Shows a large image of the teeth |
| Concave | Solar furnaces and dish solar cookers | Collects sunlight at the focus to make heat |
| Concave | Reflecting telescopes and flood lights | Collects faint light, or spreads a bright beam over a building |
| Convex | Rear-view mirrors of cars, buses and scooters | Always an erect, smaller image and a wider field of view |
| Convex | Safety mirrors at dangerous corners and on the upper deck of double-decker buses | Shows a wide area around the bend |
The round mirrors at hairpin bends on ghat roads are convex, so drivers see a wide, upright view of oncoming traffic. A torch or a bike headlight places its bulb at the focus of a concave reflector to throw a strong parallel beam.
1 marks: “A concave mirror has a radius of curvature of 30 cm. What is its focal length?” f = R/2 = 30 cm ÷ 2 = 15 cm.
2 marks: “Why is a convex mirror used as a rear-view mirror in vehicles? Give two reasons.” It always gives an erect image, so the driver sees traffic the right way up. The image is smaller, so the mirror shows a wider field of view behind the vehicle.
5 marks: “Draw ray diagrams to show the image formed by a concave mirror when the object is (i) beyond C, (ii) at F, (iii) between P and F, and by a convex mirror for an object in front of it. State the nature of each image.” Use two rays from the top of the object (parallel ray through F; ray through C back on itself). (i) Between F and C: real, inverted, smaller. (ii) At infinity: real, inverted, highly enlarged. (iii) Behind the mirror: virtual, erect, larger (dotted lines). Convex: between P and F behind the mirror: virtual, erect, smaller.
A concave mirror stays real and upside down until the object crosses F. Inside F the image jumps behind the mirror, upright and big. A convex mirror is always small, upright and behind.