Tilt Two Mirrors Together and Count How Many of You Appear
Learn to count the images formed by two inclined mirrors, name every part of a spherical mirror, tell a concave mirror from a convex one and use f equals R by two, and apply the three ray construction rules.
How many images appear when two mirrors face each other?
Stand a small object between two plane mirrors hinged at an angle and look in. You see more than one image — and the number depends only on the angle.
At you see three images. At you see five. At you see seven. And with the mirrors set parallel, facing each other, the images go on for ever, each one fainter than the last.
The count comes from a single expression:
so gives , and gives . Closing the mirrors towards parallel makes approach zero and the count grows without limit.
What is happening is that each mirror forms an image, each mirror then forms an image of the other's image, and the chain continues until the reflections run out of room. A kaleidoscope is built on exactly that.
The rest of this page leaves flat mirrors behind for curved ones, where a single mirror can enlarge, shrink or invert what it shows. That needs a vocabulary — pole, centre of curvature, focus — and three rules for drawing rays. This page covers the second part of the ICSE Class 9 Physics chapter on the reflection of light.
At you see three images. At you see five. At you see seven. And with the mirrors set parallel, facing each other, the images go on for ever, each one fainter than the last.
The count comes from a single expression:
so gives , and gives . Closing the mirrors towards parallel makes approach zero and the count grows without limit.
What is happening is that each mirror forms an image, each mirror then forms an image of the other's image, and the chain continues until the reflections run out of room. A kaleidoscope is built on exactly that.
The rest of this page leaves flat mirrors behind for curved ones, where a single mirror can enlarge, shrink or invert what it shows. That needs a vocabulary — pole, centre of curvature, focus — and three rules for drawing rays. This page covers the second part of the ICSE Class 9 Physics chapter on the reflection of light.
Formula
How do you count the images formed by two inclined mirrors?
**Divide by the angle between the mirrors and subtract one:**
Worked example 1 — perpendicular mirrors. :
Three images. Two of them are the direct images in each mirror, and the third is formed behind the corner by light that bounced off both mirrors in turn.
Worked example 2. :
Worked example 3. :
Worked example 4. :
Worked example 5 — a single flat mirror. Two mirrors opened right out to form one flat surface:
One image, as a plane mirror should give — a useful check that the formula is behaving.
Worked example 6 — finding the angle. How must two mirrors be set to give images?
Parallel mirrors are the limiting case. As falls towards zero, grows without bound, so the number of images becomes infinite. Two mirrors facing each other on opposite walls of a room show a corridor of images stretching away, and each is dimmer than the one before because a little light is absorbed at every reflection. The count is infinite and the visible count is not, which is why the corridor seems to fade out rather than go on for ever.
Perpendicular mirrors give exactly three, and they are used in a rear-view periscope arrangement and in the corner reflectors that send a beam straight back the way it came.
**The formula works cleanly when is an even whole number**, as at , and . When it comes out odd, as at or , the count depends slightly on where the object sits: an object on the bisector of the angle gives the formula's answer, and an object off the bisector can show one more. So a question should say where the object is, and the syllabus expression assumes the symmetric case.
Worked example 1 — perpendicular mirrors. :
Three images. Two of them are the direct images in each mirror, and the third is formed behind the corner by light that bounced off both mirrors in turn.
Worked example 2. :
Worked example 3. :
Worked example 4. :
Worked example 5 — a single flat mirror. Two mirrors opened right out to form one flat surface:
One image, as a plane mirror should give — a useful check that the formula is behaving.
Worked example 6 — finding the angle. How must two mirrors be set to give images?
Parallel mirrors are the limiting case. As falls towards zero, grows without bound, so the number of images becomes infinite. Two mirrors facing each other on opposite walls of a room show a corridor of images stretching away, and each is dimmer than the one before because a little light is absorbed at every reflection. The count is infinite and the visible count is not, which is why the corridor seems to fade out rather than go on for ever.
Perpendicular mirrors give exactly three, and they are used in a rear-view periscope arrangement and in the corner reflectors that send a beam straight back the way it came.
**The formula works cleanly when is an even whole number**, as at , and . When it comes out odd, as at or , the count depends slightly on where the object sits: an object on the bisector of the angle gives the formula's answer, and an object off the bisector can show one more. So a question should say where the object is, and the syllabus expression assumes the symmetric case.
What are the parts of a spherical mirror called?
A spherical mirror is a slice of a hollow sphere, silvered on one side, and every term below is a feature of that sphere.
- Pole (P) — the centre of the reflecting surface, the middle of the mirror's face. Also called the vertex
- Centre of curvature (C) — the centre of the sphere of which the mirror is a part. For a concave mirror it lies in front of the mirror; for a convex mirror it lies behind
- Radius of curvature (R) — the radius of that sphere, which is the distance
- Principal axis — the straight line through P and C, at right angles to the mirror at the pole
- Aperture — the effective width of the mirror, the diameter of its reflecting surface
- Principal focus (F) — for a concave mirror, the point on the principal axis where rays parallel to the axis actually converge after reflection. For a convex mirror, the point from which such rays appear to diverge
- Focal length (f) — the distance , from the pole to the principal focus
A concave mirror's focus is REAL and a convex mirror's focus is VIRTUAL. Hold a concave mirror in sunlight and you can find a bright hot spot on a card at its focus, because the light genuinely gathers there. Do the same with a convex mirror and there is no such spot anywhere — the reflected rays spread out, and only their backward extensions meet behind the glass. So one focus can be located with a card and the other cannot, which is the practical difference between them.
Worked identification. A concave mirror has cm. Then:
- C lies on the principal axis, cm in front of the pole
- F lies halfway between, cm in front of the pole
- The focal length is cm
The terms all describe the sphere, not the mirror. The centre of curvature is the centre of a sphere that is mostly not there, and the radius of curvature is a distance to a point in empty space. That is why a shallow mirror can have a very large radius of curvature — a slightly curved mirror is a small piece of an enormous sphere, and a plane mirror is the limiting case with infinite.
A small aperture matters for sharpness. The three ray rules and the relation hold accurately only for mirrors whose aperture is small compared with the radius of curvature. A wide mirror does not bring parallel rays to a single point, and the blur is why large reflecting surfaces are shaped as parabolas rather than spheres.
- Pole (P) — the centre of the reflecting surface, the middle of the mirror's face. Also called the vertex
- Centre of curvature (C) — the centre of the sphere of which the mirror is a part. For a concave mirror it lies in front of the mirror; for a convex mirror it lies behind
- Radius of curvature (R) — the radius of that sphere, which is the distance
- Principal axis — the straight line through P and C, at right angles to the mirror at the pole
- Aperture — the effective width of the mirror, the diameter of its reflecting surface
- Principal focus (F) — for a concave mirror, the point on the principal axis where rays parallel to the axis actually converge after reflection. For a convex mirror, the point from which such rays appear to diverge
- Focal length (f) — the distance , from the pole to the principal focus
A concave mirror's focus is REAL and a convex mirror's focus is VIRTUAL. Hold a concave mirror in sunlight and you can find a bright hot spot on a card at its focus, because the light genuinely gathers there. Do the same with a convex mirror and there is no such spot anywhere — the reflected rays spread out, and only their backward extensions meet behind the glass. So one focus can be located with a card and the other cannot, which is the practical difference between them.
Worked identification. A concave mirror has cm. Then:
- C lies on the principal axis, cm in front of the pole
- F lies halfway between, cm in front of the pole
- The focal length is cm
The terms all describe the sphere, not the mirror. The centre of curvature is the centre of a sphere that is mostly not there, and the radius of curvature is a distance to a point in empty space. That is why a shallow mirror can have a very large radius of curvature — a slightly curved mirror is a small piece of an enormous sphere, and a plane mirror is the limiting case with infinite.
A small aperture matters for sharpness. The three ray rules and the relation hold accurately only for mirrors whose aperture is small compared with the radius of curvature. A wide mirror does not bring parallel rays to a single point, and the blur is why large reflecting surfaces are shaped as parabolas rather than spheres.
How do you tell a concave mirror from a convex one?
Look at which side is silvered. A concave mirror reflects from the inner hollow surface; a convex mirror reflects from the outer bulging surface.
Concave mirror.
- The reflecting surface is the inside of the curve, curving inward away from the object — like the inside of a spoon's bowl
- It is converging: a parallel beam is brought together to a real focus in front of the mirror
- C and F both lie in front of the mirror, on the same side as the object
Convex mirror.
- The reflecting surface is the outside of the curve, bulging towards the object — like the back of a spoon
- It is diverging: a parallel beam is spread out, and only the backward extensions meet at a virtual focus behind the mirror
- C and F both lie behind the mirror
The relation between focal length and radius.
The focus sits exactly halfway between the pole and the centre of curvature, for both kinds of mirror.
Worked example 1. A concave mirror has cm, so
Worked example 2. A mirror of focal length cm has
Worked example 3. cm gives cm, and cm gives cm.
Worked example 4 — locating both points. A concave mirror of focal length cm has cm, so on the principal axis in front of the mirror:
- F is at cm
- C is at cm
- The distance is cm, equal to the focal length
So the pole, the focus and the centre of curvature are equally spaced, each apart — which makes the six object positions of the next part of this chapter easy to mark on a diagram.
The spoon test settles which is which. Look into the bowl of a spoon held close to your face and you see yourself magnified and erect — that is the concave side. Turn it over and look at the back and you see yourself small and erect — the convex side. One spoon gives both mirrors, and holding it at arm's length shows the concave side producing an inverted image, which the convex side never does.
A curved mirror is not automatically a magnifier. The concave side magnifies only when the object is closer than the focus; further away it shrinks and inverts. The convex side diminishes at every distance. So "concave magnifies" is an incomplete statement that holds only near the mirror, and the full account is the table in the next part of this chapter.
Concave mirror.
- The reflecting surface is the inside of the curve, curving inward away from the object — like the inside of a spoon's bowl
- It is converging: a parallel beam is brought together to a real focus in front of the mirror
- C and F both lie in front of the mirror, on the same side as the object
Convex mirror.
- The reflecting surface is the outside of the curve, bulging towards the object — like the back of a spoon
- It is diverging: a parallel beam is spread out, and only the backward extensions meet at a virtual focus behind the mirror
- C and F both lie behind the mirror
The relation between focal length and radius.
The focus sits exactly halfway between the pole and the centre of curvature, for both kinds of mirror.
Worked example 1. A concave mirror has cm, so
Worked example 2. A mirror of focal length cm has
Worked example 3. cm gives cm, and cm gives cm.
Worked example 4 — locating both points. A concave mirror of focal length cm has cm, so on the principal axis in front of the mirror:
- F is at cm
- C is at cm
- The distance is cm, equal to the focal length
So the pole, the focus and the centre of curvature are equally spaced, each apart — which makes the six object positions of the next part of this chapter easy to mark on a diagram.
The spoon test settles which is which. Look into the bowl of a spoon held close to your face and you see yourself magnified and erect — that is the concave side. Turn it over and look at the back and you see yourself small and erect — the convex side. One spoon gives both mirrors, and holding it at arm's length shows the concave side producing an inverted image, which the convex side never does.
A curved mirror is not automatically a magnifier. The concave side magnifies only when the object is closer than the focus; further away it shrinks and inverts. The convex side diminishes at every distance. So "concave magnifies" is an incomplete statement that holds only near the mirror, and the full account is the table in the next part of this chapter.
What are the rules for drawing rays at a spherical mirror?
Three rules, and any two of them locate an image. Each is the laws of reflection applied to a particular convenient ray.
Rule 1 — a ray parallel to the principal axis. After reflection it passes through F (concave), or appears to come from F (convex).
That is the definition of the principal focus, read as an instruction.
Rule 2 — a ray through the focus. A ray passing through F (concave), or directed towards F (convex), emerges parallel to the principal axis after reflection.
This is rule 1 run backwards. Light paths are reversible, so if parallel rays converge at F then rays from F emerge parallel.
Rule 3 — a ray through the centre of curvature. A ray passing through C (concave), or directed towards C (convex), is reflected straight back along its own path.
Why rule 3 works. A line through the centre of a sphere meets the sphere's surface at right angles — it is a radius, and a radius is the normal at that point. So the ray strikes along the normal, , and therefore and it returns the way it came. Rule 3 is not an extra fact; it is normal incidence.
Rule 4 — a ray striking the pole. It is reflected with measured about the principal axis, since the axis is the normal at the pole.
How to use them. Pick any two rays from a point on the object, follow them by the rules, and the point where the reflected rays meet — or where their backward extensions meet — is the image of that point. Two rays are enough because two lines cross at one point, and a third is worth drawing as a check.
Worked construction. A concave mirror has cm, so cm. Mark F at cm and C at cm on the principal axis. Place an object cm from the pole, beyond C.
- From the tip of the object, draw a ray parallel to the axis. By rule 1 it reflects through F
- From the same tip, draw a ray through C. By rule 3 it comes straight back
- The two reflected rays cross between F and C
So the image lies between cm and cm from the pole, and it is real, inverted and diminished — the case the next part of this chapter tabulates in full.
Which two rays to choose is a matter of convenience. Rules 1 and 3 are usually easiest because both reflected paths are known immediately. Rule 2 is awkward for an object placed at F itself, since the ray through F would have to start at the object and the object is already there — and in that case the reflected rays come out parallel and the image is at infinity.
**All the rules follow from and nothing else. They are shortcuts for rays whose behaviour can be written down without measuring an angle, chosen precisely because they save drawing normals. A ray striking anywhere else obeys the same law and is simply harder to draw**, which is why the four special rays are the ones named.
Rule 1 — a ray parallel to the principal axis. After reflection it passes through F (concave), or appears to come from F (convex).
That is the definition of the principal focus, read as an instruction.
Rule 2 — a ray through the focus. A ray passing through F (concave), or directed towards F (convex), emerges parallel to the principal axis after reflection.
This is rule 1 run backwards. Light paths are reversible, so if parallel rays converge at F then rays from F emerge parallel.
Rule 3 — a ray through the centre of curvature. A ray passing through C (concave), or directed towards C (convex), is reflected straight back along its own path.
Why rule 3 works. A line through the centre of a sphere meets the sphere's surface at right angles — it is a radius, and a radius is the normal at that point. So the ray strikes along the normal, , and therefore and it returns the way it came. Rule 3 is not an extra fact; it is normal incidence.
Rule 4 — a ray striking the pole. It is reflected with measured about the principal axis, since the axis is the normal at the pole.
How to use them. Pick any two rays from a point on the object, follow them by the rules, and the point where the reflected rays meet — or where their backward extensions meet — is the image of that point. Two rays are enough because two lines cross at one point, and a third is worth drawing as a check.
Worked construction. A concave mirror has cm, so cm. Mark F at cm and C at cm on the principal axis. Place an object cm from the pole, beyond C.
- From the tip of the object, draw a ray parallel to the axis. By rule 1 it reflects through F
- From the same tip, draw a ray through C. By rule 3 it comes straight back
- The two reflected rays cross between F and C
So the image lies between cm and cm from the pole, and it is real, inverted and diminished — the case the next part of this chapter tabulates in full.
Which two rays to choose is a matter of convenience. Rules 1 and 3 are usually easiest because both reflected paths are known immediately. Rule 2 is awkward for an object placed at F itself, since the ray through F would have to start at the object and the object is already there — and in that case the reflected rays come out parallel and the image is at infinity.
**All the rules follow from and nothing else. They are shortcuts for rays whose behaviour can be written down without measuring an angle, chosen precisely because they save drawing normals. A ray striking anywhere else obeys the same law and is simply harder to draw**, which is why the four special rays are the ones named.
Exam tip
Exam tip: check whether 360 over theta is even, and put F halfway
**Use **, and check it against giving one image — that is a free verification.
**Note whether is even or odd.** It is clean when even (, , ); when odd, say that the object is assumed to be on the bisector.
Parallel mirrors give infinitely many images, each fainter than the last because light is absorbed at every reflection.
Learn the terms as features of the SPHERE: the pole is on the mirror, but C and R describe the sphere the mirror was cut from.
A concave mirror's focus is REAL, a convex mirror's is VIRTUAL — one can be found with a card, the other cannot.
** and .** cm gives cm; cm gives cm.
Mark P, F and C equally spaced, each apart, before drawing anything.
Concave reflects from the INNER surface and converges; convex reflects from the outer surface and diverges.
State the rule you used beside each ray: parallel ray through F, ray through C returns along itself.
Explain rule 3 by normal incidence — a radius is the normal, so .
And draw only two rays to locate an image, with a third as a check — and use dotted lines for any part behind the mirror.
**Note whether is even or odd.** It is clean when even (, , ); when odd, say that the object is assumed to be on the bisector.
Parallel mirrors give infinitely many images, each fainter than the last because light is absorbed at every reflection.
Learn the terms as features of the SPHERE: the pole is on the mirror, but C and R describe the sphere the mirror was cut from.
A concave mirror's focus is REAL, a convex mirror's is VIRTUAL — one can be found with a card, the other cannot.
** and .** cm gives cm; cm gives cm.
Mark P, F and C equally spaced, each apart, before drawing anything.
Concave reflects from the INNER surface and converges; convex reflects from the outer surface and diverges.
State the rule you used beside each ray: parallel ray through F, ray through C returns along itself.
Explain rule 3 by normal incidence — a radius is the normal, so .
And draw only two rays to locate an image, with a third as a check — and use dotted lines for any part behind the mirror.
Did you know
Why a kaleidoscope needs exactly sixty degrees
A kaleidoscope is three strips of mirror taped into a tube, and the pattern it shows is startlingly regular. The regularity is not luck — it comes from the angle.
Set the mirrors as an equilateral triangle and each pair meets at . Feeding that into the formula:
Five images plus the original makes six shapes arranged around the meeting point, filling the completely with no gap and no overlap. Every piece of coloured glass appears six times in a perfect rosette.
The reason works so cleanly is that comes out as six, a whole number. The images fit round the circle exactly. Try an angle where the division is not whole — , say, giving — and the last image overlaps the first untidily, so the pattern breaks.
That is why the useful kaleidoscope angles are the ones dividing exactly: gives four-fold patterns, gives six-fold, gives eight-fold. Each is a whole-number division, and each fills the circle.
The same arithmetic decides how many mirror images of yourself appear in a lift with mirrored walls at right angles, and it is the reason a corner of three perpendicular mirrors sends a beam of light straight back to wherever it came from, whatever direction that was. Such corner reflectors are used on road studs and on cycle reflectors — the light from a headlamp returns to the driver rather than scattering away.
So one small formula, written for a physics problem about counting images, is also a rule about which angles tile a circle. The mirrors are doing geometry, and being a whole number is the condition for the tiling to close up.
Set the mirrors as an equilateral triangle and each pair meets at . Feeding that into the formula:
Five images plus the original makes six shapes arranged around the meeting point, filling the completely with no gap and no overlap. Every piece of coloured glass appears six times in a perfect rosette.
The reason works so cleanly is that comes out as six, a whole number. The images fit round the circle exactly. Try an angle where the division is not whole — , say, giving — and the last image overlaps the first untidily, so the pattern breaks.
That is why the useful kaleidoscope angles are the ones dividing exactly: gives four-fold patterns, gives six-fold, gives eight-fold. Each is a whole-number division, and each fills the circle.
The same arithmetic decides how many mirror images of yourself appear in a lift with mirrored walls at right angles, and it is the reason a corner of three perpendicular mirrors sends a beam of light straight back to wherever it came from, whatever direction that was. Such corner reflectors are used on road studs and on cycle reflectors — the light from a headlamp returns to the driver rather than scattering away.
So one small formula, written for a physics problem about counting images, is also a rule about which angles tile a circle. The mirrors are doing geometry, and being a whole number is the condition for the tiling to close up.
Exam relevance
How do spherical mirror basics feed into JEE Main and NEET?
Because the vocabulary and are assumed without restatement in the whole of ray optics, and the ray rules become the sign convention.
This is the foundation for Class 12 Physics Ray Optics and Optical Instruments, examined in JEE Main and NEET. The relation is derived there from the geometry rather than quoted, and it is used in every mirror numerical alongside
The terms named on this page are the symbols in those formulas — measured from the pole, measured to the principal focus, and the radius of the sphere. A student who cannot place P, F and C on an axis cannot set up either equation.
The Cartesian sign convention is where the ray rules become arithmetic. Distances are measured from the pole, with the direction of incident light taken as negative, so a concave mirror has negative and a convex mirror has positive. The real-against-virtual focus distinction made here is exactly what that sign records, and getting it backwards is the commonest failure in Class 12 mirror problems.
The small-aperture condition becomes spherical aberration. Class 12 explains that a wide spherical mirror does not focus parallel rays to a single point, and that a parabolic surface does — which is why a telescope mirror and a headlight reflector are parabolic. The warning given here about aperture is the qualitative version of that.
The inclined-mirror formula appears in JEE Main as a direct question, usually with an angle that makes odd, precisely to test whether the bisector condition is known. Corner reflectors and the reversibility of light paths are the related conceptual items.
Reversibility of light — used here to justify rule 2 — becomes a stated principle in Class 12 and underlies the lens formula, Snell's law applied both ways, and the fact that an object and its real image can be swapped.
For NEET Physics, mirror numericals and image-characteristic questions appear in the optics section, and the terminology is assumed. For NEET Biology, the eye's optics use the same vocabulary for its lens.
What the questions look like. For board work, expect count the images for a given angle, define each term of a spherical mirror, distinguish concave from convex by surface and by action, **use both ways, and state the ray rules with a reason for the one through C. Labelled diagrams carry much of the credit. For JEE Main and NEET, expect mirror-formula numericals, magnification, sign-convention questions, and the inclined-mirror count.
How board and competitive emphasis differ. A board paper rewards the definitions and the labelled axis** with P, F and C marked. A competitive paper never asks for a definition — it gives and with signs and wants , so the vocabulary is only the means of reading the question correctly.
The single trap that costs the most marks. Taking as equal to , or placing the focus at the centre of curvature. The focus is halfway between the pole and C, so a mirror with cm has cm — and using cm gives a self-consistent but wrong answer to every later part. The defence is to mark P, F and C on the axis as three equally spaced points before any calculation, which makes visible rather than remembered.
This is the foundation for Class 12 Physics Ray Optics and Optical Instruments, examined in JEE Main and NEET. The relation is derived there from the geometry rather than quoted, and it is used in every mirror numerical alongside
The terms named on this page are the symbols in those formulas — measured from the pole, measured to the principal focus, and the radius of the sphere. A student who cannot place P, F and C on an axis cannot set up either equation.
The Cartesian sign convention is where the ray rules become arithmetic. Distances are measured from the pole, with the direction of incident light taken as negative, so a concave mirror has negative and a convex mirror has positive. The real-against-virtual focus distinction made here is exactly what that sign records, and getting it backwards is the commonest failure in Class 12 mirror problems.
The small-aperture condition becomes spherical aberration. Class 12 explains that a wide spherical mirror does not focus parallel rays to a single point, and that a parabolic surface does — which is why a telescope mirror and a headlight reflector are parabolic. The warning given here about aperture is the qualitative version of that.
The inclined-mirror formula appears in JEE Main as a direct question, usually with an angle that makes odd, precisely to test whether the bisector condition is known. Corner reflectors and the reversibility of light paths are the related conceptual items.
Reversibility of light — used here to justify rule 2 — becomes a stated principle in Class 12 and underlies the lens formula, Snell's law applied both ways, and the fact that an object and its real image can be swapped.
For NEET Physics, mirror numericals and image-characteristic questions appear in the optics section, and the terminology is assumed. For NEET Biology, the eye's optics use the same vocabulary for its lens.
What the questions look like. For board work, expect count the images for a given angle, define each term of a spherical mirror, distinguish concave from convex by surface and by action, **use both ways, and state the ray rules with a reason for the one through C. Labelled diagrams carry much of the credit. For JEE Main and NEET, expect mirror-formula numericals, magnification, sign-convention questions, and the inclined-mirror count.
How board and competitive emphasis differ. A board paper rewards the definitions and the labelled axis** with P, F and C marked. A competitive paper never asks for a definition — it gives and with signs and wants , so the vocabulary is only the means of reading the question correctly.
The single trap that costs the most marks. Taking as equal to , or placing the focus at the centre of curvature. The focus is halfway between the pole and C, so a mirror with cm has cm — and using cm gives a self-consistent but wrong answer to every later part. The defence is to mark P, F and C on the axis as three equally spaced points before any calculation, which makes visible rather than remembered.
Key takeaways
Inclined mirrors, spherical mirror terms and ray rules: quick revision
- Number of images from two inclined mirrors: .
- gives 3; gives 5; gives 7; gives 4; gives 1 — a plane mirror, which checks the formula.
- For images, , so .
- Parallel mirrors give infinitely many images, each fainter because light is absorbed at every reflection.
- **When is odd the count depends on the object's position; the formula assumes it is on the bisector.
- Pole (P) — the centre of the reflecting surface. Centre of curvature (C) — the centre of the sphere. Radius of curvature (R) — the distance PC.
- Principal axis — the line through P and C. Aperture — the effective width of the mirror.
- Principal focus (F) — where parallel rays converge (concave, real focus) or appear to diverge from (convex, virtual focus). Focal length (f) — the distance PF.
- A concave focus can be found with a card in sunlight; a convex focus cannot.
- Concave: reflects from the inner surface, converging, with C and F in front.
- Convex: reflects from the outer surface, diverging, with C and F behind.
- and **: gives ; gives ; gives ; gives cm.
- P, F and C are equally spaced, each apart.
- The spoon test: the bowl is concave and magnifies close up; the back is convex and always diminishes.
- "Concave magnifies" holds only within the focus — beyond it the image shrinks and inverts.
- Rule 1: a ray parallel to the axis reflects through F (or appears to come from F).
- Rule 2: a ray through F reflects parallel to the axis — rule 1 reversed, since light paths are reversible.
- Rule 3: a ray through C returns along its own path, because a radius is the normal, so .
- Rule 4: a ray at the pole reflects with about the principal axis.
- Two rays locate an image; a third is a check. Use dotted lines behind the mirror.
- With cm and an object at cm, rules 1 and 3 cross between F and C, giving a real, inverted, diminished image.
- The rules hold accurately only for a small aperture — a wide spherical mirror blurs, which is why reflectors are parabolic.
Hold a spoon at arm's length, then bring it close to your eye, and note the exact moment the image flips upright — that moment is the focus.
- gives 3; gives 5; gives 7; gives 4; gives 1 — a plane mirror, which checks the formula.
- For images, , so .
- Parallel mirrors give infinitely many images, each fainter because light is absorbed at every reflection.
- **When is odd the count depends on the object's position; the formula assumes it is on the bisector.
- Pole (P) — the centre of the reflecting surface. Centre of curvature (C) — the centre of the sphere. Radius of curvature (R) — the distance PC.
- Principal axis — the line through P and C. Aperture — the effective width of the mirror.
- Principal focus (F) — where parallel rays converge (concave, real focus) or appear to diverge from (convex, virtual focus). Focal length (f) — the distance PF.
- A concave focus can be found with a card in sunlight; a convex focus cannot.
- Concave: reflects from the inner surface, converging, with C and F in front.
- Convex: reflects from the outer surface, diverging, with C and F behind.
- and **: gives ; gives ; gives ; gives cm.
- P, F and C are equally spaced, each apart.
- The spoon test: the bowl is concave and magnifies close up; the back is convex and always diminishes.
- "Concave magnifies" holds only within the focus — beyond it the image shrinks and inverts.
- Rule 1: a ray parallel to the axis reflects through F (or appears to come from F).
- Rule 2: a ray through F reflects parallel to the axis — rule 1 reversed, since light paths are reversible.
- Rule 3: a ray through C returns along its own path, because a radius is the normal, so .
- Rule 4: a ray at the pole reflects with about the principal axis.
- Two rays locate an image; a third is a check. Use dotted lines behind the mirror.
- With cm and an object at cm, rules 1 and 3 cross between F and C, giving a real, inverted, diminished image.
- The rules hold accurately only for a small aperture — a wide spherical mirror blurs, which is why reflectors are parabolic.
Hold a spoon at arm's length, then bring it close to your eye, and note the exact moment the image flips upright — that moment is the focus.