Chapter 7: Reflection of Light
In 1668, in Cambridge, Isaac Newton finished building a telescope no longer than a forearm. Every good telescope of that time used a large glass lens at the front to gather light. Newton’s had no such lens at all. At the closed end of his tube sat a small dish of polished metal, curved inward, barely wider than a coin.
Newton had a reason for this odd design. Light passing through a lens spread into coloured fringes that blurred the image, and he had convinced himself the problem could not be cured. Light bouncing off a mirror, however, showed no such fringes. So he cast a disc of a copper–tin alloy, ground its face into a shallow curve, and polished it until it shone.
The curved surface gathered the faint light of the sky and turned it back into a tiny bright image inside the tube. A second small flat mirror, tilted, threw that image sideways to an eyepiece in the wall of the tube, where the eye could reach it. The little instrument magnified far more than its size suggested, and a few years later the Royal Society in London examined it with great interest.
Two mirrors, one flat and one curved, doing two different jobs. This chapter is about exactly that difference — how light behaves when it strikes a polished surface, and what a flat mirror and a curved mirror each do with it.
Diagram
Fig. 7.0-1 — Newton’s Reflecting Telescope Simple cutaway sketch of a short tube open at one end; at the closed end a small concave metal mirror; parallel light rays entering the open end, striking the concave mirror and converging; a small flat mirror tilted at 45° in the path of the converging rays, turning them sideways through a hole in the side wall to an eyepiece; label the open end, concave mirror, flat mirror, eyepiece, and the direction of incoming light.
7.1 Reflection Of Light
Hold a torch so that its beam falls on a polished steel plate, and the beam does not stop there. It leaves the plate again and travels off in a new direction. Some part of the light that arrives at a surface is always sent back into the medium it came from.
The light that is sent back in this way is called reflected light — light returned into the same medium after striking a surface. The process itself is called reflection.
Not all of the arriving light comes back. The rest of it is dealt with in one of two ways. If the surface is opaque, that remaining light is absorbed by it. If the surface is transparent, part of the light passes through and part is absorbed.
Why We See Things
Reflection is the reason the world is visible to us. An object can be seen only when light from it enters the eye.
A few objects give out light of their own. Such luminous bodies — the Sun, a burning lamp, a glowing filament — are seen directly, because the light entering the eye starts at the object itself.
Most things around you give out no light at all. A non-luminous object such as a book, a wall or a chair is seen only because light falling on it is reflected from its surface and some of that reflected light reaches the eye. Switch off every source of light in a windowless room and the objects do not vanish, but they become invisible, because there is nothing left for them to reflect.
Surfaces differ greatly in how much light they send back. A dark cloth returns very little; a highly polished and silvered surface, such as a plane mirror — a flat, smooth reflecting surface — returns almost all of the light falling on it. Even a good plane mirror, however, does not reflect the whole of the incident light; a small part is always absorbed.
7.1.1 How A Plane Mirror Is Made
A plane mirror is not a sheet of metal. It is made from a glass plate a few millimetres thick, and the two faces of that plate do different jobs, as shown in Fig. 7.1-1.
One face is polished to a high degree of smoothness. This becomes the front surface of the mirror, and it is the side that faces you. The other face, at the back, is silvered — a thin layer of silver, mercury or another suitable material is deposited on it.
Diagram
Fig. 7.1-1 — The Two Surfaces Of A Plane Mirror Enlarged edge-on section of a plane mirror drawn as a thick vertical strip; from left to right label: polished surface (front, facing the incoming light), the body of the glass plate, the silvered surface at the back, and the opaque protective coating behind the silvering; an arrow on the left showing light entering at the polished surface and a second arrow showing it returning after reflection at the silvered surface.
The silvering is delicate, so it is protected by a layer of opaque material coated over it. This coating does not reflect light; it only shields the silver behind.
Light therefore enters through the polished front surface, travels through the thin glass, and is strongly reflected at the silvered back surface before returning to your eye.
7.1.2 Kinds Of Reflection
Whether a reflected beam stays neat or spreads out in all directions depends entirely on the surface it struck. On this basis reflection is of two kinds: regular reflection and irregular reflection.
7.1.2.1 Regular Reflection
Regular reflection occurs when a beam of light falls on a smooth and polished surface, such as a plane mirror.
In Fig. 7.1-2, a parallel beam of light arrives at a plane mirror. Every ray of the beam meets the same flat surface, so every ray is turned through the same amount. The reflected beam therefore leaves as a parallel beam too, travelling in one fixed direction.
Diagram
Fig. 7.1-2 — Regular Reflection At A Plane Mirror Four parallel rays, drawn with arrowheads, striking the flat surface of a plane mirror at the same slant and leaving as four parallel reflected rays in a single fixed direction; label the plane mirror and mark the incident beam and the reflected beam.
Because the whole reflected beam goes one way, it can be seen only from one particular direction. Stand anywhere else and you see nothing of it. This is why the bright patch thrown by a mirror on a sunny day is visible from one place and invisible a step to the side.
7.1.2.2 Irregular Reflection
Irregular reflection occurs when a beam of light falls on a rough or ordinary surface, such as the wall of a room or the page of a book.
A wall or a page looks smooth, but it is not. Seen closely, it is quite uneven, carrying a great many small projections. Each tiny part of such a surface is tilted differently from its neighbours.
In Fig. 7.1-3, the rays of a parallel beam fall at different points on a rough surface. Every ray still obeys the laws of reflection at the small patch it strikes. But since those patches face in different directions, the rays leave in different directions. This spreading is called irregular reflection, or diffused reflection.
Diagram
Fig. 7.1-3 — Irregular Reflection At A Rough Surface Four parallel rays with arrowheads striking a surface drawn as an irregular, bumpy line; each reflected ray leaving in a clearly different direction so that the reflected light spreads widely; label the rough surface and mark the incident beam and the scattered reflected rays.
The reflected light therefore spreads over a wide area and follows no particular direction, and so it can be seen from anywhere in the room. The two kinds of reflection are set side by side in Table 7.1.
Notice the last row. Irregular reflection is not lawless reflection. The laws are obeyed just as strictly on a rough wall as on a mirror; it is the surface that is uneven, not the physics.
7.3 The Laws Of Reflection
A light ray does not leave a surface in just any direction. Whatever the surface, and however the ray arrives, it obeys two rules while getting reflected. These two rules are called the laws of reflection.
The First Law
The direction in which a ray leaves is fixed by the direction in which it arrived. If a ray comes in steeply, it goes out steeply; if it comes in almost grazing the surface, it leaves almost grazing it. Stated exactly: the angle of incidence is equal to the angle of reflection.
\[\\angle i = \\angle r\]
where \(i\) is the angle of incidence and \(r\) the angle of reflection, both measured from the normal at the point of incidence and both in degrees. For the ray drawn earlier in Fig. 7.2-1, this law says that \(\\angle AON = \\angle BON\).
The Second Law
The first law fixes the size of the turn, but not the plane in which the ray leaves. A ray could obey \(\\angle i = \\angle r\) and still swing sideways out of the page. The second law rules that out: the incident ray, the reflected ray and the normal at the point of incidence all lie in the same plane.
In Fig. 7.2-1, the ray \(AO\), the normal \(ON\) and the ray \(OB\) are all in one plane — the plane of the page. Nothing steps out of it.
7.3.1 A Ray Incident Normally On A Plane Mirror
One case is worth looking at on its own, because it looks like an exception and is not.
Suppose a ray falls on a plane mirror along the normal itself, as in Fig. 7.3-1. The ray then makes no angle at all with the normal, so the angle of incidence is \(i = 0^\\circ\). By the first law, the angle of reflection must also be \(r = 0^\\circ\).
Diagram
Fig. 7.3-1 — A Ray Incident Normally On A Plane Mirror A horizontal plane mirror MM₁ drawn as a hatched line. A single vertical ray AO arrives from above and meets the mirror at O, perpendicular to it. The reflected ray is drawn along the same vertical line going back up, with arrowheads showing both directions to make clear that the path is retraced. Label the mirror MM₁, the point of incidence O, the incident ray AO and the reflected ray OA, and mark i = 0°, r = 0°.
A reflected ray at \(0^\\circ\) to the normal is a ray travelling back along the normal. So the ray simply returns the way it came: the ray \(AO\) is reflected along \(OA\) and retraces its own path.
7.4 Experimental Verification Of The Laws Of Reflection
The two laws were stated in the last section. They can be checked on a drawing board with nothing more costly than four pins, a small mirror and a sheet of paper. The set-up is shown in Fig. 7.4-1 and the order of the steps in FC 7.1.
Setting Up The Apparatus
Fix a sheet of white paper on a drawing board and draw a straight line \(MM\_1\) on it. Take a point \(O\) near the middle of this line, and draw a line \(OA\) such that \(\\angle MOA\) is less than \(90^\\circ\) — say \(\\angle MOA = 60^\\circ\). Then draw the normal \(ON\) on the line \(MM\_1\) at the point \(O\).
Now stand a small plane mirror vertically, held by a stand, with its silvered surface resting along the line \(MM\_1\). The line \(MM\_1\) on the paper is therefore the position of the reflecting surface itself.
Notice that \(\\angle MOA\) is the angle the incident ray makes with the mirror, not with the normal. Since \(ON\) is perpendicular to \(MM\_1\), the angle of incidence is \(i = 90^\\circ - \\angle MOA\). For \(\\angle MOA = 60^\\circ\), the angle of incidence is \(30^\\circ\).
Diagram
Fig. 7.4-1 — Verification Of The Laws Of Reflection A drawing board with a white sheet on it. A line MM₁ runs across the sheet with a small plane mirror standing vertically on a stand, its silvered face on MM₁. A point O is marked at the middle of MM₁, with the dashed normal ON drawn perpendicular to MM₁ at O. A line OA runs up to the left from O, carrying two pins P and Q stuck upright about 5 cm apart. To the lower left, a line OB runs from O carrying two more upright pins R and S, with an eye symbol beyond S looking along OB towards the mirror. Behind the mirror, mark the images P′ and Q′ as small circles. Label the mirror, the stand, the pins P, Q, R, S, the images P′ and Q′, the normal ON, and the angles i and r on either side of ON.
Taking The Reading
Fix two pins \(P\) and \(Q\) vertically on the line \(OA\), about \(5\\ \\text{cm}\) apart. Now keep your eye on the other side of the normal, but still on the same side of the mirror, and look into the mirror. You will see the images \(P'\) and \(Q'\) of the two pins.
Fix a third pin \(R\) so that it is in line with the images \(P'\) and \(Q'\) as seen in the mirror. Then fix a fourth pin \(S\) so that it too is in line with the pin \(R\) as well as with the images \(P'\) and \(Q'\). When all four line up, your eye is looking straight along the path the reflected light is taking.
Draw small circles on the paper around the positions of the pins. Remove the pins and join the point \(O\) to the pin points \(S\) and \(R\) by a line \(OB\).
Flowchart
FC 7.1 — Steps In Verifying The Laws Of Reflection Fix paper on board and draw MM₁ → mark O and draw the incident line OA → draw the normal ON at O → stand the mirror with its silvered face on MM₁ → fix pins P and Q on OA → look into the mirror and see the images P′ and Q′ → fix pin R in line with P′ and Q′ → fix pin S in line with R, P′ and Q′ → circle the pin positions and remove the pins → join O to S and R to get the reflected ray OB → measure ∠AON and ∠BON → repeat for other directions of OA
In Fig. 7.4-1, \(AO\) is now the incident ray and \(OB\) the reflected ray. The angle \(\\angle AON = i\) is the angle of incidence and \(\\angle BON = r\) is the angle of reflection. Measure both angles with a protractor and record them.
The whole experiment is then repeated for \(\\angle MOA\) equal to \(50^\\circ\), \(40^\\circ\) and \(30^\\circ\), which give angles of incidence of \(40^\\circ\), \(50^\\circ\) and \(60^\\circ\).
The Observations
The four sets of readings are entered in Table 7.2.
Table 7.2 — Observations for the reflection of light at a plane mirror
| S. No. | Angle of incidence \(i = \\angle AON\) (in degrees) | Angle of reflection \(r = \\angle BON\) (in degrees) |
|---|---|---|
| 1. | 30 | 30 |
| 2. | 40 | 40 |
| 3. | 50 | 50 |
| 4. | 60 | 60 |
In every case the angle of incidence comes out equal to the angle of reflection. This verifies the first law of reflection.
What The Same Experiment Says About The Second Law
The second law is verified by the arrangement itself, and it is easy to miss.
The experiment is performed on a flat drawing board, with the mirror standing normal to the plane of the board on which the white sheet is fixed. The lower tips of all four pins rest on that same sheet, so they all lie in one plane — the plane of the paper.
Since the incident ray, the reflected ray and the normal have all been drawn on that one sheet, they all lie in one plane. This verifies the second law of reflection.
7.5 Formation Of An Image By Reflection
From each point of an illuminated object, rays of light travel out in all directions. A mirror catches some of them, and what it does with them decides where the image lies.
To find the position of the image of an object formed by a mirror after reflection, we need to consider at least two rays of light incident on the mirror from one point of the object. One ray is not enough, because a single reflected ray is a line, and a line by itself does not fix a point.
Each of the two incident rays gets reflected obeying the laws of reflection. The point where the two reflected rays actually meet, or where they appear to meet when produced backwards, gives the position of the image of that point of the object.
Repeat this for other points of the object and we get the position of the image of each of them. Joining these points gives the complete image of the object.
An image can be of two types: a real image and a virtual image.
7.5.1 Real Image
Sometimes the reflected rays really do come together. They travel to a point, cross there, and go on past it. Light energy is genuinely arriving at that point, so a screen held there catches a bright patch.
An image formed in this way is a real image, and it is always inverted with respect to the object. It is formed when light rays, after reflection, actually intersect. For example, the image of a distant object formed by a concave mirror is real.
Diagram
Fig. 7.5-1 — Formation Of A Real Image Schematic: a point object O on the left; two rays leaving O and striking a curved (concave) reflecting surface on the right; the two reflected rays drawn as solid lines converging and crossing at a point I in front of the mirror; a small vertical screen drawn at I showing a bright spot. All rays solid, with arrowheads showing the direction of travel. Label the object O, the mirror, the reflected rays, the real image I and the screen.
7.5.2 Virtual Image
At other times the reflected rays never meet at all. They leave the mirror spreading apart, and however far they travel they only get farther from one another. No light ever arrives at a single point, so no screen can catch anything.
Yet the eye still sees an image. If those spreading rays are produced backwards, geometrically they do intersect, at a point behind the mirror. The eye, receiving the diverging rays, judges them to have come from that point, and the image is seen there.
An image formed in this way is a virtual image, and it is always erect. It cannot be obtained on a screen. For example, the image of an object formed by a plane mirror or by a convex mirror is virtual.
Diagram
Fig. 7.5-2 — Formation Of A Virtual Image Schematic: a point object O in front of a vertical plane mirror drawn as a hatched line; two rays leaving O and striking the mirror; the two reflected rays drawn as solid lines diverging away from the mirror towards an eye symbol on the left; both reflected rays produced backwards as dashed lines behind the mirror, meeting at a point I. Label the object O, the mirror, the reflected rays, the dashed produced-back rays, the virtual image I and the eye.
7.5.3 Distinction Between A Real And A Virtual Image
The two kinds of image are compared feature by feature in Table 7.3.
7.6 Image Of A Point Object Formed By A Plane Mirror
We now apply the method of the last section to the simplest possible object — a single point.
In Fig. 7.6-1, let \(MM\_1\) be a plane mirror in front of which a point object \(O\) is placed. From the object \(O\), rays of light travel out in all directions. To show how the image is formed, we pick two of them.
Let \(OA\) and \(OB\) be two rays incident from the object \(O\). At \(A\) the normal to the mirror is \(AN\_1\), and at \(B\) it is \(BN\_2\). Each ray obeys the first law of reflection at the point where it lands, so \[\\angle CAN\_1 = \\angle OAN\_1 \\qquad \\text{and} \\qquad \\angle DBN\_2 = \\angle OBN\_2\] and the two rays are reflected along the directions \(AC\) and \(BD\) respectively.
Diagram
Fig. 7.6-1 — Image Of A Point Object In A Plane Mirror A horizontal plane mirror MM₁ drawn as a hatched line, with a point object O marked above it on the left. Two rays OA and OB leave O and strike the mirror at A and B. Dashed normals AN₁ and BN₂ are drawn perpendicular to the mirror at A and B. The reflected rays AC and BD are drawn going up to the right, diverging, with an eye symbol placed between C and D looking back at the mirror. Both reflected rays are produced backwards as dashed lines below the mirror, meeting at a point I directly below O. Mark the equal angles at A and at B. Label O, A, B, C, D, I, the normals AN₁ and BN₂, and the mirror MM₁.
Now look at the reflected rays \(AC\) and \(BD\). They are moving apart. Seen from a position between \(C\) and \(D\), they appear to come from some point \(I\) behind the mirror, and that point \(I\) is the image of the object \(O\).
To locate \(I\) on paper, the reflected rays \(AC\) and \(BD\) are produced backwards, and the point where they meet gives the position of the image \(I\).
The image is virtual, because the reflected rays \(AC\) and \(BD\) do not actually meet at \(I\). They only appear, to our eye, to come from the point \(I\).
7.7 Image Of An Extended Object Formed By A Plane Mirror
A real object is not a point. It has length, so it is an extended object, and its image has to be built up point by point.
In Fig. 7.7-1, let \(MM\_1\) be a plane mirror in front of which an extended object \(AB\) is placed. From all points of the object, light rays travel in all directions. We consider only two rays incident on the mirror from each of the two end points \(A\) and \(B\).
Locating The Image Of Each End
Let \(AP\) and \(AQ\) be the two rays incident on the mirror from the point \(A\) of the object. They get reflected from the mirror as \(PP'\) and \(QQ'\) respectively. These reflected rays, when produced backwards, meet at a point \(A'\). Thus \(A'\) is the virtual image of the point \(A\).
Similarly, let \(BR\) and \(BS\) be the two rays incident on the mirror from the point \(B\). They are reflected as \(RR'\) and \(SS'\), and these reflected rays meet at a point \(B'\) when produced backwards. Thus \(B'\) is the virtual image of the point \(B\).
Diagram
Fig. 7.7-1 — Image Of An Extended Object In A Plane Mirror A vertical plane mirror MM₁ drawn as a hatched line on the right. To its left, an extended object AB drawn as an arrow, with A the upper end and B the lower end. From A, two rays AP and AQ strike the mirror at P and Q and are reflected as PP′ and QQ′, drawn diverging to the left. From B, two rays BR and BS strike the mirror at R and S and are reflected as RR′ and SS′. All four reflected rays are produced backwards as dashed lines behind the mirror; those from A meet at A′ and those from B meet at B′. Draw A′B′ as the image arrow behind the mirror. Mark N as the foot of the perpendicular from B to the mirror, and show BN and B′N as equal lengths. Label the object AB, the image A′B′, the points P, Q, R, S and the mirror MM₁.
In the same way, virtual images are formed between \(A'\) and \(B'\) for all the other points of the object \(AB\). Thus \(A'B'\) is the virtual image of the object \(AB\).
What The Image Looks Like
Three things about this image can be read straight off the diagram.
The image is erect — the end \(A'\) is on the same side as the end \(A\), so the image is not turned over.
The image is of size equal to that of the object. A plane mirror neither enlarges nor shrinks.
The normal distance of each point of the image behind the mirror is the same as the normal distance of the corresponding object point in front of the mirror. In Fig. 7.7-1, this means \(BN = B'N\), where \(N\) is the foot of the perpendicular dropped from \(B\) on the mirror. This last result is important enough to be proved on its own, which is done in the next section.
7.8 Position Of The Image In A Plane Mirror
The last section showed that the image of each point lies behind the mirror on the perpendicular from that point. How far behind? Exactly as far as the object is in front:
The image \(I\) is as far behind the mirror as the object \(O\) is in front of it. That is, the perpendicular distance of the image from the mirror is equal to the perpendicular distance of the object from the mirror.
This is not an assumption. It follows from the first law of reflection alone, and the proof is short.
The Proof
Fig. 7.8-1 shows the formation of the image of a point object \(O\) by a plane mirror \(MM\_1\).
A ray \(OF\) is incident normally on the mirror at the point \(F\). Since \(\\angle i = 0^\\circ\), we get \(\\angle r = 0^\\circ\), so this ray is reflected back along the same path, that is, along \(FO\).
The other incident ray \(OA\) gets reflected along \(AC\), where \(AN\) is the normal drawn at the point \(A\) on the mirror. The reflected rays \(FO\) and \(AC\) meet at a point \(I\) when they are produced backwards. The point \(I\) is the virtual image of the point object \(O\). We are to prove that \(IF = OF\).
Diagram
Fig. 7.8-1 — Proof Of The Position Of The Image A horizontal plane mirror MM₁ drawn as a hatched line. A point object O is marked above the mirror on the left. A vertical ray OF drops from O to the point F on the mirror and is reflected straight back along FO, shown with arrowheads both ways. A second ray OA leaves O and strikes the mirror at A to the right of F; a dashed normal AN is drawn upward at A, and the reflected ray AC leaves A up to the right. The reflected rays are produced backwards as dashed lines below the mirror, meeting at the point I vertically below F. Mark the angle i between OA and AN and the angle r between AN and AC. Draw the dashed line IA. Label O, F, A, C, I, N, and the mirror MM₁, and mark OF and IF as the two distances to be compared.
For the incident ray \(OA\) reflected as \(AC\),
\(\\angle OAN\) = angle of incidence \(i\)
\(\\angle CAN\) = angle of reflection \(r\)
By the law of reflection, angle of incidence = angle of reflection,
\[\\angle OAN = \\angle CAN \\qquad \\dots (7.2)\]
Both the normal \(AN\) at the point \(A\) and the ray \(OF\) are perpendicular to the same mirror, so \(AN\) and \(OF\) are parallel to each other. This is what lets the next two steps be taken, since a pair of parallel lines cut by a transversal gives equal alternate angles and equal corresponding angles.
With \(OA\) as the transversal, \(\\angle OAN = \\angle AOF\) (alternate angles)
With \(IC\) as the transversal, \(\\angle CAN = \\angle AIF\) (corresponding angles)
Therefore, from equation (7.2),
\[\\angle AOF = \\angle AIF \\qquad \\dots (7.3)\]
Now consider the triangles \(AOF\) and \(AIF\):
\(\\angle AOF = \\angle AIF\), from equation (7.3)
\(\\angle AFO = \\angle AFI \\ (= 90^\\circ)\), since \(OF\) is normal to the mirror
and \(FA\) is the common side.
Therefore the triangles \(AOF\) and \(AIF\) are congruent.
Hence \[OF = IF \\qquad \\dots (7.4)\]
Since \(OF\) is the normal drawn from the object \(O\) to the mirror, the normal distance of the object from the mirror is equal to the normal distance of the image from the mirror. This is what we set out to prove.
The Result In Words
The image is situated on the normal drawn from the object to the mirror, and it is as far behind the mirror as the object is in front of it.
7.9 Lateral Inversion
Stand in front of a mirror wearing a ring on a finger of your left hand. In the image, the ring appears on a finger of the right hand, as shown in Fig. 7.9-1. A shirt pocket stitched on the left appears on the right in the image. The image is not upside down — it is left-right swapped.
This interchange of the left and right sides is called lateral inversion.
Diagram
Fig. 7.9-1 — A Ring Seen In A Plane Mirror Three panels in a row. Left panel: a sketch of a left hand, palm facing the viewer, with a ring drawn on one finger; label it RING. Middle panel: a vertical plane mirror drawn as a hatched strip; label PLANE MIRROR. Right panel: the image — the same hand with the ring now on the corresponding finger of the other side; label RING and label the whole panel IMAGE.
Letters In A Mirror
Fig. 7.9-2 shows how the image of the letter P is formed in a plane mirror. The letter P appears in a plane mirror with its loop on the other side, and the letter F appears with its arms turned the other way.
Diagram
Fig. 7.9-2 — Lateral Inversion Of A Letter A vertical plane mirror drawn as a hatched strip down the middle. To its left, a large capital letter P standing upright. From the top and bottom of the letter, pairs of rays are drawn striking the mirror and reflecting back towards an eye placed to the lower left; each reflected ray is produced backwards as a dashed line. Behind the mirror, the dashed produced-back rays meet to form the image of the letter, drawn left–right reversed and outlined with a dashed line. Label the object letter, the mirror, and the laterally inverted image.
Not every letter shows the effect. In letters such as A, H, I, M, O, T, U, V, W, X and Y the lateral inversion is not noticeable, since the image remains unchanged. Each of these letters is symmetrical about a vertical line passing through its mid point, so swapping left for right leaves the shape exactly as it was.
It is because of lateral inversion that the text of a page is difficult to read in its image formed by a plane mirror. Turn the situation around, however, and it becomes useful: text written in laterally inverted letters is read easily in the image formed by a plane mirror.
One more point, which will matter later in this chapter: the image formed by a spherical mirror is also laterally inverted.
7.10 Characteristics Of The Image Formed By A Plane Mirror
Everything established in the last four sections can now be gathered into one short list. The image formed by a plane mirror has the following characteristics:
it is upright, or erect;
it is virtual;
it is of the same size as the object; and
it is laterally inverted.
Its location is given by the fact that the image is situated at the same perpendicular distance behind the mirror as the object is in front of it.
These four points, together with the location, answer every ordinary question about a plane mirror. Notice how little is left free: whatever the object, and wherever it stands, the plane mirror does the same four things to it.
7.10.1 Shift Of The Image When The Object Moves
Since the image always keeps step with the object, moving the object must move the image.
If the object is shifted by a distance \(d\) towards the mirror, the image also shifts by the same distance \(d\) towards the mirror, from the other side. The object has come \(d\) nearer and the image has come \(d\) nearer, so the separation between the object and the image decreases by \(2d\).
The same holds if the object moves away: the separation then increases by \(2d\).
In the same way, if an object moves with a speed \(v\) towards a mirror, its image moves with the speed \(v\) towards the mirror as well, from behind it. The two are approaching each other, so relative to the object the image appears to move with a speed \(2v\). Here \(v\) is measured in m s⁻¹.
A Worked Case
Fig. 7.10-1 follows a boy who steps back from a plane mirror.
A boy standing at \(A\), a distance \(5\\ \\text{m}\) away from a plane mirror, finds his image at \(A'\), at a distance \(10\\ \\text{m}\) from him. Now if he moves to position \(B\), a distance \(2\\ \\text{m}\) still farther away from the mirror, his distance from the mirror becomes \(7\\ \\text{m}\). His image \(B'\) is then \(7\\ \\text{m}\) behind the mirror, and so appears to be at a distance of \(14\\ \\text{m}\) from where he is standing.
Diagram
Fig. 7.10-1 — Displacement Of The Image When The Object Moves A vertical plane mirror drawn as a hatched line down the middle. On the left, two stick figures: one at A, 5 m from the mirror, and one at B, 7 m from the mirror, with B farther to the left; label the left side OBJECT. On the right, two mirror-image stick figures drawn in dashed outline: one at A′, 5 m behind the mirror, and one at B′, 7 m behind it; label the right side IMAGE. Mark the distances 5 m and 5 m on either side of the mirror between A and A′, 7 m and 7 m between B and B′, and the overall separations 10 m (A to A′) and 14 m (B to B′).
The boy moved back by \(2\\ \\text{m}\), the image moved back by \(2\\ \\text{m}\), and the gap between them grew by \(4\\ \\text{m}\) — that is, by \(2d\).
7.11 Images Formed In Two Inclined Mirrors
A single plane mirror gives one image. Set two plane mirrors at an angle to each other, put an object in between, and you see many.
The reason is that light does not stop after one reflection. The rays reflected from one mirror fall on the other mirror and are reflected again. In other words, the image formed by one mirror acts as an object for the other mirror. This continues till no more reflection can occur on any mirror.
Where do all these images lie? Not scattered anywhere, but on a definite circle: the object and the images formed by the two inclined mirrors lie on the circumference of a circle whose centre lies at the point of intersection of the two mirrors, and whose radius is equal to the distance of the object from that point of intersection.
Diagram
Fig. 7.11-1 — Object And Images In Two Inclined Mirrors Two plane mirrors drawn as hatched lines meeting at a point O and inclined at an angle θ, with the angle marked. An object P is placed between them, at some distance from O. A circle is drawn with centre O and radius OP, passing through P. Several images are marked as small dots on the circumference of this circle, on either side of the two mirrors, each labelled I₁, I₂, I₃ and so on. Label the two mirrors, the point of intersection O, the angle θ, the object P and the circle.
How Many Images?
The number of images formed depends on the angle \(\\theta\) between the two mirrors, through the number \[n = \\frac{360^\\circ}{\\theta}\] where \(\\theta\) is the angle between the mirrors in degrees and \(n\) is a pure number with no unit. Two cases are possible.
Case (1): when \(n\) is odd. The count then depends on where the object is placed.
the number of images is \(n\), when the object is placed asymmetrically between the mirrors;
the number of images is \(n - 1\), when the object is placed symmetrically, that is, on the bisector of the angle between the mirrors.
For example, if \(\\theta\) is \(72^\\circ\), then \[n = \\frac{360^\\circ}{72^\\circ} = 5\] So the images formed will be \(n = 5\) for the object placed asymmetrically between the mirrors, and \(n - 1 = 4\) if the object is placed symmetrically on the bisector, because two images now overlap.
Case (2): when \(n\) is even. The number of images is always \(n - 1\), for all positions of the object in between the mirrors.
For example, if the angle between two mirrors is \(60^\\circ\), then \[n = \\frac{360^\\circ}{60^\\circ} = 6\] and the number of images is \(n - 1 = 5\). That is, five images will be formed for all positions of the object in between the mirrors.
We shall now consider image formation in two special cases:
when the two mirrors are parallel to each other, and
when the two mirrors are perpendicular to each other.
7.12 Multiple Images Formed By Two Parallel Mirrors
Take the two mirrors of the last section and open them out until they are parallel. The angle between them is then \(\\theta = 0^\\circ\), and \[n = \\frac{360^\\circ}{\\theta} = \\frac{360^\\circ}{0^\\circ} = \\infty\]
Dividing by zero gives no finite answer, and that is exactly the physical situation: there is no last reflection. So the number of images of an object kept in between two parallel mirrors will be infinite.
For two mirrors kept parallel to each other, an infinite number of images are formed for an object kept in between them.
How The Series Is Built Up
In Fig. 7.12-1, \(AM\) and \(BM\_1\) are the two plane mirrors placed parallel to each other, and an object \(P\) is placed in between them.
The image of \(P\) formed by reflection from mirror \(AM\) is \(I\_1\), and by reflection from mirror \(BM\_1\) it is \(I'\). Two mirrors, so two first images, one behind each.
Now \(I\_1\) acts as a virtual object for mirror \(BM\_1\), and the image \(I\_2\) is formed. Similarly, \(I'\) acts as the virtual object for mirror \(AM\) and the image \(I''\) is formed. Each new image becomes an object for the opposite mirror, and in this manner many images are formed, in two endless sets.
Diagram
Fig. 7.12-1 — Images In Two Parallel Mirrors Two vertical plane mirrors AM and BM₁ drawn as hatched strips facing each other, a short distance apart, with an object P marked as a point between them on a horizontal line. Along that horizontal line, mark the images: I₁ just behind mirror AM and I₂ farther behind it, and I′ just behind mirror BM₁ and I″ farther behind it, spacing them so that each is as far behind its mirror as the corresponding object is in front. Draw the ray path for one set only: two rays from P striking BM₁, reflecting to AM, reflecting again, and emerging towards an eye symbol; produce the emerging rays backwards as dashed lines to meet at I₂. Label both mirrors, the object P, all four images, and the horizontal line as the perpendicular from the base of the object on the mirrors.
Fig. 7.12-1 shows the ray diagram for the formation of one set of images only, that is, \(I\_1\), \(I\_2\), \(I\_3\) and so on. The formation of the other set, \(I'\), \(I''\) and so on, can be obtained in the same way. All the images lie on the perpendicular drawn from the base of the object on the mirrors.
Why You Never Actually See An Infinite Number
The brightness of the remote images keeps on decreasing. At each reflection some light is absorbed by the mirror, so the light that has bounced back and forth six or seven times carries very little energy left. Thus not more than a few images are seen, however carefully you look.
Uses And A Note
Use: in a showroom and in a barber’s shop, mirrors are arranged in this manner, so that a customer can see the back of the head as well as the front.
Note: a thick plane mirror also forms a multiple number of images, due to repeated reflections within the glass itself, between the front surface and the back reflecting surface. Out of these, the second image, formed by reflection at the back reflecting surface, is the brightest, since the silvered back is by far the better reflector.
7.13 Images Formed By Two Perpendicular Mirrors
The second special case is the right angle. When two mirrors are kept perpendicular to each other, \(\\theta = 90^\\circ\), so \[n = \\frac{360^\\circ}{\\theta} = \\frac{360^\\circ}{90^\\circ} = 4\]
Here \(n\) is even, so by Case (2) of Section 7.11 the number of images formed is \(n - 1 = 3\), whatever the position of the object between the mirrors.
For two mirrors kept perpendicular to each other, three images are formed for an object kept in between them.
Where The Three Images Come From
In Fig. 7.13-1, \(AO\) and \(OB\) are the two plane mirrors kept mutually perpendicular to each other, and \(P\) is an object placed in between them.
The image of the object \(P\) formed by reflection from the mirror \(OB\) is \(P\_2\), and by reflection from the mirror \(OA\) it is \(P\_1\). So far, two images — one for each mirror.
Now the image \(P\_2\) acts as the virtual object for the mirror \(OA\), since the rays after reflection from mirror \(OB\) fall on the mirror \(OA\). For this, the image formed by mirror \(OA\) is \(P\_3\).
Similarly, \(P\_1\) acts as the virtual object for the mirror \(OB\), since the rays after reflection from mirror \(OA\) fall on the mirror \(OB\). For this, the image formed by mirror \(OB\) is \(P\_4\).
And here is the point of the whole case: \(P\_4\) is at the same point as \(P\_3\). The image \(P\_4\) of \(P\_1\) in mirror \(OB\) overlaps with the image \(P\_3\) of \(P\_2\) in mirror \(OA\). Two twice-reflected images fall on top of each other and are seen as one. As such, three images of the object are formed in this case.
Diagram
Fig. 7.13-1 — Images In Two Perpendicular Mirrors Two plane mirrors AO and OB drawn as hatched lines meeting at right angles at O, one vertical and one horizontal, forming the corner of a quadrant. An object P is placed inside the right angle. A circle is drawn with centre O and radius OP, passing through P. On this circle mark three images: P₁ directly across the mirror OA, P₂ directly across the mirror OB, and P₃ (labelled also as P₄) diagonally opposite P. Draw the ray diagram for the formation of P₂ and P₃ only: two rays from P striking OB, reflecting onto OA, reflecting again and emerging to an eye symbol, with the emerging rays produced backwards as dashed lines to P₃. Label the mirrors, O, P, the three images and the circle of radius OP.
Fig. 7.13-1 shows the ray diagram for the formation of the images \(P\_2\) and \(P\_3\). In the same way we can draw the ray diagram for the formation of the images \(P\_1\) and \(P\_4\).
Geometrically it can be seen that the images \(P\_1\), \(P\_2\), \(P\_3\) and the object \(P\) all lie on a circle whose centre is at \(O\), the point of intersection of the two mirrors \(OA\) and \(OB\), and whose radius is \(OP\). This is the circle result of Section 7.11, showing up again in a particular case.
7.14 Uses Of A Plane Mirror
Plane mirrors find wide applications in our daily life. Some of them are given below.
(1) As a looking glass. The most common and wide use of all. The image is erect, of the same size and at the same distance behind the mirror, so what you see is a faithful — if laterally inverted — copy of yourself.
(2) In the optician’s room, to increase the effective length of the room. A plane mirror is kept on the front wall and the sign board of letters on the opposite wall, just behind the patient. The patient does not read the board directly; the patient reads its image. Since the image lies as far behind the mirror as the board is in front of it, the board is seen at nearly double the length of the room. A small room is thus made to serve as a long one.
(3) In the barber’s shop, for seeing the hair at the back of the head. Here two mirrors facing each other are fixed on opposite walls, at the front and back of the viewer, so that the customer sees the back of the head in the front mirror.
(4) In a periscope. Two parallel plane mirrors, each inclined at \(45^\\circ\) with the vertical walls of a long tube, are placed facing each other, as shown in Fig. 7.14-1. Light entering the top opening strikes the upper mirror at \(45^\\circ\), is turned through a right angle down the tube, meets the lower mirror and is turned through a right angle again into the eye. The viewer can therefore see over an obstacle without being seen.
Diagram
Fig. 7.14-1 — A Simple Periscope A tall narrow tube drawn in section, with an opening near the top on the left side and an opening near the bottom on the right side. A plane mirror is fixed near the top, tilted at 45° to the vertical wall, and a second plane mirror near the bottom, tilted at 45° and parallel to the first. Rays of light from a distant object on the left enter the top opening horizontally, strike the upper mirror, travel vertically down the tube, strike the lower mirror, and leave horizontally through the bottom opening towards an eye symbol. Mark the 45° angle at each mirror, and label the tube, both mirrors, the object side and the eye.
(5) In a kaleidoscope. Three plane mirrors inclined with each other at \(60^\\circ\) are used. If small coloured bangle pieces are kept between the mirrors, beautiful hexagonal patterns are seen, because each piece is reflected repeatedly by the three mirrors. On rotating the tube, the pieces shift and the designs of the patterns change.
(6) In solar heating devices. In a solar cooker, a solar water heater and the like, a plane mirror is used to reflect the incident light rays from the Sun on to the substance to be heated, so that the substance receives sunlight from two directions instead of one.
7.15 Spherical Mirrors
Everything so far has been about flat mirrors. Newton’s telescope in the opening story used a mirror that was not flat, and the rest of this chapter is about mirrors of that kind.
A spherical mirror is made by silvering a part of a hollow glass sphere. Take a hollow glass ball, cut a small piece out of it, silver one of its two faces, and what is left is a mirror whose reflecting surface is curved.
The Two Types
A cut piece of a sphere has two faces — one hollow, one bulging. Which of them is silvered decides everything about the mirror, because light is reflected from the face that is not silvered and coated.
Depending on whether the inner or the outer surface of the sphere is silvered, spherical mirrors are of two types: the concave mirror and the convex mirror.
In Fig. 7.15-1, the right part \(X\) of a hollow sphere is silvered and coated on its outer surface, and it forms a concave mirror. The left part \(Y\) of the same sphere is silvered and coated on its inner surface, and it forms a convex mirror.
Diagram
Fig. 7.15-1 — The Two Kinds Of Spherical Mirror Cut From One Sphere A large circle drawn with a dashed line, representing a hollow sphere, with a vertical dashed diameter AB down the middle. The right-hand arc, labelled X, is drawn thick with shading on its outer (right) side to show silvering there, and labelled CONCAVE MIRROR with its reflecting surface marked on the inner (left) side. The left-hand arc, labelled Y, is drawn thick with shading on its inner (right) side to show silvering there, and labelled CONVEX MIRROR with its reflecting surface marked on the outer (left) side. Label ‘silvered surface’ and ‘reflecting surface’ on each arc.
7.15.1 Concave Mirror
A concave mirror is made by silvering the outer, or bulging, surface of the piece of a hollow sphere, such that the reflection takes place from the hollow, or concave, surface, as shown in Fig. 7.15-2(a).
So the surface that faces you and does the reflecting is the caved-in one. The inside of a steel bowl, or the inner side of a spoon, is a rough concave mirror.
7.15.2 Convex Mirror
A convex mirror is made by silvering the inner surface of the piece of a hollow sphere, such that the reflection takes place from the outer, or bulging, surface, as shown in Fig. 7.15-2(b).
Here the surface that faces you and does the reflecting is the one that swells outward, like the back of a spoon.
Diagram
Fig. 7.15-2 — Concave And Convex Mirrors Two side-by-side sketches, each showing a curved strip in section with horizontal arrows of light arriving from the left. (a) CONCAVE MIRROR: the strip curves away from the light so that its hollow face is towards the incoming rays; shade the right-hand (outer) face and label it ‘silvered surface’; label the left-hand face ‘reflecting surface’. (b) CONVEX MIRROR: the strip curves towards the light so that its bulging face is towards the incoming rays; shade the left-hand (inner) face and label it ‘silvered surface’; label the right-hand face ‘reflecting surface’. Label the incoming rays LIGHT in both sketches.
7.17 Reflection Of A Light Ray From A Spherical Mirror
Nothing new happens to light at a curved mirror. From a spherical mirror, reflection of light follows the same laws of reflection as for a plane surface: the angle of incidence \(i\) is equal to the angle of reflection \(r\), and the incident ray, the reflected ray and the normal at the point of incidence all lie in the same plane.
The only difficulty is the normal. On a plane mirror the normal is the same perpendicular everywhere. On a curved mirror the surface tilts from point to point, so the normal has to be found afresh at each point of incidence — and Section 7.16 has already told us how.
Finding The Reflected Ray
In a spherical mirror, to obtain the direction of the reflected ray for a given incident ray, first a normal is drawn at the point of incidence. For this, a line is drawn joining the point of incidence to the centre of curvature \(C\).
This line really is the normal there. At the point of incidence the mirror behaves like the tangent drawn to the sphere at that point, and a radius of a sphere always meets the tangent at right angles. So the line to \(C\) is perpendicular to the mirror surface at the point where the ray lands.
Then a line is drawn on the other side of the normal, making an angle equal to the angle of incidence. This line represents the reflected ray.
Worked Construction
In Fig. 7.17-1, we are to find the direction of the reflected ray for an incident ray \(AD\), on a concave mirror and on a convex mirror, for which \(C\) is the position of the centre of curvature in each case.
For this, the point of incidence \(D\) is joined to the centre of curvature \(C\) to get the normal \(DC\) on the mirror at the point of incidence \(D\). Then the reflected ray \(DB\) is drawn such that \[\\angle BDC = \\angle ADC\] in the concave mirror, and \[\\angle BDC' = \\angle ADC'\] in the convex mirror, where \(C'\) marks the direction of the centre of curvature behind the convex mirror.
Diagram
Fig. 7.17-1 — Reflection Of A Ray Of Light From A Spherical Mirror Two side-by-side sketches. In (a), a concave mirror drawn as a curved arc with its hollow face to the left, its principal axis horizontal, the pole at the arc’s mid point and the centre of curvature C marked on the axis in front of the mirror. An incident ray AD comes in from the lower left and meets the mirror at a point D above the axis. A dashed line DC is drawn from D to C and marked as the normal. The reflected ray DB leaves D on the other side of DC, with the two angles marked i between AD and DC, and r between DC and DB, shown equal. Caption (a) CONCAVE MIRROR. In (b), a convex mirror drawn as a curved arc with its bulging face to the left, principal axis horizontal, and the centre of curvature C marked on the axis behind the mirror. An incident ray AD meets the mirror at D; the normal is drawn as a dashed line from D through the mirror to C and labelled DC′. The reflected ray DB leaves D in front of the mirror, with the equal angles i and r marked on either side of the normal. Caption (b) CONVEX MIRROR.
Note where the centre of curvature sits in the two cases. For the concave mirror it lies in front, so the normal runs forward from the mirror; for the convex mirror it lies behind, so the normal has to be drawn through the mirror and produced backwards to reach \(C\). In both cases the rule is the same — join the point of incidence to \(C\).
7.18 Focus And Focal Length
Light from a very distant source — the Sun, or a star — arrives at a mirror as a beam of parallel rays. What a spherical mirror does with such a beam is the single most useful thing about it, and it is what gives the mirror its focus.
7.18.1 Focus Of A Concave Mirror
In Fig. 7.18-1, rays of light are incident parallel to the principal axis on a concave mirror. Each ray gets reflected from the mirror obeying the law of reflection, angle of incidence \(i\) = angle of reflection \(r\), for which the dotted line joining the point of incidence to the centre of curvature \(C\) acts as the normal at that point.
Because the mirror is hollow, the rays that strike near the top are tilted one way and those that strike near the bottom the other way, so all of them are turned inwards. The concave mirror converges the rays on reflection, and the rays after reflection pass through a single point \(F\) on the principal axis.
This point is called the focus of the concave mirror, represented by the symbol \(F\). A concave mirror has a real focus, because the reflected rays actually meet at this point — a screen held there would show a bright spot.
Diagram
Fig. 7.18-1 — Focus And Focal Length Of A Concave Mirror A concave mirror drawn as a curved arc with its hollow face to the left and the principal axis horizontal through its pole P. Four rays arrive from the left, parallel to the principal axis, striking the mirror at different heights. At two of these points, dotted normals are drawn from the point of incidence to the centre of curvature C on the axis, with the equal angles i and r marked. All four reflected rays are drawn converging and crossing the axis at a single point F between C and P. Label the pole P, the centre of curvature C, the focus F, the principal axis, and mark the distance PF with a double-headed arrow labelled f.
The distance of the focus \(F\) from the pole \(P\) of the mirror is called the focal length of the mirror. That is, focal length \(f = PF\). It is measured in metre, though centimetre is usual in problems.
7.18.2 Focus Of A Convex Mirror
In Fig. 7.18-2, rays of light are incident parallel to the principal axis on a convex mirror. Here the surface bulges towards the light, so the normals lean outwards, and each ray after reflection is turned away from the axis.
The reflected rays therefore diverge and do not meet at any point. But they appear to come from a point \(F\) on the principal axis, behind the mirror. This point is called the focus of the convex mirror, and it is obtained geometrically, when the reflected rays are produced backwards.
A convex mirror thus has a virtual focus: no light ever reaches \(F\), and no screen can catch anything there.
Diagram
Fig. 7.18-2 — Focus And Focal Length Of A Convex Mirror A convex mirror drawn as a curved arc with its bulging face to the left and the principal axis horizontal through its pole P. Four rays arrive from the left parallel to the principal axis and strike the mirror at different heights. At two points of incidence, dotted normals are drawn from the point of incidence through the mirror to the centre of curvature C behind it, with equal angles i and r marked. The four reflected rays are drawn diverging away to the left, and each is produced backwards as a dashed line behind the mirror; all the dashed lines meet at a point F on the axis between P and C. Label P, F, C, the principal axis, and mark the distance PF with a double-headed arrow labelled f.
The distance of the focus \(F\) from the pole \(P\) of the mirror is called its focal length, that is, focal length \(f = PF\), exactly as before.
7.18.3 The Focal Plane
The focus is one point on the axis. But light does not always arrive along the axis.
A plane passing through the focus and normal to the principal axis of the mirror is called the focal plane.
A parallel beam of light inclined to the principal axis gets focused at a point in this plane, as shown in Fig. 7.18-3(a) and Fig. 7.18-3(b). It is not focused at \(F\) itself — \(F\) is reserved for a beam parallel to the axis — but at another point of the same plane.
Which point? It is the point where the ray passing through the centre of curvature meets the focal plane. That particular ray of the beam is easy to trace, since it strikes the mirror along a normal and comes straight back along its own path, so where it cuts the focal plane fixes the meeting point of the whole beam.
Diagram
Fig. 7.18-3 — The Focal Plane Two sketches, one above the other. In (a), a concave mirror with its hollow face to the left, principal axis horizontal, pole P, centre of curvature C and focus F marked on the axis; a vertical dashed line through F, perpendicular to the axis, labelled FOCAL PLANE. Four parallel rays arrive from the upper left, inclined to the principal axis, and strike the mirror; their reflected rays all converge to a single point on the focal plane above the axis. One of the four rays is drawn passing through C and reflecting back along its own path, and this ray is shown meeting the focal plane at that same convergence point. Caption (a) Concave mirror. In (b), a convex mirror with its bulging face to the left, pole P, and F and C marked on the axis behind the mirror; a vertical dashed focal plane through F behind the mirror. Four parallel rays inclined to the axis strike the mirror and diverge after reflection; each reflected ray is produced backwards as a dashed line, and all the dashed lines meet at one point on the focal plane. Caption (b) Convex mirror.
7.19 Convenient Rays For Constructing A Ray Diagram
From each point of an object an infinite number of rays travel out in all directions. To find the position and nature of the image formed due to reflection from a spherical mirror by drawing, we need to consider at least two rays incident on the mirror from the same point of the object.
Any two rays would do, in principle. In practice we choose from a short list, because for these particular rays the reflected direction is known at once, without measuring any angle. Any two of the following are taken as the convenient incident rays:
- a ray passing through the centre of curvature,
- a ray parallel to the principal axis,
- a ray passing through the focus,
- a ray incident at the pole,
- rays parallel to each other,
- rays through a single point on the focal plane.
7.19.1 A Ray Passing Through The Centre Of Curvature
A line joining the centre of curvature to any point on the surface of the mirror is normal to the mirror at that point. Therefore a ray \(AD\) passing through the centre of curvature \(C\), or appearing to pass through the centre of curvature \(C\) in a convex mirror, is incident normally on the spherical mirror.
Since its angle of incidence is zero, the angle of reflection will also be zero, and the ray \(AD\) gets reflected along its own path \(DA\), as shown in Fig. 7.19-1.
A ray passing through, or directed towards, the centre of curvature of a spherical mirror is reflected back along its own path.
Diagram
Fig. 7.19-1 — A Ray Through The Centre Of Curvature Two side-by-side sketches. In (a), a concave mirror with hollow face to the left, principal axis horizontal, pole P and centre of curvature C marked on the axis in front of the mirror. A ray A comes from the upper left, passes through C, and meets the mirror at D below the axis; the reflected ray is drawn back along DA with arrowheads showing both directions. Caption (a) CONCAVE MIRROR. In (b), a convex mirror with bulging face to the left, pole P, and C marked on the axis behind the mirror. A ray A comes from the upper left towards the mirror, meeting it at D, its direction pointing towards C; draw the continuation from D to C as a dashed line, and the reflected ray back along DA with arrowheads both ways. Caption (b) CONVEX MIRROR.
7.19.2 A Ray Parallel To The Principal Axis
This is the rule that defines the focus, seen from the other side.
A ray of light \(AD\) incident parallel to the principal axis, after reflection passes through the focus \(F\) in a concave mirror, or appears to come from the focus \(F\) in a convex mirror, along \(DB\), as shown in Fig. 7.19-2.
A ray incident parallel to the principal axis, after reflection from a spherical mirror, either passes through the focus or appears to be coming from the focus.
Diagram
Fig. 7.19-2 — A Ray Parallel To The Principal Axis Two side-by-side sketches. In (a), a concave mirror with hollow face to the left, principal axis horizontal, pole P, focus F and centre of curvature C marked on the axis in front. A ray A arrives horizontally from the left, parallel to the axis, meets the mirror at D above the axis; a dashed normal DC is drawn and the equal angles i and r marked; the reflected ray DB passes through F and continues to B below the axis. Caption (a) CONCAVE MIRROR. In (b), a convex mirror with bulging face to the left, pole P, and F and C marked on the axis behind the mirror. A ray A arrives horizontally, meets the mirror at D above the axis, and the reflected ray DB is drawn going up to the left; produce DB backwards as a dashed line to pass through F behind the mirror. Mark the dashed normal to C and the equal angles i and r. Caption (b) CONVEX MIRROR.
7.19.3 A Ray Passing Through The Focus
Reverse the last rule and it still holds, since light paths work equally well in either direction.
A ray of light \(AD\) incident on the mirror passing through the focus \(F\) in a concave mirror, or converging at the focus \(F\) in a convex mirror, after reflection becomes parallel to the principal axis as \(DB\), as shown in Fig. 7.19-3.
A ray either incident from the focus, or converging at the focus, after reflection from a spherical mirror becomes parallel to the principal axis.
Diagram
Fig. 7.19-3 — A Ray Through The Focus Two side-by-side sketches. In (a), a concave mirror with hollow face to the left, principal axis horizontal, pole P, focus F and centre of curvature C on the axis in front. A ray A comes from the lower left, passes through F, and meets the mirror at D; the reflected ray DB leaves horizontally, parallel to the principal axis. Mark the dashed normal from D to C and the equal angles i and r. Caption (a) CONCAVE MIRROR. In (b), a convex mirror with bulging face to the left, pole P, F and C on the axis behind the mirror. A ray A is directed towards F, meeting the mirror at D before reaching it; its continuation towards F is drawn as a dashed line. The reflected ray DB leaves horizontally, parallel to the axis. Mark the dashed normal and the equal angles. Caption (b) CONVEX MIRROR.
7.19.4 A Ray Incident At The Pole
At the pole, the principal axis itself is the normal, because the axis runs from \(P\) straight to \(C\).
A ray \(AP\) incident at the pole \(P\) of the mirror gets reflected along a path \(PB\) such that the angle of incidence \(\\angle APC\) is equal to the angle of reflection \(\\angle BPC\) in a concave mirror, or \(\\angle APC' = \\angle BPC'\) in a convex mirror, as shown in Fig. 7.19-4.
For a ray incident at the pole of a spherical mirror, the reflected ray is at an angle of reflection equal to the angle of incidence, with the principal axis as the normal.
Diagram
Fig. 7.19-4 — A Ray Incident At The Pole Two side-by-side sketches. In (a), a concave mirror with hollow face to the left and the principal axis drawn horizontally through the pole P, with C marked on the axis in front of the mirror. A ray A arrives from the upper left and strikes the mirror exactly at P; the reflected ray PB leaves to the lower left. Mark the angle i between AP and the axis, and the angle r between the axis and PB, shown equal. Caption (a) CONCAVE MIRROR. In (b), a convex mirror with bulging face to the left, pole P and C marked on the axis behind the mirror. A ray A strikes the mirror at P from the upper left and the reflected ray PB leaves to the lower left, with the equal angles i and r marked on either side of the principal axis. Caption (b) CONVEX MIRROR.
7.19.5 Rays Parallel To Each Other
There are two conditions for parallel rays.
Rays parallel to each other and parallel to the principal axis are reflected in the manner discussed above in Section 7.19.2 — they all pass through, or appear to come from, the focus.
Rays parallel to each other but inclined to the principal axis will be reflected such that these rays pass through, or appear to pass through, a single focal point on the focal plane of the spherical mirror. This is the case already drawn in Fig. 7.18-3(a) and Fig. 7.18-3(b).
7.19.6 Rays Through A Single Point On The Focal Plane
This is the last case turned round.
When rays through a single point on the focal plane, or appearing to come from a single point in the focal plane, are incident on a spherical mirror, they become parallel to each other after reflection, as shown in Fig. 7.19-5.
Diagram
Fig. 7.19-5 — Rays Through A Point On The Focal Plane Two sketches, one above the other. In (a), a concave mirror with hollow face to the left, principal axis horizontal, pole P, focus F and centre of curvature C on the axis; a vertical dashed line through F perpendicular to the axis, labelled FOCAL PLANE. Four rays spread out from a single point on the focal plane above the axis, strike the mirror at four different heights, and after reflection leave as four rays parallel to one another, inclined to the principal axis, going off to the upper left. Caption (a) Concave mirror. In (b), a convex mirror with bulging face to the left, pole P, and F and C on the axis behind the mirror, with a vertical dashed focal plane through F. Four rays are drawn heading towards a single point on that focal plane behind the mirror, shown as dashed continuations beyond the mirror; after reflection the four rays leave parallel to one another, inclined to the axis. Caption (b) Convex mirror.
The Four Common Rays At A Glance
Table 7.4 collects the four rays used most often in the constructions of the next two sections.
7.20 Ray Diagrams For Images Formed By A Concave Mirror
We shall now find the position, size and nature of the image by drawing the ray diagram for a small linear object placed on the principal axis of a concave mirror at different positions. Seven positions are worth taking one by one, because the image changes completely as the object walks in from far away towards the mirror.
7.20.1 Object At Infinity
When the object, for example the Sun, is at infinity, the rays of light reaching the concave mirror even from the different points of the object subtend nearly the same angle at each point of the mirror, so they can be treated to be parallel to each other.
In Fig. 7.20-1, let two rays \(AD\) and \(BE\) from the same point of the object be incident on the concave mirror, parallel to its principal axis \(CP\). After reflection from the concave mirror they pass through its focus \(F\) as \(DF\) and \(EF\) respectively. The two reflected rays \(DF\) and \(EF\) meet at the focus \(F\). Hence a real and point image is formed at the focus.
Diagram
Fig. 7.20-1 — Object At Infinity A concave mirror drawn as a curved arc with its hollow face to the left, principal axis horizontal through the pole P, with C and F marked on the axis in front of the mirror. Two rays A and B arrive from the far left, parallel to the principal axis, one above the axis striking the mirror at D and one below it striking at E. Both reflected rays are drawn crossing the axis at F, meeting there as a point. Label A, B, D, E, C, F, P and the principal axis.
When the object is at infinity, the image is at the focus \(F\). It is (i) real, (ii) inverted and (iii) diminished to a point.
7.20.2 Object At A Far Distance
The object is now at such a distance that rays from the same point of the object can be taken parallel to each other, but rays coming from different points of the object are inclined at different angles at the mirror. This is the case of a tree, or a building, rather than a star.
Consider two rays \(AP\) and \(BD\) from the same point of such an object, incident on a concave mirror parallel to each other, as shown in Fig. 7.20-2.
The incident ray \(AP\) striking at the pole \(P\) is reflected along \(PA\_1\), such that \(\\angle APC = \\angle A\_1PC\), that is, \(\\angle i = \\angle r\), with the principal axis as the normal.
The other ray \(BD\) is reflected along \(DA\_1\) such that \(\\angle BDC = \\angle A\_1DC\), since \(DC\) is normal at the point \(D\).
The reflected rays \(PA\_1\) and \(DA\_1\) intersect at a point \(A\_1\), which is the image of that point of the object. The point \(A\_1\) lies on the focal plane of the mirror.
Diagram
Fig. 7.20-2 — Object At A Far Distance A concave mirror drawn as a curved arc with hollow face to the left, principal axis horizontal, pole P, and C and F marked on the axis. Two rays A and B arrive from the upper left, parallel to each other but inclined to the principal axis; ray A strikes the mirror at the pole P and ray B strikes it at a point D above the axis. A dashed normal DC is drawn from D to C. The reflected ray from P is drawn going down to the right of the axis and the reflected ray from D is drawn downwards, the two crossing at a point A₁ just below the axis near F. Mark the equal angles at P and at D. Draw a short vertical dashed line through F to indicate the focal plane, and show A₁ lying on it. Label A, B, D, P, C, F and A₁.
Similarly, the rays from other points of the object also converge in the focal plane, thus forming the image along \(A\_1F\). Hence \(A\_1F\) is an inverted, real and highly diminished image formed in the focal plane of the concave mirror.
When the object is at a far distance, the image is in the focal plane of the mirror. It is (i) real, (ii) inverted, and (iii) highly diminished.
7.20.3 Object Beyond The Centre Of Curvature
In Fig. 7.20-3, \(AB\) is an object placed beyond the centre of curvature \(C\). Consider two light rays from the point \(A\) of the object.
A ray \(AD\) incident parallel to the principal axis, after reflection from the concave mirror passes through its focus \(F\) along \(DA\_1\).
Another ray \(AE\) passing through the centre of curvature \(C\) is incident normally on the mirror, so it gets reflected back as \(EA\).
The two reflected rays \(DA\_1\) and \(EA\) intersect at a point \(A\_1\) to form the image of the point \(A\). Similarly, the image is formed at points between \(A\_1\) and \(B\_1\) for the other points of the object between \(A\) and \(B\).
So the image \(A\_1B\_1\) is obtained between the focus \(F\) and the centre of curvature \(C\). The image is real, inverted and diminished.
Diagram
Fig. 7.20-3 — Object Beyond The Centre Of Curvature A concave mirror drawn as a curved arc with hollow face to the left, principal axis horizontal through pole P, with C and F marked on the axis. The object AB is drawn as an upright arrow on the axis to the left of C, with B on the axis and A at the top. From A, a ray AD runs parallel to the axis to the mirror at D and reflects through F, continuing to A₁. A second ray AE runs from A through C to the mirror at E and reflects back along EA. The two reflected rays cross at A₁, below the axis between F and C. Draw the inverted image A₁B₁ as a downward arrow from the axis at B₁ to A₁. Label A, B, D, E, A₁, B₁, C, F, P.
When the object is beyond the centre of curvature \(C\), the image is between the focus \(F\) and the centre of curvature \(C\). It is (i) real, (ii) inverted, and (iii) diminished.
7.20.4 Object At The Centre Of Curvature
In Fig. 7.20-4, \(AB\) is an object placed at the centre of curvature \(C\).
An incident ray of light \(AD\), being parallel to the principal axis, passes through its focus \(F\) along \(DA\_1\) after reflection.
The other ray \(AE\), being incident on the concave mirror through the focus \(F\), becomes parallel to the principal axis as \(EA\_1\) after reflection.
The two reflected rays \(DA\_1\) and \(EA\_1\) meet at a point \(A\_1\) to form the image. Similarly, the image is formed for the other points of the object, and \(B\_1A\_1\) is the image of the object \(AB\), formed at the centre of curvature \(C\) itself. It is real, inverted and of the same size as the object.
Diagram
Fig. 7.20-4 — Object At The Centre Of Curvature A concave mirror with hollow face to the left, principal axis horizontal, pole P, C and F marked on the axis. The object AB is drawn as an upright arrow standing at C, with B on the axis at C and A at the top. From A, a ray AD runs parallel to the axis to the mirror at D and reflects through F, continuing to A₁. A second ray AE runs from A through F to the mirror at E and reflects parallel to the axis as EA₁. The two reflected rays meet at A₁ below the axis, level with C. Draw the inverted image B₁A₁ as a downward arrow at C, the same length as the object. Label A, B, D, E, A₁, B₁, C, F, P.
When the object is at the centre of curvature \(C\), the image is also at the centre of curvature \(C\). It is (i) real, (ii) inverted, and (iii) the same size as that of the object.
7.20.5 Object Between The Centre Of Curvature And The Focus
In Fig. 7.20-5, \(AB\) is an object placed between \(C\) and \(F\).
An incident ray of light \(AD\) from the point \(A\) of the object is parallel to the principal axis, so it passes through the focus \(F\) after reflection, along \(DA\_1\).
Another ray \(AE\), being incident on the concave mirror through the focus \(F\), becomes parallel to the principal axis as \(EA\_1\) after reflection.
The two reflected rays \(DA\_1\) and \(EA\_1\) intersect at a point \(A\_1\) to form the image of the point \(A\) of the object. Similarly, the image is formed on \(A\_1B\_1\) for the other points on the object \(AB\), and the image \(A\_1B\_1\) is formed beyond the centre of curvature \(C\), which is real, inverted and magnified.
Diagram
Fig. 7.20-5 — Object Between The Centre Of Curvature And The Focus A concave mirror with hollow face to the left, principal axis horizontal, pole P on the right, with C and F marked on the axis to the left of P. The object AB is drawn as a short upright arrow standing on the axis between C and F, B on the axis and A at the top. From A, a ray AD runs parallel to the axis to the mirror at D and reflects through F towards A₁. A second ray AE runs from A through F to the mirror at E and reflects parallel to the axis as EA₁. The two reflected rays cross at A₁ below the axis, beyond C. Draw the inverted image A₁B₁ as a downward arrow at that position, longer than the object. Label A, B, D, E, A₁, B₁, C, F, P.
When the object is between the centre of curvature \(C\) and the focus \(F\), the image is beyond the centre of curvature \(C\). It is (i) real, (ii) inverted, and (iii) magnified.
7.20.6 Object At The Focus
In Fig. 7.20-6, \(AB\) is an object placed at the focus \(F\).
A ray of light \(AD\) is incident on the mirror parallel to the principal axis, and so it gets reflected along \(DF\), passing through its focus \(F\).
The other ray \(AE\), moving in the direction \(CA\), appears to be coming from the centre of curvature \(C\), so it gets reflected back along the same path as \(EC\).
The two reflected rays \(DF\) and \(EA\) are parallel to each other. Parallel rays are taken to meet only at infinity, so the image is formed at infinity. In this case the image is real, inverted and highly magnified.
Diagram
Fig. 7.20-6 — Object At The Focus A concave mirror with hollow face to the left, principal axis horizontal, pole P, C and F marked on the axis. The object AB is drawn as a short upright arrow standing on the axis at F, with B on the axis and A at the top. From A, a ray AD runs parallel to the axis to the mirror at D and reflects downwards through F and onwards. A second ray runs from C through A to the mirror at E and is reflected straight back along the same line towards C. The two reflected rays are drawn clearly parallel to each other, travelling away to the left and downwards. Label A, B, D, E, C, F, P.
When the object is at the focus \(F\), the image is at infinity. It is (i) real, (ii) inverted, and (iii) highly magnified.
7.20.7 Object Between The Focus And The Pole
In Fig. 7.20-7, \(AB\) is an object placed between the pole \(P\) and the focus \(F\).
A light ray \(AD\) incident parallel to the principal axis gets reflected along \(DF\), passing through its focus \(F\), and the incident ray \(AE\) coming from the centre of curvature \(C\) gets reflected back along \(EA\) passing through \(C\).
The two reflected rays \(DF\) and \(EC\) do not intersect each other, but they appear to diverge from a point \(A\_1\) when produced backwards. For an eye between \(C\) and \(F\), the reflected rays appear to come from \(A\_1\), which is the virtual image of \(A\).
Similarly, images are formed for the other points of the object, and the image \(A\_1B\_1\) is formed behind the mirror. It is virtual, upright and magnified.
Diagram
Fig. 7.20-7 — Object Between The Focus And The Pole A concave mirror with hollow face to the left, principal axis horizontal, pole P on the right, with C and F marked on the axis to the left of P. The object AB is drawn as a short upright arrow standing on the axis between F and P, B on the axis and A at the top. From A, a ray AD runs parallel to the axis to the mirror at D and reflects downwards through F. A second ray from C through A meets the mirror at E and reflects back along the same line. The two reflected rays diverge and are produced backwards as dashed lines behind the mirror, meeting at A₁. Draw the upright image A₁B₁ behind the mirror as a dashed upward arrow, taller than the object, with B₁ on the axis. Label A, B, D, E, A₁, B₁, C, F, P.
When the object is between the focus \(F\) and the pole \(P\), the image is behind the mirror. It is (i) virtual, (ii) upright, and (iii) magnified.
Note: In this case, as the object \(AB\) moves towards the pole \(P\), the image \(A\_1B\_1\) formed behind the mirror also shifts towards the pole \(P\) and its size keeps on decreasing, but it remains bigger than the object \(AB\). Finally, when the object \(AB\) comes very close to the pole \(P\) of the mirror, the virtual image will also be very close to the pole and is almost of the same size as that of the object.
7.20.8 Position, Size And Nature Of The Image At A Glance
Table 7.5 gives the position, size and nature of the image formed by a concave mirror corresponding to the different positions of the object.
Table 7.5 — Position, size and nature of the image formed by a concave mirror
| No. | Position of the object | Position of the image | Size of the image | Nature of the image |
|---|---|---|---|---|
| 1 | At infinity | At the focus | Diminished to a point | Real and inverted |
| 2 | At a very far distance | In the focal plane | Highly diminished | Real and inverted |
| 3 | Beyond the centre of curvature | Between the centre of curvature and the focus | Diminished | Real and inverted |
| 4 | At the centre of curvature | At the centre of curvature | Same size | Real and inverted |
| 5 | Between the centre of curvature and the focus | Beyond the centre of curvature | Magnified | Real and inverted |
| 6 | At the focus | At infinity | Highly magnified | Real and inverted |
| 7 | Between the focus and the pole | Behind the mirror | Magnified | Virtual and upright |
The Inference
In the case of a concave mirror, the image formed can be real as well as virtual; it can be diminished, of the same size, as well as magnified. The nature and size of the image depends on the position of the object.
For an object situated beyond the focus, the image is always real and inverted, whereas for an object situated between the focus and the pole, the image is upright and virtual.
The image is diminished when the object is beyond the centre of curvature, but it becomes magnified as the object comes within the centre of curvature. The image is of the same size as the object when the object is at the centre of curvature.
7.21 Ray Diagram For The Image Formed By A Convex Mirror
A concave mirror needed seven cases. A convex mirror needs only one, because it treats every object in the same way.
In Fig. 7.21-1, \(AB\) is a linear object placed in front of a convex mirror. Consider two rays starting from a point \(A\) of the object.
A ray of light \(AD\), incident parallel to the principal axis, is reflected along \(DM\) such that it appears to diverge from the focus \(F\) behind the mirror.
The other ray \(AE\), travelling towards the centre of curvature \(C\), falls normally on the mirror, so it is reflected back as \(EA\) along the same path.
The two reflected rays \(DM\) and \(EA\) meet at a point \(A\_1\) when they are produced backwards. In other words, the reflected rays appear to come from a point \(A\_1\), which is the virtual image of \(A\).
Diagram
Fig. 7.21-1 — Image Formation In A Convex Mirror A convex mirror drawn as a curved arc with its bulging face to the left, principal axis horizontal through the pole P, with F and C marked on the axis behind the mirror, F nearer the mirror than C. The object AB is drawn as an upright arrow in front of the mirror, B on the axis and A at the top. From A, a ray AD runs parallel to the axis to the mirror at D and is reflected up to the left along DM; produce DM backwards as a dashed line through the mirror to pass through F. A second ray from A is drawn heading towards C, meeting the mirror at E, and reflected straight back along EA; its continuation towards C behind the mirror is dashed. The two dashed produced-back lines meet at A₁ behind the mirror, between P and F. Draw the upright image A₁B₁ as a small dashed upward arrow behind the mirror, with B₁ on the axis. Label A, B, D, E, M, A₁, B₁, P, F, C.
Similarly, for the rays incident from the other points of the object, a virtual image is formed along \(A\_1B\_1\), which is situated between the pole and the focus. The image is virtual, upright and diminished.
When the object is in front of a convex mirror, the image is between the pole \(P\) and the focus \(F\), on the other side of the mirror. It is (i) virtual, (ii) upright, and (iii) diminished.
Note: When the object in front of a convex mirror is at a distance equal to the focal length of the mirror, the image is exactly at the midpoint between its pole and focus.
The Inference
In a convex mirror, the image formed is always virtual, upright and diminished. It is always situated between its pole and focus, irrespective of the distance of the object in front of the mirror.
That is a strong statement, and it is worth seeing what it costs the mirror. The whole space in front of the mirror, from infinity right up to the pole, is squeezed into the short gap between \(P\) and \(F\) behind it. As the object comes closer to the mirror from a far distance, its image shifts from the focus towards the pole and increases in size — but it never grows as large as the object itself.
The two positions worth tabulating are given in Table 7.6.
Table 7.6 — Position, size and nature of the image formed by a convex mirror
| No. | Position of the object | Position of the image | Size of the image | Nature of the image |
|---|---|---|---|---|
| 1 | At infinity | At the focus | Diminished to a point | Virtual and upright |
| 2 | At any other point | Between the focus and the pole | Diminished | Virtual and upright |
7.22 Relationship Between Focal Length And Radius Of Curvature
The focus was found by drawing rays; the centre of curvature came from the parent sphere. They are not independent. For a spherical mirror of small aperture, the focus lies exactly halfway between the pole and the centre of curvature.
The focal length of a spherical mirror is equal to half of its radius of curvature. That is,
\[f = \\frac{1}{2}R \\qquad \\dots (7.5)\]
where \(f\) is the focal length of the mirror and \(R\) its radius of curvature, both measured from the pole and both in the same unit — metre, or centimetre in most problems. The relation holds for a mirror of small aperture, and it holds for both a concave and a convex mirror.
The proof is given below. The syllabus asks for the relation and its use, not for its derivation, but the argument is short and it shows exactly where the condition of small aperture enters.
Proof For A Concave Mirror
In Fig. 7.22-1, \(P\) is the pole, \(C\) is the centre of curvature and \(F\) is the focus of the mirror. The distance \(PF\) is equal to the focal length \(f\) and the distance \(PC\) is equal to the radius of curvature \(R\) of the mirror.
A ray of light \(BD\), parallel to the principal axis \(PC\), is incident on the concave mirror at the point \(D\). After reflection, it goes along \(DQ\), passing through its focus \(F\). The normal at \(D\) is the line \(DC\).
Diagram
Fig. 7.22-1 — Relation Between f And R For A Concave Mirror A concave mirror drawn as a curved arc with hollow face to the left, principal axis horizontal through pole P on the right, with C and F marked on the axis, C farther from the mirror than F. A ray B arrives horizontally from the left, parallel to the axis, and strikes the mirror at a point D above the axis. A dashed normal DC is drawn from D to C. The reflected ray leaves D and crosses the axis at F, continuing to Q below the axis. Mark the angle between BD and DC as θ, the angle between DC and DF as θ, and the angle DCF at C as θ, all shown equal. Mark the distance PF as f with a double-headed arrow and PC as R with a longer double-headed arrow. Label B, D, Q, P, F, C.
Since \(BD\) is parallel to \(PC\) and \(DC\) cuts them both, \[\\angle BDC = \\angle DCF \\qquad \\text{(alternate angles)}\]
and by the law of reflection, with \(DC\) as the normal, \[\\angle BDC = \\angle CDF \\qquad (\\angle i = \\angle r)\]
Hence \[\\angle DCF = \\angle CDF\]
So in the triangle \(FDC\), the angles at \(C\) and at \(D\) are equal. Therefore \(\\triangle FDC\) is isosceles, and the sides opposite these equal angles are equal: \[DF = FC\]
Now comes the condition. If the aperture of the mirror is small, the point \(D\) is very close to the point \(P\), so the distance \(DF\) is very nearly the distance \(PF\), and we may write \(DF = PF\).
Therefore \[PF = FC\] \[\\text{or} \\qquad PF + PF = PF + FC\] \[\\text{or} \\qquad 2PF = PC\] \[\\text{or} \\qquad PF = \\frac{1}{2}PC, \\qquad \\text{that is,} \\qquad f = \\frac{1}{2}R\]
Proof For A Convex Mirror
In Fig. 7.22-2, a ray of light \(BD\) is incident on the convex mirror, parallel to the principal axis \(PC\). After reflection it appears to come from the focus \(F\) behind the mirror. At the point of incidence \(D\), the normal on the mirror is \(DC\), produced through the mirror to the centre of curvature \(C\) behind it. Let \(DN\) be the part of that normal in front of the mirror, and let \(DQ\) be the reflected ray.
Diagram
Fig. 7.22-2 — Relation Between f And R For A Convex Mirror A convex mirror drawn as a curved arc with bulging face to the left, principal axis horizontal through pole P, with F and C marked on the axis behind the mirror, F nearer to P. A ray B arrives horizontally from the left, parallel to the axis, and strikes the mirror at a point D above the axis. The normal at D is drawn as a full line from C through D and produced in front of the mirror to a point N. The reflected ray DQ leaves D up to the left. Mark the angle between BD and DN as θ, the angle between DN and DQ as θ, and the angle DCF at C as θ, all equal. Mark PF as f and PC as R with double-headed arrows. Label B, D, N, Q, P, F, C.
Since \(BD\) is parallel to \(PC\), \[\\angle BDN = \\angle FCD \\qquad \\text{(corresponding angles)}\]
By the law of reflection, with \(DN\) along the normal, \[\\angle BDN = \\angle NDQ \\qquad (\\angle i = \\angle r)\]
and \[\\angle NDQ = \\angle CDF \\qquad \\text{(vertically opposite angles)}\]
Hence \[\\angle FCD = \\angle CDF\]
So \(\\triangle FDC\) is isosceles, and therefore \[DF = FC\]
If the aperture of the mirror is small, the point \(D\) is very close to the point \(P\), so \(DF = PF\). Therefore \[PF = FC\] \[\\text{or} \\qquad PF + PF = PF + FC\] \[\\text{or} \\qquad 2PF = PC\] \[\\text{or} \\qquad PF = \\frac{1}{2}PC, \\qquad \\text{that is,} \\qquad f = \\frac{1}{2}R\]
Thus, for a spherical mirror, both concave and convex, the focal length is half of its radius of curvature.
7.23 Cartesian Sign Convention For The Measurement Of Distances
Up to here, distances have been described in words — in front of the mirror, behind it, above the axis. To put numbers into a formula we need something stricter. To specify the position of the object and the image, we need a reference point and a sign convention.
We follow the cartesian sign convention, so called because it treats the mirror like a graph, with the pole as the origin and the principal axis as the \(x\)-axis. According to it, the rules are:
(i) All distances are measured from the pole of the mirror, taken as origin. The rays are made incident from the left.
(ii) The distances measured along the principal axis in the direction of the incident light are positive, while those opposite to the incident light are negative.
(iii) The distances above the principal axis are taken positive and those below the principal axis are taken negative.
Rule (iii) is what gives an inverted image a negative height, and it is the reason a magnification can come out negative later in this chapter.
Diagram
Fig. 7.23-1 — Sign Convention For Spherical Mirrors Two sketches, one above the other. In (a), a concave mirror drawn as a curved arc with hollow face to the left; a horizontal principal axis through the pole, and a vertical line through the pole marked YY₁, so that the pole is the origin of a pair of axes. An object arrow stands on the axis to the left of the mirror, and a horizontal arrow beside it labelled INCIDENT LIGHT points to the right. Mark the axis to the right of the pole POSITIVE and the axis to the left NEGATIVE, with the note that the positive direction is the direction of the incident light. Mark the vertical direction above the axis POSITIVE and below it NEGATIVE. Caption (a) CONCAVE MIRROR. In (b), the same arrangement for a convex mirror drawn with its bulging face to the left, with the identical labelling of positive and negative directions. Caption (b) CONVEX MIRROR.
What The Convention Does To The Focal Length
Apply rule (ii) to the two mirrors and the focal lengths come out with opposite signs, automatically.
For a concave mirror, the focus lies in front of the mirror, on the same side as the incoming light. Look back at Fig. 7.18-1: the distance \(PF\) is measured opposite to the direction of the incident rays. This convention therefore gives the focal length of a concave mirror to be negative.
For a convex mirror, the focus lies behind the mirror. In Fig. 7.18-2, the distance \(PF\) is measured in the direction of the incident rays, so the focal length of a convex mirror is positive.
For convenience, the object is always placed to the left of the mirror, so that the graphical convention of sign comes into operation as described.
The convention has to be applied without exception, and applied at the moment the values are written down — not afterwards. A sign put in later is a sign that will be forgotten.
7.24 Formulae For A Spherical Mirror
Ray diagrams show where the image is. Two formulae let us calculate it instead, and with the sign convention of the last section in place, they can now be written down and used.
7.24.1 The Mirror Formula
Move an object nearer a mirror and its image moves too. The mirror formula is the exact statement of how the three distances involved — object, image and focus — depend on one another.
The expression relating the distance of the object \(u\), the distance of the image \(v\) and the focal length \(f\) for a spherical mirror is called the formula for a spherical mirror. It is given as
\[\\frac{1}{f} = \\frac{1}{u} + \\frac{1}{v} \\qquad \\dots (7.6)\]
where \(u\) is the distance of the object from the pole, \(v\) the distance of the image from the pole and \(f\) the focal length, all measured from the pole and all in the same unit, metre or centimetre. The relation holds for a spherical mirror of small aperture, concave or convex alike, and every quantity in it must be substituted with its proper sign.
Which Signs To Expect
In the above relation, the values of the known quantities out of \(u\), \(v\) and \(f\) are substituted with their proper sign, as follows.
For a concave mirror, the values of \(u\) and \(f\) are always negative, while the value of \(v\) is negative for a real image and positive for a virtual image.
For a convex mirror, the value of \(u\) is always negative, and the values of \(v\) and \(f\) are always positive.
Two notes make the reason plain.
In the case of mirrors, plane or spherical, the virtual image is obtained on the other side of the object, that is, behind the mirror, so \(u\) is negative and \(v\) is positive.
In a concave mirror, a real image is formed on the side of the object, that is, in front of the mirror, so both \(u\) and \(v\) are negative.
The signs are collected in Table 7.7.
7.24.2 Linear Magnification
The second formula answers the other question: not where the image is, but how big.
If the length of the object and the image are measured perpendicular to the principal axis, the ratio of the length of the image to the length of the object is called linear magnification. If the length of the image is \(I\) and that of the object is \(O\), then
\[\\text{magnification } m = \\frac{\\text{length of image } (I)}{\\text{length of object } (O)}\]
and it can be shown that this ratio is fixed by the two distances, as
\[m = \\frac{I}{O} = -\\frac{v}{u} \\qquad \\dots (7.7)\]
where \(I\) and \(O\) are the lengths of the image and the object, in the same unit, and \(u\) and \(v\) are the object and image distances with their proper signs. \(m\) itself is a pure number and carries no unit.
The minus sign in equation (7.7) is not a bookkeeping device. It is what makes the sign of \(m\) report the orientation of the image.
For a real image, \(u\) and \(v\) are both negative, so the linear magnification \(m\) is negative. For a virtual image, \(u\) is negative and \(v\) is positive, so the linear magnification \(m\) is positive. This matches what the ray diagrams showed: a real image is always inverted, while a virtual image is always erect.
7.25 Uses Of Spherical Mirrors
Every use of a spherical mirror is one of the cases of Sections 7.20 and 7.21, put to work. It is worth asking, for each device below, which row of Table 7.5 or Table 7.6 it is using.
7.25.1 Uses Of A Concave Mirror
(1) As a shaving mirror. When a concave mirror is held near the face, such that the face is between the pole and the focus of the mirror, it gives an upright and magnified image, as shown earlier in Fig. 7.20-7. Even the tiny hairs on the face can then be seen clearly.
For this purpose, a concave mirror of large focal length is used, so that the face always lies between its focus and the pole, and of large aperture, so as to view the entire face. A mirror of short focal length would put the face outside \(F\) at the first careless movement, and the image would flip over and become inverted.
(2) As a reflector. In a torch, a search light, and the head light of automobiles and cycles, a concave polished metallic surface is used as a reflector to obtain a parallel beam of light.
For this, the source of light — the bulb — is placed at the focus of the concave reflector. The rays of light from the bulb fall on the concave reflector and, after reflection, form a parallel beam, as shown in Fig. 7.25-1. This is Section 7.19.3 read forwards: a ray from the focus comes back parallel to the principal axis. A parallel beam does not spread out, so the light reaches far down the road instead of being wasted sideways.
Diagram
Fig. 7.25-1 — A Concave Mirror As A Reflector A concave mirror drawn as a curved arc with its hollow face to the left, principal axis horizontal. A small bulb is drawn on the axis at the focus F, labelled BULB, with F marked. Several rays are drawn spreading out from the bulb to different points on the mirror, and each reflected ray is drawn leaving horizontally to the left, all parallel to the principal axis and to one another. Label the bulb, F, the concave mirror and the emerging parallel beam.
(3) As a doctor’s head mirror. If a parallel beam of light is incident on a concave mirror, the mirror focuses the beam to a point, as shown earlier in Fig. 7.18-3(a).
This fact enables it to be used as a doctor’s head mirror, to concentrate a light beam on a small area of the body part — such as the teeth, nose, throat or ear — to be examined. A parallel beam of light is made to fall on a concave mirror attached to the band tied at the forehead of the doctor examining the body part, and the concave mirror then focuses the beam on the body part.
7.25.2 Uses Of A Convex Mirror
(1) As a reflector in street lamps. A convex polished metallic surface is used in street lamps as a reflector, so as to diverge the light over a larger area. Here the spreading that makes a convex mirror useless for forming a real image is exactly what is wanted: one lamp lights a long stretch of road.
(2) As a rear view mirror. A convex mirror diverges the incident light beam and always forms a virtual, small and erect image behind the mirror, between its pole and focus. This fact enables the driver to use it as a rear view mirror in vehicles, to see all the traffic approaching from behind.
A plane mirror can also be used for the purpose. But a convex mirror provides a much wider field of view as compared to a plane mirror of the same size, as the ray diagrams in Fig. 7.25-2 show. Here \(D\) is the position of the driver of the vehicle.
Diagram
Fig. 7.25-2 — Why A Convex Mirror Gives A Wider Field Of View Two side-by-side sketches, each with the driver’s eye marked D on the left, labelled DRIVER. In (a), a plane mirror is drawn as a vertical hatched strip of a certain height; two rays are drawn from D to the top edge and to the bottom edge of the mirror, each reflected with the angles i and r marked equal; the two reflected rays are produced outwards to enclose a narrow cone, labelled FIELD OF VIEW, and the image point behind the mirror is marked IMAGE. Caption (a) PLANE MIRROR. In (b), a convex mirror of the same height is drawn as a curved arc bulging towards D; two rays from D strike its top and bottom edges, with normals drawn to the centre of curvature and the equal angles i and r marked; the two reflected rays are produced outwards to enclose a much wider cone, labelled FIELD OF VIEW, and the image is marked behind the mirror. Caption (b) CONVEX MIRROR. Draw the two cones so that the convex one is visibly the wider.
7.26 Distinguishing A Plane, Concave And Convex Mirror Without Touching
Three mirrors of the same size lie on a table, and you are not allowed to run a finger over them to feel the curve. They can still be told apart at a glance, because each does something different to the image of your own face.
To distinguish between a plane mirror, a concave mirror and a convex mirror, the given mirror is held near the face and the image is seen. There can be the following three cases.
Case (i): If the image is upright, of the same size, and it does not change in size on moving the mirror towards or away from the face, the mirror is plane.
Case (ii): If the image is upright, magnified, and it increases in size on a small movement of the mirror away from the face, the mirror is concave.
Case (iii): If the image is upright, diminished, and it decreases in size on a small movement of the mirror away from the face, the mirror is convex.
Note that in all three cases the image is upright, so uprightness alone settles nothing. It is the size, and the way the size responds to a small movement, that separates the three.
Flowchart
FC 7.2 — Identifying A Mirror From The Image Of Your Face Hold the mirror close to the face and look at the image → is the image the same size as the face? → if yes, plane mirror → if no, is the image magnified? → if magnified, move the mirror slightly away; the image grows larger → concave mirror → if diminished, move the mirror slightly away; the image grows smaller → convex mirror
The three tests are set side by side in Table 7.8.
The middle column is worth a second look, because it is the one with a limit. The concave mirror behaves in this way only while the face is between its pole and its focus. Move the mirror far enough away and the face passes outside the focus, at which point the image turns inverted — which is itself a further confirmation that the mirror is concave, since neither of the other two can ever invert.
7.27 Difference Between A Concave And A Convex Mirror
The two spherical mirrors have now been met from every side — how each is made, what each does to a beam, and what kind of image each forms. Table 7.9 gathers those differences into one place.
Read the last two rows together and the whole difference comes down to one thing. A concave mirror bends light inwards, so where the image lands depends on where the object stands, and the answer changes as the object moves. A convex mirror bends light outwards, so the reflected rays never meet at all, and the answer never changes.
7.28 Summary
Reflection is the return of light into the same medium after it strikes a surface. The rest of the light is absorbed if the surface is opaque, and partly transmitted and partly absorbed if it is transparent.
Luminous bodies are seen directly; non-luminous bodies are seen only by the light they reflect.
A plane mirror is a glass plate polished on the front and silvered at the back, with the silvering protected by an opaque coating. Light is reflected mainly at the silvered surface.
Regular reflection occurs at a smooth polished surface: the reflected beam stays parallel and is seen from one direction only. Irregular reflection occurs at a rough surface: the reflected light spreads in all directions and is seen from anywhere. It is this diffused light that lets us see objects around us.
The terms of reflection are the incident ray, the point of incidence, the reflected ray, the normal, the angle of incidence \(i\), the angle of reflection \(r\), the plane of incidence and the plane of reflection. Both \(i\) and \(r\) are measured from the normal, in degrees.
The laws of reflection are: (i) \(\\angle i = \\angle r\), where \(i\) is the angle of incidence and \(r\) the angle of reflection, both in degrees; and (ii) the incident ray, the reflected ray and the normal at the point of incidence lie in the same plane. They hold at every surface, flat or curved.
A ray incident normally on a mirror has \(i = 0^\\circ\), so \(r = 0^\\circ\), and it retraces its own path.
The laws are verified on a drawing board using four pins: two fixed on the incident line and two lined up with the images of the first two. The measured angles come out equal, and the fact that all four pin tips lie on one sheet verifies the second law.
To locate an image, at least two rays from one point of the object must be traced. The image lies where the reflected rays actually meet, or where they appear to meet when produced backwards.
A real image is formed by the actual intersection of the reflected rays, can be taken on a screen, and is inverted. A virtual image is formed where the rays only appear to meet, cannot be taken on a screen, and is erect.
For a plane mirror, the image is situated on the normal drawn from the object to the mirror, and it is as far behind the mirror as the object is in front of it.
Lateral inversion is the interchange of the left and right sides in the image formed by a plane mirror. Letters symmetrical about a vertical line, such as A, H, I, M, O, T, U, V, W, X and Y, show no noticeable change.
The image formed by a plane mirror is (i) upright, (ii) virtual, (iii) of the same size as the object, and (iv) laterally inverted.
If the object is shifted a distance \(d\) towards a plane mirror, the image also shifts \(d\) towards the mirror, and the object–image separation decreases by \(2d\).
For two mirrors inclined at an angle \(\\theta\) in degrees, let \[n = \\frac{360^\\circ}{\\theta}\] where \(n\) is a pure number. If \(n\) is odd, the number of images is \(n\) for an object placed asymmetrically and \(n-1\) for one placed symmetrically. If \(n\) is even, the number of images is \(n-1\) for all positions.
The object and its images in two inclined mirrors lie on a circle whose centre is the point of intersection of the mirrors and whose radius is the distance of the object from that point.
Two parallel mirrors give an infinite number of images, of which only a few are seen, because some light is absorbed at each reflection. Two perpendicular mirrors give three images, since two of the four coincide.
Plane mirrors are used as looking glasses, in an optician’s room, in a barber’s shop, in a periscope, in a kaleidoscope, and in solar heating devices.
A spherical mirror is a reflecting surface which is a part of a sphere. A concave mirror reflects from its hollow surface; a convex mirror reflects from its bulging surface.
For a spherical mirror: the centre of curvature \(C\) is the centre of the parent sphere; the radius of curvature \(R\), in metre, is its radius; the pole \(P\) is the geometric centre of the mirror surface; the aperture is the surface area through which light enters; and the principal axis is the line \(PC\). The normal at any point of the mirror is the line joining that point to \(C\).
The focus \(F\) of a concave mirror is the point on the principal axis through which rays parallel to the axis pass after reflection — a real focus. The focus of a convex mirror is the point from which such rays appear to come after reflection — a virtual focus. In both cases the focal length is \(f = PF\), in metre.
The focal plane is the plane through the focus, normal to the principal axis. A parallel beam inclined to the axis is focused at a point in this plane.
The convenient rays for a ray diagram are: a ray through \(C\), reflected back along its own path; a ray parallel to the principal axis, reflected through \(F\) or appearing to come from \(F\); a ray through \(F\), reflected parallel to the axis; and a ray at the pole, reflected with the principal axis as normal.
For a concave mirror the image is real and inverted for every object position beyond the focus, and virtual, upright and magnified for an object between the focus and the pole. It is diminished beyond \(C\), the same size at \(C\), and magnified between \(C\) and \(F\).
For a convex mirror the image is always virtual, upright and diminished, and always lies between the pole and the focus, whatever the position of the object.
The focal length of a spherical mirror is half its radius of curvature: \[f = \\frac{1}{2}R\] where \(f\) is the focal length and \(R\) the radius of curvature, both from the pole and in the same unit. It holds for a mirror of small aperture.
By the cartesian sign convention, all distances are measured from the pole; distances measured in the direction of the incident light are positive and those opposite to it are negative; distances above the principal axis are positive and those below it are negative. Hence \(f\) is negative for a concave mirror and positive for a convex mirror, and \(u\) is always negative.
The mirror formula is \[\\frac{1}{f} = \\frac{1}{u} + \\frac{1}{v}\] where \(u\) is the object distance, \(v\) the image distance and \(f\) the focal length, all measured from the pole in the same unit and all substituted with their proper signs.
The linear magnification is \[m = \\frac{I}{O} = -\\frac{v}{u}\] where \(I\) and \(O\) are the lengths of the image and the object in the same unit, and \(u\) and \(v\) carry their signs. \(m\) is a pure number; it is negative for a real, inverted image and positive for a virtual, erect image.
A concave mirror is used as a shaving mirror, as a reflector in a torch, a search light and a headlight with the bulb at its focus, and as a doctor’s head mirror. A convex mirror is used as a reflector in street lamps and as a rear view mirror, because it gives a wider field of view.
Held near the face, a plane mirror gives an image of unchanged size, a concave mirror a magnified image that grows as the mirror moves away, and a convex mirror a diminished image that shrinks as the mirror moves away.
Flowchart
FC 7.3 — The Image In A Concave Mirror As The Object Moves In Object at infinity: real, inverted, point image at F → object at a far distance: real, inverted, highly diminished image in the focal plane → object beyond C: real, inverted, diminished image between F and C → object at C: real, inverted, same-size image at C → object between C and F: real, inverted, magnified image beyond C → object at F: real, inverted, highly magnified image at infinity → object between F and P: virtual, upright, magnified image behind the mirror
7.29 Key Terms
Aperture — the plane surface area of a mirror through which light rays enter and fall on it; its width is given by the diameter of the aperture, in metre (m).
Cartesian sign convention — the rule that all distances are measured from the pole, that distances along the direction of the incident light are positive and those opposite to it negative, and that distances above the principal axis are positive and those below it negative.
Centre of curvature — the centre of the sphere of which a spherical mirror is a part; marked \(C\).
Concave mirror — a spherical mirror silvered on its bulging surface, so that reflection takes place at its hollow surface; it converges light.
Convex mirror — a spherical mirror silvered on its hollow surface, so that reflection takes place at its bulging surface; it diverges light.
Extended object — an object having length, whose image has to be built up point by point.
Focal length — the distance of the focus \(F\) from the pole \(P\) of a mirror, measured in metre (m); it is half the radius of curvature.
Focal plane — the plane passing through the focus and normal to the principal axis of a mirror.
Focus — the point on the principal axis through which rays incident parallel to the axis pass after reflection, or from which they appear to come; marked \(F\).
Image — the picture of an object formed where the reflected rays meet, or appear to meet on being produced backwards.
Irregular reflection — reflection at a rough or ordinary surface, in which the reflected rays travel in different directions and the light is diffused.
Lateral inversion — the interchange of the left and right sides in the image of an object formed in a plane mirror.
Laws of reflection — the two rules obeyed at every reflecting surface: the angle of incidence equals the angle of reflection, and the incident ray, the reflected ray and the normal at the point of incidence lie in the same plane.
Linear magnification — the ratio of the length of the image to the length of the object, equal to \(-v/u\); it is a pure number with no unit.
Luminous — giving out light of its own, as the Sun or a flame does.
Mirror formula — the relation \(\\dfrac{1}{f} = \\dfrac{1}{u} + \\dfrac{1}{v}\) between the object distance, the image distance and the focal length of a spherical mirror.
Non-luminous — giving out no light of its own, and seen only by the light it reflects.
Normal — the perpendicular drawn to a reflecting surface at the point of incidence, from which the angles of incidence and reflection are measured.
Plane mirror — a flat, smooth reflecting surface, made from a glass plate polished in front and silvered at the back.
Pole — the geometric centre of the surface of a spherical mirror; marked \(P\), and the origin from which all distances are measured.
Principal axis — the straight line joining the pole of a mirror to its centre of curvature.
Radius of curvature — the radius of the sphere of which a spherical mirror is a part, measured in metre (m); marked \(R\).
Real focus — a focus at which the reflected rays actually meet, as in a concave mirror.
Real image — an image formed by the actual intersection of the reflected rays; it can be obtained on a screen and is inverted.
Reflected light — the part of the light returned into the same medium after striking a surface.
Reflection — the return of light into the same medium after it strikes a surface.
Regular reflection — reflection at a smooth polished surface, in which a parallel incident beam remains parallel after reflection.
Spherical mirror — a reflecting surface which is a part of a sphere.
Virtual focus — a focus from which the reflected rays only appear to come, as in a convex mirror.
Virtual image — an image formed where the reflected rays appear to meet on being produced backwards; it cannot be obtained on a screen and is erect.
7.30 Points To Ponder
A plane mirror always forms an image the same size as the object, yet a small mirror on a wall can show you a whole room. How can a mirror show something far larger than itself, and what does that suggest about the difference between the size of an image and the amount of it your eye can receive?
A concave mirror can form a real image, and a plane mirror never can, however it is turned or moved. What is it about a flat surface that makes a real image impossible for it?
Stand a metre from a plane mirror and your image stands a metre behind it, in a space that may be solid brick. Physics is quite content with this, while nothing is actually there. What does that tell you about what a virtual image is a statement about — the world, or the light arriving at your eye?
The relation \(f = R/2\) was derived only for a mirror of small aperture. What do you think would happen to the sharpness of the image formed by a concave mirror of very large aperture, and why does the derivation break down for it?
Two mirrors inclined at \(60^\\circ\) give five images. As the angle between them is made smaller and smaller, the count rises without limit, and at \(0^\\circ\) it is infinite. Yet nobody has ever seen an infinite number of images. Where does the mathematics part company with what the eye can do?
A convex mirror gives a wide field of view but a diminished image, and it cannot be used to judge distance accurately. If you were designing the mirrors for a vehicle, how would you weigh a wider view against a truer sense of how far away the traffic behind is?
7.31 Objective Questions
- The angle of reflection is measured between the reflected ray and the: (1 mark)
- mirror surface
- normal at the point of incidence
- incident ray
- principal axis
- Light falling on the page of a book is reflected: (1 mark)
- regularly, and the laws of reflection are obeyed
- regularly, and the laws of reflection are not obeyed
- irregularly, and the laws of reflection are obeyed
- irregularly, and the laws of reflection are not obeyed
- A ray strikes a plane mirror along the normal. The angle between the incident ray and the reflected ray is: (1 mark)
- \(0^\\circ\)
- \(45^\\circ\)
- \(90^\\circ\)
- \(180^\\circ\)
- Two plane mirrors are inclined at \(90^\\circ\). The number of images of an object placed between them is: (1 mark)
- 2
- 3
- 4
- infinite
- For a spherical mirror, the normal at any point on its surface passes through the: (1 mark)
- pole
- focus
- centre of curvature
- mid point of the aperture
- An object is placed at the centre of curvature of a concave mirror. The image formed is: (1 mark)
- at the focus, real and diminished
- at the centre of curvature, real and of the same size
- behind the mirror, virtual and magnified
- at infinity, real and highly magnified
- Using the cartesian sign convention, the focal length of a convex mirror and the object distance are respectively: (1 mark)
- positive and negative
- negative and negative
- positive and positive
- negative and positive
- The magnification produced by a mirror is \(+0.4\). The image is: (1 mark)
- real, inverted and diminished
- real, erect and magnified
- virtual, erect and diminished
- virtual, inverted and magnified
For each of the following, choose from: a. Both A and R are true and R is the correct explanation of A b. Both A and R are true but R is not the correct explanation of A c. A is true but R is false d. A is false but R is true
Assertion (A): The image formed by a plane mirror cannot be taken on a screen. (1 mark) Reason (R): The reflected rays from a plane mirror meet only when produced backwards.
Assertion (A): A convex mirror is preferred as a rear view mirror in vehicles. (1 mark) Reason (R): A convex mirror forms a magnified image of the traffic behind.
Assertion (A): A ray directed towards the centre of curvature of a spherical mirror returns along its own path. (1 mark) Reason (R): Such a ray strikes the mirror along the normal, so its angle of incidence is zero.
Assertion (A): The number of images formed by two parallel mirrors is infinite, but only a few are seen. (1 mark) Reason (R): Some light is absorbed at each reflection, so the remote images grow fainter.
Match each position of an object in front of a concave mirror with the description of its image. (4 marks)
| Position of the object | Description of the image |
|---|---|
| a. At infinity | i. Behind the mirror, virtual and magnified |
| b. Between the centre of curvature and the focus | ii. At the focus, real and diminished to a point |
| c. At the focus | iii. Beyond the centre of curvature, real and magnified |
| d. Between the focus and the pole | iv. At infinity, real and highly magnified |
- Fill in the blanks.
- The perpendicular drawn to a reflecting surface at the point of incidence is called the …………. . (1 mark)
- For a mirror of small aperture, the focal length is …………. of the radius of curvature. (1 mark)
- In two mirrors inclined at an angle, the object and its images all lie on a …………. whose centre is the point of intersection of the mirrors. (1 mark)
- State whether the following statements are true or false.
- A virtual image is always erect. (1 mark)
- Irregular reflection takes place because the laws of reflection are not obeyed at a rough surface. (1 mark)
- The image formed by a convex mirror lies between its pole and its focus for every position of the object. (1 mark)
- Name the following.
- The mirror that always forms a virtual, erect and diminished image. (1 mark)
- The point on the principal axis at which a beam parallel to the axis converges after reflection from a concave mirror. (1 mark)
- The kind of reflection that enables you to see the walls of a room. (1 mark)
- Give one word or phrase for each of the following.
- The interchange of the left and right sides in the image formed by a plane mirror. (1 mark)
- The geometric centre of the surface of a spherical mirror. (1 mark)
- The ratio of the length of the image to the length of the object. (1 mark)
- State the SI unit of the following.
- Radius of curvature. (1 mark)
- Focal length. (1 mark)
- Linear magnification. (1 mark)
- Pick the odd one out and give a reason for your choice.
- Shaving mirror, torch reflector, doctor’s head mirror, street lamp reflector. (2 marks)
- Pole, focus, centre of curvature, aperture. (2 marks)
7.32 Check Your Concepts
Define reflection of light. (2 marks)
State the two laws of reflection of light. (2 marks)
Name the SI unit of focal length, and state what the focal length of a spherical mirror measures. (2 marks)
Give a reason why the letters on the front of an ambulance are painted laterally inverted. (2 marks)
A ray of light strikes a plane mirror and retraces its own path. State its angle of incidence and its angle of reflection. (2 marks)
Differentiate between regular and irregular reflection, stating the basis of difference. (3 marks)
Differentiate between a real and a virtual image, stating the basis of difference. (3 marks)
Explain why a convex mirror is used as a reflector in street lamps. (3 marks)
State the cartesian sign convention for the measurement of distances at a spherical mirror, and hence state the sign of the focal length of a concave mirror. (3 marks)
Why does a plane mirror never form a real image, however far the object is placed from it? (3 marks)
Write the mirror formula and explain the meaning of each symbol used in it, together with its unit. (3 marks)
Explain, with an example, why a concave mirror used as a shaving mirror must have a large focal length. (4 marks)
State the number of images formed when an object is placed between two plane mirrors inclined at \(45^\\circ\), and explain how you obtained it. (4 marks)
What would happen to the beam from a torch if its bulb were shifted from the focus of its concave reflector to a point between the focus and the pole? Explain your answer. (4 marks)
Describe an experiment, with the apparatus and the observations, to verify the two laws of reflection. (5 marks)
Discuss how the position, size and nature of the image formed by a concave mirror change as an object is brought from infinity towards the pole of the mirror. (5 marks)
7.33 Diagrams, Graphs & Constructions
Draw a ray diagram showing the reflection of a ray of light at a plane mirror. Label the incident ray, the reflected ray, the normal, the angle of incidence and the angle of reflection. (3 marks)
Redraw the arrangement of Fig. 7.6-1 for a point object placed in front of a plane mirror. Taking two rays from the object, complete the diagram to locate the image, mark the position of the eye, and state the nature of the image. (4 marks)
Copy Fig. 7.16-1 for a concave mirror and label the pole, the centre of curvature, the radius of curvature, the principal axis and the aperture. (3 marks)
Draw a diagram, on the lines of Fig. 7.19-2, to show the path of a ray incident parallel to the principal axis on (i) a concave mirror and (ii) a convex mirror. Mark the focus in each case. (4 marks)
Draw the ray diagram for an object placed between the centre of curvature and the focus of a concave mirror, as in Fig. 7.20-5, and state three characteristics of the image formed. (4 marks)
Draw the ray diagram for an object placed in front of a convex mirror, as in Fig. 7.21-1, and state three characteristics of the image formed. (4 marks)
Draw and label the apparatus used to verify the laws of reflection with four pins on a drawing board, as shown in Fig. 7.4-1. (4 marks)
With reference to Fig. 7.25-2, draw the two diagrams that show why a convex mirror gives a wider field of view than a plane mirror of the same size. (4 marks)
Draw a ray diagram to show how a concave reflector in a torch produces a parallel beam, marking the position of the bulb. (3 marks)
Sketch the arrangement of two mirrors and the object in a periscope, marking the angle each mirror makes with the wall of the tube and the path of a ray from the object to the eye. (4 marks)
7.34 Practice With Numericals
A ray of light strikes a plane mirror, making an angle of \(20^\\circ\) with the mirror surface. Find (i) the angle of incidence and (ii) the angle between the incident ray and the reflected ray. (2 marks)
The radius of curvature of a concave mirror is \(25\\ \\text{cm}\). Find its focal length. (2 marks)
The focal length of a convex mirror is \(18\\ \\text{cm}\). Find its radius of curvature. (2 marks)
Two plane mirrors are inclined at an angle of \(36^\\circ\). Find the number of images formed for an object placed between them. (2 marks)
A man stands \(1.8\\ \\text{m}\) in front of a plane mirror fixed on a wall. He then walks \(0.4\\ \\text{m}\) towards the mirror. Find the new distance between the man and his image. (3 marks)
An object is placed \(45\\ \\text{cm}\) in front of a concave mirror of focal length \(15\\ \\text{cm}\). Find the position of the image and its magnification, and state its nature. (3 marks)
An object \(4.0\\ \\text{cm}\) high is placed \(30\\ \\text{cm}\) in front of a convex mirror of focal length \(20\\ \\text{cm}\). Find the position and the height of the image. (3 marks)
An object \(5.0\\ \\text{cm}\) high is placed \(24\\ \\text{cm}\) from a concave mirror of focal length \(8\\ \\text{cm}\). Find the position and the height of the image, and state whether it is erect or inverted. (3 marks)
A concave mirror forms a real image three times the size of an object placed \(20\\ \\text{cm}\) in front of it. Find the position of the image and the focal length of the mirror. (4 marks)
A concave mirror forms a virtual image twice the size of an object placed \(10\\ \\text{cm}\) in front of it. Find the position of the image and the focal length of the mirror. (4 marks)
A concave mirror has a focal length of \(0.20\\ \\text{m}\). An object is placed \(60\\ \\text{cm}\) in front of it. Find the position of the image and its magnification. (4 marks)