Chapter Nine

RAY OPTICS

AND OPTICAL

INSTRUMENTS

9.1 INTRODUCTION

Nature has endowed the human eye (retina) with the sensitivity to detect

electromagnetic waves within a small range of the electromagnetic

spectrum. Electromagnetic radiation belonging to this region of the

spectrum (wavelength of about 400 nm to 750 nm) is called light. It is

mainly through light and the sense of vision that we know and interpret

the world around us.

There are two things that we can intuitively mention about light from

common experience. First, that it travels with enormous speed and second,

that it travels in a straight line. It took some time for people to realise that

the speed of light is finite and measurable. Its presently accepted value

in vacuum is c = 2.99792458 × 10

m s

–1

. For many purposes, it suffices

to take c = 3 × 10

m s

–1

. The speed of light in vacuum is the highest

speed attainable in nature.

The intuitive notion that light travels in a straight line seems to

contradict what we have learnt in Chapter 8, that light is an

electromagnetic wave of wavelength belonging to the visible part of the

spectrum. How to reconcile the two facts? The answer is that the

wavelength of light is very small compared to the size of ordinary objects

that we encounter commonly (generally of the order of a few cm or larger).

In this situation, as you will learn in Chapter 10, a light wave can be

considered to travel from one point to another, along a straight line joining

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them. The path is called a ray of light, and a bundle of such rays

constitutes a beam of light.

In this chapter, we consider the phenomena of reflection, refraction

and dispersion of light, using the ray picture of light. Using the basic

laws of reflection and refraction, we shall study the image formation by

plane and spherical reflecting and refracting surfaces. We then go on to

describe the construction and working of some important optical

instruments, including the human eye.

PARTICLE MODEL OF LIGHT

Newton’s fundamental contributions to mathematics, mechanics, and gravitation often blind

us to his deep experimental and theoretical study of light. He made pioneering contributions

in the field of optics. He further developed the corpuscular model of light proposed by

Descartes. It presumes that light energy is concentrated in tiny particles called corpuscles.

He further assumed that corpuscles of light were massless elastic particles. With his

understanding of mechanics, he could come up with a simple model of reflection and

refraction. It is a common observation that a ball bouncing from a smooth plane surface

obeys the laws of reflection. When this is an elastic collision, the magnitude of the velocity

remains the same. As the surface is smooth, there is no force acting parallel to the surface,

so the component of momentum in this direction also remains the same. Only the component

perpendicular to the surface, i.e., the normal component of the momentum, gets reversed

in reflection. Newton argued that smooth surfaces like mirrors reflect the corpuscles in a

similar manner.

In order to explain the phenomena of refraction, Newton postulated that the speed of

the corpuscles was greater in water or glass than in air. However, later on it was discovered

that the speed of light is less in water or glass than in air.

In the field of optics, Newton – the experimenter, was greater than Newton – the theorist.

He himself observed many phenomena, which were difficult to understand in terms of

particle nature of light. For example, the colours observed due to a thin film of oil on water.

Property of partial reflection of light is yet another such example. Everyone who has looked

into the water in a pond sees image of the face in it, but also sees the bottom of the pond.

Newton argued that some of the corpuscles, which fall on the water, get reflected and some

get transmitted. But what property could distinguish these two kinds of corpuscles? Newton

had to postulate some kind of unpredictable, chance phenomenon, which decided whether

an individual corpuscle would be reflected or not. In explaining other phenomena, however,

the corpuscles were presumed to behave as if they are identical. Such a dilemma does not

occur in the wave picture of light. An incoming wave can be divided into two weaker waves

at the boundary between air and water.

9.2 REFLECTION OF LIGHT BY SPHERICAL MIRRORS

We are familiar with the laws of reflection. The angle of reflection (i.e., the

angle between reflected ray and the normal to the reflecting surface or

the mirror) equals the angle of incidence (angle between incident ray and

the normal). Also that the incident ray, reflected ray and the normal to

the reflecting surface at the point of incidence lie in the same plane

(Fig. 9.1). These laws are valid at each point on any reflecting surface

whether plane or curved. However, we shall restrict our discussion to the

special case of curved surfaces, that is, spherical surfaces. The normal in

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this case is to be taken as normal to the tangent

to surface at the point of incidence. That is, the

normal is along the radius, the line joining the

centre of curvature of the mirror to the point of

incidence.

We have already studied that the geometric

centre of a spherical mirror is called its pole while

that of a spherical lens is called its optical centre.

The line joining the pole and the centre of curvature

of the spherical mirror is known as the principal

axis. In the case of spherical lenses, the principal

axis is the line joining the optical centre with its

principal focus as you will see later.

9.2.1 Sign convention

To derive the relevant formulae for reflection by spherical mirrors and

refraction by spherical lenses, we must first adopt a sign convention for

measuring distances. In this book, we shall follow the Cartesian sign

convention. According to this

convention, all distances are measured

from the pole of the mirror or the optical

centre of the lens. The distances

measured in the same direction as the

incident light are taken as positive and

those measured in the direction

opposite to the direction of incident

light are taken as negative (Fig. 9.2).

The heights measured upwards with

respect to x-axis and normal to the

principal axis (x-axis) of the mirror/

lens are taken as positive (Fig. 9.2). The

heights measured downwards are

taken as negative.

With a common accepted convention, it turns out that a single formula

for spherical mirrors and a single formula for spherical lenses can handle

all different cases.

9.2.2 Focal length of spherical mirrors

Figure 9.3 shows what happens when a parallel beam of light is incident

on (a) a concave mirror, and (b) a convex mirror. We assume that the rays

are

paraxial, i.e., they are incident at points close to the pole P of the mirror

and make small angles with the principal axis. The reflected rays converge

at a point F on the principal axis of a concave mirror [Fig. 9.3(a)].

For a convex mirror, the reflected rays appear to diverge from a point F

on its principal axis [Fig. 9.3(b)]. The point F is called the principal focus

of the mirror. If the parallel paraxial beam of light were incident, making

some angle with the principal axis, the reflected rays would converge (or

appear to diverge) from a point in a plane through F normal to the principal

axis. This is called the focal plane of the mirror [Fig. 9.3(c)].

FIGURE 9.1 The incident ray, reflected ray

and the normal to the reflecting surface lie

in the same plane.

FIGURE 9.2 The Cartesian Sign Convention.

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The distance between the focus F and the pole P of the mirror is called

the focal length of the mirror, denoted by f. We now show that f = R/2,

where R is the radius of curvature of the mirror. The geometry

of reflection of an incident ray is shown in Fig. 9.4.

Let C be the centre of curvature of the mirror. Consider a

ray parallel to the principal axis striking the mirror at M. Then

CM will be perpendicular to the mirror at M. Let

be the angle

of incidence, and MD be the perpendicular from M on the

principal axis. Then,

∠MCP =

and ∠MFP = 2

Now,

tan

and tan 2

(9.1)

For small

, which is true for paraxial rays, tan

≈

tan 2

≈ 2

θ.

Therefore, Eq. (9.1) gives

= 2

or, FD =

(9.2)

Now, for small

, the point D is very close to the point P.

Therefore, FD = f and CD = R. Equation (9.2) then gives

f = R/2 (9.3)

9.2.3 The mirror equation

If rays emanating from a point actually meet at another point after

reflection and/or refraction, that point is called the image of the first

point. The image is real if the rays actually converge to the point; it is

FIGURE 9.3 Focus of a concave and convex mirror.

FIGURE 9.4 Geometry of

reflection of an incident ray on

(a) concave spherical mirror,

and (b) convex spherical mirror.

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virtual if the rays do not actually meet but appear

to diverge from the point when produced

backwards. An image is thus a point-to-point

correspondence with the object established

through reflection and/or refraction.

In principle, we can take any two rays

emanating from a point on an object, trace their

paths, find their point of intersection and thus,

obtain the image of the point due to reflection at a

spherical mirror. In practice, however, it is

convenient to choose any two of the following rays:

(i) The ray from the point which is parallel to the

principal axis. The reflected ray goes through

the focus of the mirror.

(ii) The ray passing through the centre of

curvature of a concave mirror or appearing to pass through it for a

convex mirror. The reflected ray simply retraces the path.

(iii) The ray passing through (or directed towards) the focus of the concave

mirror or appearing to pass through (or directed towards) the focus

of a convex mirror. The reflected ray is parallel to the principal axis.

(iv) The ray incident at any angle at the pole. The reflected ray follows

laws of reflection.

Figure 9.5 shows the ray diagram considering three rays. It shows

the image A′B′ (in this case, real) of an object AB formed by a concave

mirror. It does not mean that only three rays emanate from the point A.

An infinite number of rays emanate from any source, in all directions.

Thus, point A′ is image point of A if every ray originating at point A and

falling on the concave mirror after reflection passes through the point A′.

We now derive the mirror equation or the relation between the object

distance (u), image distance (v) and the focal length (f ).

From Fig. 9.5, the two right-angled triangles A′B′F and MPF are

similar. (For paraxial rays, MP can be considered to be a straight line

perpendicular to CP.) Therefore,

B A B F

PM FP

′ ′ ′

B A B F

BA FP

′ ′ ′

(

∵

PM = AB) (9.4)

Since ∠ APB = ∠ A′PB′, the right angled triangles A′B′P and ABP are

also similar. Therefore,

B A B P

′ ′ ′

(9.5)

Comparing Eqs. (9.4) and (9.5), we get

B P – FP

B F B P

FP FP BP

′

′ ′

= =

(9.6)

Equation (9.6) is a relation involving magnitude of distances. We now

apply the sign convention. We note that light travels from the object to

the mirror MPN. Hence this is taken as the positive direction. To reach

FIGURE 9.5 Ray diagram for image

formation by a concave mirror.

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the object AB, image A′B′ as well as the focus F from the pole P, we have

to travel opposite to the direction of incident light. Hence, all the three

will have negative signs. Thus,

B′ P = –v, FP = –f, BP = –u

Using these in Eq. (9.6), we get

– –

–

v f v

f u

–

v f v

f u

= +1

Dividing it by v, we get

1 1 1

v u f

+ =

(9.7)

This relation is known as the mirror equation.

The size of the image relative to the size of the object is another

important quantity to consider. We define linear magnification (m) as the

ratio of the height of the image (h′) to the height of the object (h):

m =

′

(9.8)

h and h

′

will be taken positive or negative in accordance with the accepted

sign convention. In triangles A′B′P and ABP, we have,

B A B P

BA BP

′ ′ ′

With the sign convention, this becomes

– –

h v

h u

′

–

so that

m =

–

h v

h u

′

(9.9)

We have derived here the mirror equation, Eq. (9.7), and the

magnification formula, Eq. (9.9), for the case of real, inverted image formed

by a concave mirror. With the proper use of sign convention, these are,

in fact, valid for all the cases of reflection by a spherical mirror (concave

or convex) whether the image formed is real or virtual. Figure 9.6 shows

the ray diagrams for virtual image formed by a concave and convex mirror.

You should verify that Eqs. (9.7) and (9.9) are valid for these cases as

well.

FIGURE 9.6 Image formation by (a) a concave mirror with object between

P and F, and (b) a convex mirror.

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EXAMPLE 9.3

EXAMPLE 9.2

EXAMPLE 9.1

Example 9.1 Suppose that the lower half of the concave mirror’s

reflecting surface in Fig. 9.5 is covered with an opaque (non-reflective)

material. What effect will this have on the image of an object placed

in front of the mirror?

Solution You may think that the image will now show only half of the

object, but taking the laws of reflection to be true for all points of the

remaining part of the mirror, the image will be that of the whole object.

However, as the area of the reflecting surface has been reduced, the

intensity of the image will be low (in this case, half).

Example 9.2 A mobile phone lies along the principal axis of a concave

mirror, as shown in Fig. 9.7. Show by suitable diagram, the formation

of its image. Explain why the magnification is not uniform. Will the

distortion of image depend on the location of the phone with respect

to the mirror?

FIGURE 9.7

Solution

The ray diagram for the formation of the image of the phone is shown

in Fig. 9.7. The image of the part which is on the plane perpendicular

to principal axis will be on the same plane. It will be of the same size,

i.e., B′C = BC. You can yourself realise why the image is distorted.

Example 9.3 An object is placed at (i) 10 cm, (ii) 5 cm in front of a

concave mirror of radius of curvature 15 cm. Find the position, nature,

and magnification of the image in each case.

Solution

The focal length f = –15/2 cm = –7.5 cm

(i) The object distance u = –10 cm. Then Eq. (9.7) gives

– – .

1 1 1

10 7 5

+ =

10 7 5

2 5

−

= – 30 cm

The image is 30 cm from the mirror on the same side as the object.

Also, magnification m =

( 30)

– – – 3

( 10)

−

= =

−

The image is magnified, real and inverted.

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EXAMPLE 9.4 EXAMPLE 9.3

(ii) The object distance u = –5 cm. Then from Eq. (9.7),

1 1 1

5 7.5

+ =

− −

( )

. –

5 7 5

15 cm

7 5 5

This image is formed at 15 cm behind the mirror. It is a virtual image.

Magnification m =

– – 3

( 5)

= =

−

The image is magnified, virtual and erect.

Example 9.4 Suppose while sitting in a parked car, you notice a

jogger approaching towards you in the side view mirror of R = 2 m. If

the jogger is running at a speed of 5 m s

–1

, how fast the image of the

jogger appear to move when the jogger is (a) 39 m, (b) 29 m, (c) 19 m,

and (d) 9 m away.

Solution

From the mirror equation, Eq. (9.7), we get

u f

−

For convex mirror, since R = 2 m, f = 1 m. Then

for u = –39 m,

( 39) 1 39

39 1 40

− ×

= =

− −

Since the jogger moves at a constant speed of 5 m s

–1

, after 1 s the

position of the image v (for u = –39 + 5 = –34) is (34/35 )m.

The shift in the position of image in 1 s is

1365 1360

39 34 5 1

40 35 1400 1400 280

−

− = = =

Therefore, the average speed of the image when the jogger is between

39 m and 34 m from the mirror, is (1/280) m s

–1

Similarly, it can be seen that for u = –29 m, –19 m and –9 m, the

speed with which the image appears to move is

–1 –1 –1

1 1 1

m s , m s and m s ,

150 60 10

respectively.

Although the jogger has been moving with a constant speed, the speed

of his/her image appears to increase substantially as he/she moves

closer to the mirror. This phenomenon can be noticed by any person

sitting in a stationary car or a bus. In case of moving vehicles, a

similar phenomenon could be observed if the vehicle in the rear is

moving closer with a constant speed.

9.3 REFRACTION

When a beam of light encounters another transparent medium, a part of

light gets reflected back into the first medium while the rest enters the

other. A ray of light represents a beam. The direction of propagation of an

obliquely incident (0°< i < 90°) ray of light that enters the other medium,

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changes at the interface of the two media. This

phenomenon is called refraction of light. Snell

experimentally obtained the following laws of

refraction:

(i) The incident ray, the refracted ray and the

normal to the interface at the point of

incidence, all lie in the same plane.

(ii) The ratio of the sine of the angle of incidence

to the sine of angle of refraction is constant.

Remember that the angles of incidence (i ) and

refraction (r ) are the angles that the incident

and its refracted ray make with the normal,

espectively. We have

sin

(9.10)

where n

is a constant, called the refractive index of the second medium

with respect to the first medium. Equation (9.10) is the well-known Snell’s

law of refraction. We note that n

is a characteristic of the pair of media

(and also depends on the wavelength of light), but is independent of the

angle of incidence.

From Eq. (9.10), if n

> 1, r < i, i.e., the refracted ray bends towards

the normal. In such a case medium 2 is said to be optically denser (or

denser, in short) than medium 1. On the other hand, if n

<1, r > i, the

refracted ray bends away from the normal. This is the case when incident

ray in a denser medium refracts into a rarer medium.

Note: Optical density should not be confused with mass density,

which is mass per unit volume. It is possible that mass density of

an optically denser medium may be less than that of an optically

rarer medium (optical density is the ratio of the speed of light in

two media). For example, turpentine and water. Mass density of

turpentine is less than that of water but its optical density is higher.

If n

is the refractive index of medium 2

with respect to medium 1 and n

the refractive

index of medium 1 with respect to medium 2,

then it should be clear that

(9.11)

It also follows that if n

is the refractive

index of medium 3 with respect to medium 2

then

= n

× n

, where n

is the refractive

index of medium 3 with respect to medium 1.

Some elementary results based on the laws

of refraction follow immediately. For a

rectangular slab, refraction takes place at two

interfaces (air-glass and glass-air). It is easily seen from Fig. 9.9 that

= i

, i.e., the emergent ray is parallel to the incident ray—there is no

FIGURE 9.8 Refraction and reflection of light.

FIGURE 9.9 Lateral shift of a ray refracted

through a parallel-sided slab.

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EXAMPLE 9.5

deviation, but it does suffer lateral displacement/

shift with respect to the incident ray. Another familiar

observation is that the bottom of a tank filled with

water appears to be raised (Fig. 9.10). For viewing

near the normal direction, it can be shown that the

apparent depth (h

) is real depth (h

) divided by

the refractive index of the medium (water).

The refraction of light through the atmosphere

is responsible for many interesting phenomena. For

example, the Sun is visible a little before the actual

sunrise and until a little after the actual sunset

due to refraction of light through the atmosphere

(Fig. 9.11). By actual sunrise we mean the actual

crossing of the horizon by the sun. Figure 9.11

shows the actual and apparent positions of the Sun

with respect to the horizon. The figure is highly

exaggerated to show the effect. The refractive index

of air with respect to vacuum is 1.00029. Due to

this, the apparent shift in the direction of the Sun

is by about half a degree and the corresponding

time difference between actual sunset and apparent

sunset is about 2 minutes (see Example 9.5). The

apparent flattening (oval shape) of the Sun at sunset

and sunrise is also due to the same phenomenon.

FIGURE 9.10 Apparent depth for

(a) normal, and (b) oblique viewing.

FIGURE 9.11 Advance sunrise and delayed sunset due to

atmospheric refraction.

Example 9.5 The earth takes 24 h to rotate once about its axis. How

much time does the sun take to shift by 1° when viewed from

the earth?

Solution

Time taken for 360° shift = 24 h

Time taken for 1° shift = 24/360 h = 4 min.

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9.4 TOTAL INTERNAL REFLECTION

When light travels from an optically denser medium to a rarer medium

at the interface, it is partly reflected back into the same medium and

partly refracted to the second medium. This reflection is called the internal

reflection.

When a ray of light enters from a denser medium to a rarer medium,

it bends away from the normal, for example, the ray AO

B in Fig. 9.12.

The incident ray AO

is partially reflected (O

C) and partially transmitted

B) or refracted, the angle of refraction (r) being larger than the angle of

incidence (i). As the angle of incidence increases, so does the angle of

refraction, till for the ray AO

, the angle of refraction is π/2. The refracted

ray is bent so much away from the normal that it grazes the surface at

the interface between the two media. This is shown by the ray AO

D in

Fig. 9.12. If the angle of incidence is increased still further (e.g., the ray

), refraction is not possible, and the incident ray is totally reflected.

THE DROWNING CHILD, LIFEGUARD AND SNELL’S

LAW

Consider a rectangular swimming pool PQSR; see figure here. A lifeguard sitting at G

outside the pool notices a child drowning at a point C. The guard wants to reach the

child in the shortest possible time. Let SR be the

side of the pool between G and C. Should he/she

take a straight line path GAC between G and C or

GBC in which the path BC in water would be the

shortest, or some other path GXC? The guard knows

that his/her running speed v

on ground is higher

than his/her swimming speed v

Suppose the guard enters water at X. Let GX =l

and XC =l

. Then the time taken to reach from G to

C would be

1 2

l l

v v

= +

To make this time minimum, one has to

differentiate it (with respect to the coordinate of X) and find the point X when t is a

minimum. On doing all this algebra (which we skip here), we find that the guard should

enter water at a point where Snell’s law is satisfied. To understand this, draw a

perpendicular LM to side SR at X. Let ∠GXM = i and ∠CXL = r. Then it can be seen that t

is minimum when

sin

r v

In the case of light v

, the ratio of the velocity of light in vacuum to that in the

medium, is the refractive index n of the medium.

In short, whether it is a wave or a particle or a human being, whenever two mediums

and two velocities are involved, one must follow Snell’s law if one wants to take the

shortest time.

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This is called total internal reflection. When

light gets reflected by a surface, normally

some fraction of it gets transmitted. The

reflected ray, therefore, is always less intense

than the incident ray, howsoever smooth the

reflecting surface may be. In total internal

reflection, on the other hand, no

transmission of light takes place.

The angle of incidence corresponding to

an angle of refraction 90°, say ∠AO

N, is

called the critical angle (i

) for the given pair

of media. We see from Snell’s law [Eq. (9.10)]

that if the relative refractive index of the

refracting medium is less than one then,

since the maximum value of sin r is unity,

there is an upper limit to the value of sin i for which the law can be

satisfied, that is, i = i

such that

sin i

= n

(9.12)

For values of i larger than i

, Snell’s law of refraction cannot be

satisfied, and hence no refraction is possible.

The refractive index of denser medium 1 with respect to rarer medium

2 will be n

= 1/sini

. Some typical critical angles are listed in Table 9.1.

FIGURE 9.12 Refraction and internal reflection

of rays from a point A in the denser medium

(water) incident at different angles at the interface

with a rarer medium (air).

A demonstration for total internal reflection

All optical phenomena can be demonstrated very easily with the use of a

laser torch or pointer, which is easily available nowadays. Take a glass

beaker with clear water in it. Add a few drops of milk or any other

suspension to water and stir so that water becomes a little turbid. Take

a laser pointer and shine its beam through the turbid water. You will

find that the path of the beam inside the water shines brightly.

Shine the beam from below the beaker such that it strikes at the

upper water surface at the other end. Do you find that it undergoes partial

reflection (which is seen as a spot on the table below) and partial refraction

[which comes out in the air and is seen as a spot on the roof; Fig. 9.13(a)]?

Now direct the laser beam from one side of the beaker such that it strikes

the upper surface of water more obliquely [Fig. 9.13(b)]. Adjust the

direction of laser beam until you find the angle for which the refraction

TABLE 9.1 CRITICAL ANGLE OF SOME TRANSPARENT MEDIA WITH RESPECT TO AIR

Substance medium Refractive index Critical angle

Water 1.33 48.75

Crown glass 1.52 41.14

Dense flint glass 1.62 37.31

Diamond 2.42 24.41

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above the water surface is totally absent and the beam is totally reflected

back to water. This is total internal reflection at its simplest.

Pour this water in a long test tube and shine the laser light from top,

as shown in Fig. 9.13(c). Adjust the direction of the laser beam such that

it is totally internally reflected every time it strikes the walls of the tube.

This is similar to what happens in optical fibres.

Take care not to look into the laser beam directly and not to point it

at anybody’s face.

9.4.1 Total internal reflection in nature and

its technological applications

(i) Mirage: On hot summer days, the air near the ground becomes hotter

than the air at higher levels. The refractive index of air increases with

its density. Hotter air is less dense, and has smaller refractive index

than the cooler air. If the air currents are small, that is, the air is still,

the optical density at different layers of air increases with height. As a

result, light from a tall object such as a tree, passes through a medium

whose refractive index decreases towards the ground. Thus, a ray of

light from such an object successively bends away from the normal

and undergoes total internal reflection, if the angle of incidence for

the air near the ground exceeds the critical angle. This is shown in

Fig. 9.14(b). To a distant observer, the light appears to be coming

from somewhere below the ground. The observer naturally assumes

that light is being reflected from the ground, say, by a pool of water

near the tall object. Such inverted images of distant tall objects cause

an optical illusion to the observer. This phenomenon is called mirage.

This type of mirage is especially common in hot deserts. Some of you

might have noticed that while moving in a bus or a car during a hot

summer day, a distant patch of road, especially on a highway, appears

to be wet. But, you do not find any evidence of wetness when you

reach that spot. This is also due to mirage.

FIGURE 9.13

Observing total

internal reflection in

water with a laser

beam (refraction due

to glass of beaker

neglected being very

thin).

FIGURE 9.14 (a) A tree is seen by an observer at its place when the air above the ground is

at uniform temperature, (b) When the layers of air close to the ground have varying

temperature with hottest layers near the ground, light from a distant tree may

undergo total internal reflection, and the apparent image of the tree may create

an illusion to the observer that the tree is near a pool of water.

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(ii) Diamond: Diamonds are known for their

spectacular brilliance. Their brilliance

is mainly due to the total internal

reflection of light inside them. The critical

angle for diamond-air interface (≅ 24.4°)

is very small, therefore once light enters

a diamond, it is very likely to undergo

total internal reflection inside it.

Diamonds found in nature rarely exhibit

the brilliance for which they are known.

It is the technical skill of a diamond

cutter which makes diamonds to

sparkle so brilliantly. By cutting the

diamond suitably, multiple total

internal reflections can be made

to occur.

(iii) Prism: Prisms designed to bend light by

90° or by 180° make use of total internal

reflection [Fig. 9.15(a) and (b)]. Such a

prism is also used to invert images

without changing their size [Fig. 9.15(c)].

In the first two cases, the critical angle i

for the material of the prism

must be less than 45°. We see from Table 9.1 that this is true for both

crown glass and dense flint glass.

(iv) Optical fibres: Nowadays optical fibres are extensively used for

transmitting audio and video signals through long distances. Optical

fibres too make use of the phenomenon of total internal reflection.

Optical fibres are fabricated with high quality composite glass/quartz

fibres. Each fibre consists of a core and cladding. The refractive index

of the material of the core is higher than that of the cladding.

When a signal in the form of light is

directed at one end of the fibre at a suitable

angle, it undergoes repeated total internal

reflections along the length of the fibre and

finally comes out at the other end (Fig. 9.16).

Since light undergoes total internal reflection

at each stage, there is no appreciable loss in

the intensity of the light signal. Optical fibres

are fabricated such that light reflected at one

side of inner surface strikes the other at an

angle larger than the critical angle. Even if the

fibre is bent, light can easily travel along its

length. Thus, an optical fibre can be used to act as an optical pipe.

A bundle of optical fibres can be put to several uses. Optical fibres

are extensively used for transmitting and receiving electrical signals which

are converted to light by suitable transducers. Obviously, optical fibres

can also be used for transmission of optical signals. For example, these

are used as a ‘light pipe’ to facilitate visual examination of internal organs

like esophagus, stomach and intestines. You might have seen a commonly

FIGURE 9.15 Prisms designed to bend rays by

90° and 180° or to invert image without changing

its size make use of total internal reflection.

FIGURE 9.16 Light undergoes successive total

internal reflections as it moves through an

optical fibre.

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available decorative lamp with fine plastic fibres with their free ends

forming a fountain like structure. The other end of the fibres is fixed over

an electric lamp. When the lamp is switched on, the light travels from the

bottom of each fibre and appears at the tip of its free end as a dot of light.

The fibres in such decorative lamps are optical fibres.

The main requirement in fabricating optical fibres is that there should

be very little absorption of light as it travels for long distances inside

them. This has been achieved by purification and special preparation of

materials such as quartz. In silica glass fibres, it is possible to transmit

more than 95% of the light over a fibre length of 1 km. (Compare with

what you expect for a block of ordinary window glass 1 km thick.)

9.5 REFRACTION AT SPHERICAL SURFACES

AND BY LENSES

We have so far considered refraction at a plane interface. We shall now

consider refraction at a spherical interface between two transparent media.

An infinitesimal part of a spherical surface can be regarded as planar

and the same laws of refraction can be applied at every point on the

surface. Just as for reflection by a spherical mirror, the normal at the

point of incidence is perpendicular to the tangent plane to the spherical

surface at that point and, therefore, passes through its centre of curvature.

We first consider refraction by a single spherical surface and follow it by

thin lenses. A thin lens is a transparent optical medium bounded by two

surfaces; at least one of which should be spherical. Applying the formula

for image formation by a single spherical surface successively at the two

surfaces of a lens, we shall obtain the lens maker’s formula and then the

lens formula.

9.5.1 Refraction at a spherical surface

Figure 9.17 shows the geometry of formation of image I of an object O on

the principal axis of a spherical surface with centre of curvature C, and

radius of curvature R. The rays are incident from a medium of refractive

index n

, to another of refractive index n

. As before, we take the aperture

(or the lateral size) of the surface to be small

compared to other distances involved, so that small

angle approximation can be made. In particular,

NM will be taken to be nearly equal to the length of

the perpendicular from the point N on the principal

axis. We have, for small angles,

tan ∠NOM =

tan ∠NCM =

tan ∠NIM =

FIGURE 9.17 Refraction at a spherical

surface separating two media.

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Now, for ∆NOC, i is the exterior angle. Therefore, i = ∠NOM + ∠NCM

i =

MN MN

OM MC

(9.13)

Similarly,

r = ∠NCM – ∠NIM

i.e., r =

MN MN

MC MI

−

(9.14)

Now, by Snell’s law

sin i = n

sin r

or for small angles

i = n

LIGHT SOURCES AND PHOTOMETRY

It is known that a body above absolute zero temperature emits electromagnetic radiation.

The wavelength region in which the body emits the radiation depends on its absolute

temperature. Radiation emitted by a hot body, for example, a tungsten filament lamp

having temperature 2850 K are partly invisible and mostly in infrared (or heat) region.

As the temperature of the body increases radiation emitted by it is in visible region. The

sun with temperature of about 5500 K emits radiation whose energy versus wavelength

graph peaks approximately at 550 nm corresponding to green light and is almost in the

middle of the visible region. The energy versus wavelength distribution graph for a given

body peaks at some wavelength, which is inversely proportional to the absolute

temperature of that body.

The measurement of light as perceived by human eye is called photometry. Photometry

is measurement of a physiological phenomenon, being the stimulus of light as received

by the human eye, transmitted by the optic nerves and analysed by the brain. The main

physical quantities in photometry are (i) the luminous intensity of the source,

(ii) the luminous flux or flow of light from the source, and (iii) illuminance of the surface.

The SI unit of luminous intensity (I) is candela (cd). The candela is the luminous intensity,

in a given direction, of a source that emits monochromatic radiation of frequency

540 × 10

Hz and that has a radiant intensity in that direction of 1/683 watt per steradian.

If a light source emits one candela of luminous intensity into a solid angle of one steradian,

the total luminous flux emitted into that solid angle is one lumen (lm). A standard

100 watt incadescent light bulb emits approximately 1700 lumens.

In photometry, the only parameter, which can be measured directly is illuminance. It

is defined as luminous flux incident per unit area on a surface (lm/m

or lux). Most light

meters measure this quantity. The illuminance E, produced by a source of luminous

intensity I, is given by E = I/r

, where r is the normal distance of the surface from the

source. A quantity named luminance (L), is used to characterise the brightness of emitting

or reflecting flat surfaces. Its unit is cd/m

(sometimes called ‘nit’ in industry). A good

LCD computer monitor has a brightness of about 250 nits.

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EXAMPLE

9.6

Substituting i and r from Eqs. (9.13) and (9.14), we get

1 2 2 1

OM MI MC

n n n n

−

+ =

(9.15)

Here, OM, MI and MC represent magnitudes of distances. Applying the

Cartesian sign convention,

OM = –u, MI = +v, MC = +R

Substituting these in Eq. (9.15), we get

2 1 2 1

n n n n

v u R

−

− =

(9.16)

Equation (9.16) gives us a relation between object and image distance

in terms of refractive index of the medium and the radius of

curvature of the curved spherical surface. It holds for any curved

spherical surface.

Example 9.6 Light from a point source in air falls on a spherical

glass surface (n = 1.5 and radius of curvature = 20 cm). The distance

of the light source from the glass surface is 100 cm. At what position

the image is formed?

Solution

We use the relation given by Eq. (9.16). Here

u = – 100 cm, v = ?, R = + 20 cm, n

= 1, and n

= 1.5.

We then have

1.5 1 0.5

100 20v

+ =

or v = +100 cm

The image is formed at a distance of 100 cm from the glass surface,

in the direction of incident light.

9.5.2 Refraction by a lens

Figure 9.18(a) shows the geometry of image formation by a double convex

lens. The image formation can be seen in terms of two steps:

(i) The first refracting surface forms the image I

of the object O

[Fig. 9.18(b)]. The image I

acts as a virtual object for the second surface

that forms the image at I [Fig. 9.18(c)]. Applying Eq. (9.15) to the first

interface ABC, we get

1 2 2 1

1 1

OB BI BC

n n n n

−

+ =

(9.17)

A similar procedure applied to the second interface* ADC gives,

2 1 2 1

1 2

DI DI DC

n n n n

−

− + =

(9.18)

* Note that now the refractive index of the medium on the right side of ADC is n

while on its left it is n

. Further DI

is negative as the distance is measured

against the direction of incident light.

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For a thin lens, BI

= DI

. Adding

Eqs. (9.17) and (9.18), we get

1 1

2 1

1 2

1 1

( )

OB DI BC DC

n n

 

+ = − +

 

 

(9.19)

Suppose the object is at infinity, i.e.,

OB → ∞ and DI = f, Eq. (9.19) gives

2 1

1 2

1 1

( )

BC DC

n n

 

= − +

 

 

(9.20)

The point where image of an object

placed at infinity is formed is called the

focus F, of the lens and the distance f gives

its focal length. A lens has two foci, F and

F′, on either side of it (Fig. 9.19). By the

sign convention,

= + R

= –R

So Eq. (9.20) can be written as

(

)

21 21

1 2 1

1 1 1

n n

f R R n

   

= − − =

   

   

∵

(9.21)

Equation (9.21) is known as the lens

maker’s formula. It is useful to design

lenses of desired focal length using surfaces

of suitable radii of curvature. Note that the

formula is true for a concave lens also. In

that case R

is negative, R

positive and

therefore, f is negative.

From Eqs. (9.19) and (9.20), we get

1 1 1

OB DI

n n n

+ =

(9.22)

Again, in the thin lens approximation, B and D are both close to the

optical centre of the lens. Applying the sign convention,

BO = – u, DI = +v, we get

1 1 1

v u f

− =

(9.23)

Equation (9.23) is the familiar thin lens formula. Though we derived

it for a real image formed by a convex lens, the formula is valid for both

convex as well as concave lenses and for both real and virtual images.

It is worth mentioning that the two foci, F and F′, of a double convex

or concave lens are equidistant from the optical centre. The focus on the

side of the (original) source of light is called the first focal point, whereas

the other is called the second focal point.

To find the image of an object by a lens, we can, in principle, take any

two rays emanating from a point on an object; trace their paths using

FIGURE 9.18 (a) The position of object, and the

image formed by a double convex lens,

(b) Refraction at the first spherical surface and

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EXAMPLE 9.7

the laws of refraction and find the point where

the refracted rays meet (or appear to meet). In

practice, however, it is convenient to choose any

two of the following rays:

(i) A ray emanating from the object parallel to

the principal axis of the lens after refraction

passes through the second principal focus

F′ (in a convex lens) or appears to diverge (in

a concave lens) from the first principal focus F.

(ii) A ray of light, passing through the optical

centre of the lens, emerges without any

deviation after refraction.

(iii) (a) A ray of light passing through the first

principal focus of a convex lens [Fig. 9.19(a)]

emerges parallel to the principal axis after

refraction.

(b) A ray of light incident on a concave lens

appearing to meet the principal axis at

second focus point emerges parallel to the

principal axis after refraction [Fig. 9.19(b)].

Figures 9.19(a) and (b) illustrate these rules

for a convex and a concave lens, respectively.

You should practice drawing similar ray diagrams for different positions

of the object with respect to the lens and also verify that the lens formula,

Eq. (9.23), holds good for all cases.

Here again it must be remembered that each point on an object gives

out infinite number of rays. All these rays will pass through the same

image point after refraction at the lens.

Magnification (m) produced by a lens is defined, like that for a mirror,

as the ratio of the size of the image to that of the object. Proceeding in the

same way as for spherical mirrors, it is easily seen that for a lens

m =

′

(9.24)

When we apply the sign convention, we see that, for erect (and virtual)

image formed by a convex or concave lens, m is positive, while for an

inverted (and real) image, m is negative.

Example 9.7 A magician during a show makes a glass lens with

n = 1.47 disappear in a trough of liquid. What is the refractive index

of the liquid? Could the liquid be water?

Solution

The refractive index of the liquid must be equal to 1.47 in order to

make the lens disappear. This means n

= n

. This gives 1/f =0 or

f → ∞. The lens in the liquid will act like a plane sheet of glass. No,

the liquid is not water. It could be glycerine.

9.5.3 Power of a lens

Power of a lens is a measure of the convergence or divergence, which a

lens introduces in the light falling on it. Clearly, a lens of shorter focal

FIGURE 9.19 Tracing rays through (a)

convex lens (b) concave lens.

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EXAMPLE 9.8

length bends the incident light more, while converging it

in case of a convex lens and diverging it in case of a

concave lens. The power P of a lens is defined as the

tangent of the angle by which it converges or diverges a

beam of light parallel to the principal axis falling at unit

distance from the optical centre (Fig. 9.20).

tan ; , tan

δ δ

= = =

if 1

for small

value of

. Thus,

P =

(9.25)

The SI unit for power of a lens is dioptre (D): 1D = 1m

–1

. The power of

a lens of focal length of 1 metre is one dioptre. Power of a lens is positive

for a converging lens and negative for a diverging lens. Thus, when an

optician prescribes a corrective lens of power + 2.5 D, the required lens is

a convex lens of focal length + 40 cm. A lens of power of – 4.0 D means a

concave lens of focal length – 25 cm.

Example 9.8 (i) If f = 0.5 m for a glass lens, what is the power of the

lens? (ii) The radii of curvature of the faces of a double convex lens

are 10 cm and 15 cm. Its focal length is 12 cm. What is the refractive

index of glass? (iii) A convex lens has 20 cm focal length in air. What

is focal length in water? (Refractive index of air-water = 1.33, refractive

index for air-glass = 1.5.)

Solution

(i) Power = +2 dioptre.

(ii) Here, we have f = +12 cm, R

= +10 cm, R

= –15 cm.

Refractive index of air is taken as unity.

We use the lens formula of Eq. (9.22). The sign convention has to

be applied for f, R

and R

Substituting the values, we have

= − −

−













( )n

This gives n = 1.5.

(iii) For a glass lens in air, n

= 1.5, n

= 1, f = +20 cm. Hence, the lens

formula gives

0 5

1 1

1 2

= −













R R

For the same glass lens in water, n

= 1.5, n

= 1.33. Therefore,

1 33

1 5 1 33

1 1

1 2

( . . )

f R R

= − −













(9.26)

Combining these two equations, we find f = + 78.2 cm.

9.5.4 Combination of thin lenses in contact

Consider two lenses A and B of focal length f

and f

placed in contact

with each other. Let the object be placed at a point O beyond the focus of

FIGURE 9.20 Power of a lens.

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the first lens A (Fig. 9.21). The first lens produces

an image at I

. Since image I

is real, it serves as a

virtual object for the second lens B, producing the

final image at I. It must, however, be borne in mind

that formation of image by the first lens is presumed

only to facilitate determination of the position of the

final image. In fact, the direction of rays emerging

from the first lens gets modified in accordance with

the angle at which they strike the second lens. Since

the lenses are thin, we assume the optical centres of the lenses to be

coincident. Let this central point be denoted by P.

For the image formed by the first lens A, we get

1 1

1 1 1

v u f

− =

(9.27)

For the image formed by the second lens B, we get

1 2

1 1 1

v v f

− =

(9.28)

Adding Eqs. (9.27) and (9.28), we get

1 2

1 1 1 1

v u f f

− = +

(9.29)

If the two lens-system is regarded as equivalent to a single lens of

focal length f, we have

1 1 1

v u f

− =

so that we get

1 2

1 1 1

f f f

= +

(9.30)

The derivation is valid for any number of thin lenses in contact. If

several thin lenses of focal length f

, f

,... are in contact, the effective

focal length of their combination is given by

1 2 3

1 1 1 1

f f f f

= + + +

… (9.31)

In terms of power, Eq. (9.31) can be written as

P = P

+ P

+ … (9.32)

where P is the net power of the lens combination. Note that the sum in

Eq. (9.32) is an algebraic sum of individual powers, so some of the terms

on the right side may be positive (for convex lenses) and some negative

(for concave lenses). Combination of lenses helps to obtain diverging or

converging lenses of desired magnification. It also enhances sharpness

of the image. Since the image formed by the first lens becomes the object

for the second, Eq. (9.25) implies that the total magnification m of the

combination is a product of magnification (m

, m

,...) of individual

lenses

m = m

... (9.33)

FIGURE 9.21 Image formation by a

combination of two thin lenses in contact.

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EXAMPLE 9.9

Such a system of combination of lenses is commonly used in designing

lenses for cameras, microscopes, telescopes and other optical instruments.

Example 9.9 Find the position of the image formed by the lens

combination given in the Fig. 9.22.

FIGURE 9.22

Solution Image formed by the first lens

1 1 1

v u f

− =

1 1 1

30 10

− =

−

or v

= 15 cm

The image formed by the first lens serves as the object for the second.

This is at a distance of (15 – 5) cm = 10 cm to the right of the second

lens. Though the image is real, it serves as a virtual object for the

second lens, which means that the rays appear to come from it for

the second lens.

1 1 1

10 10

− =

−

or v

= ∞

The virtual image is formed at an infinite distance to the left of the

second lens. This acts as an object for the third lens.

3 3 3

1 1 1

v u f

− =

1 1 1

= +

∞

or v

= 30 cm

The final image is formed 30 cm to the right of the third lens.

9.6 REFRACTION THROUGH A PRISM

Figure 9.23 shows the passage of light through a triangular prism ABC.

The angles of incidence and refraction at the first face AB are i and r

while the angle of incidence (from glass to air) at the second face AC is r

and the angle of refraction or emergence e. The angle between the

emergent ray RS and the direction of the incident ray PQ is called the

angle of deviation,

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In the quadrilateral AQNR, two of the angles

(at the vertices Q and R) are right angles.

Therefore, the sum of the other angles of the

quadrilateral is 180°.

∠A + ∠QNR = 180°

From the triangle QNR,

+ r

+ ∠QNR = 180°

Comparing these two equations, we get

+ r

= A (9.34)

The total deviation

is the sum of deviations

at the two faces,

= (i – r

) + (e – r

)

that is,

= i + e – A (9.35)

Thus, the angle of deviation depends on the angle of incidence. A plot

between the angle of deviation and angle of incidence is shown in

Fig. 9.24. You can see that, in general, any given value of

, except for

i = e, corresponds to two values i and hence of e. This, in fact, is expected

from the symmetry of i and e in Eq. (9.35), i.e.,

remains the same if i

and e are interchanged. Physically, this is related

to the fact that the path of ray in Fig. 9.23 can be

traced back, resulting in the same angle of

deviation. At the minimum deviation D

, the

refracted ray inside the prism becomes parallel

to its base. We have

δ = D

, i = e which implies r

= r

Equation (9.34) gives

2r = A or r =

(9.36)

In the same way, Eq. (9.35) gives

= 2i – A, or i = (A + D

)/2 (9.37)

The refractive index of the prism is

sin[( )/ 2]

sin[ /2]

A Dn

n A

= =

(9.38)

The angles A and D

can be measured

experimentally. Equation (9.38) thus provides a

method of determining refractive index of the material of the prism.

For a small angle prism, i.e., a thin prism, D

is also very small, and

we get

(

)

sin[( )/2]

sin[ /2] /2

A D

A A

= 

= (n

–1)A

It implies that, thin prisms do not deviate light much.

FIGURE 9.23 A ray of light passing through

a triangular glass prism.

FIGURE 9.24 Plot of angle of deviation (

)

versus angle of incidence (i) for a

triangular prism.

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9.7 SOME NATURAL PHENOMENA DUE TO SUNLIGHT

The Interplay of light with things around us gives rise to several beautiful

phenomena. The spectacles of colours that we see around us all the time

is possible due to sunlight.

While studying dispersion of visible (or white) light by a prism

(Class X) and the electromagnetic spectrum (Chapter 8, Class XII), we

got to know that colour is associated with the frequency of light or the

wavelength of light in the given medium. In the visible spectrum, red

light is at the long wavelength end (~700 nm) while the violet light is at

the short wavelength end (~ 400 nm). Dispersion takes place because

the refractive index of medium for different frequencies (colours) is

different. For example, the bending of red component of white light is

least while it is most for the violet. Equivalently, red light travels faster

than violet light in a glass prism. Table 9.2 gives the refractive indices for

different wavelength for crown glass and flint glass. Thick lenses could

be assumed as made of many prisms, therefore, thick lenses show

chromatic aberration due to dispersion of light. When white light passes

through thick lenses, red and blue colours focus at different points. This

phenomenon is known as chromatic aberration.

Colour Wavelength (nm) Crown glass Flint glass

Violet 396.9 1.533 1.663

Blue 486.1 1.523 1.639

Yellow 589.3 1.517 1.627

Red 656.3 1.515 1.622

TABLE 9.2 REFRACTIVE INDICES FOR DIFFERENT WAVELENGTHS

The variation of refractive index with wavelength may be more

pronounced in some media than the other. In vacuum, of course, the

speed of light is independent of wavelength. Thus, vacuum (or air

approximately) is a non-dispersive medium in which all colours travel

with the same speed. This also follows from the fact that sunlight reaches

us in the form of white light and not as its components. On the other

hand, glass is a dispersive medium.

The blue of the sky, white clouds, the red-hue at sunrise and sunset,

the rainbow, the brilliant colours of some pearls, shells, and wings of

birds, are just a few of the natural wonders we are used to. We describe

some of them here from the point of view of physics.

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9.7.1 The rainbow

The rainbow is an example of the dispersion of sunlight by the water

drops in the atmosphere. This is a phenomenon due to combined effect

of dispersion, refraction and reflection of sunlight by spherical water

droplets of rain. The conditions for observing a rainbow are that the Sun

should be shining in one part of the sky (say near western horizon) while

it is raining in the opposite part of the sky (say eastern horizon).

An observer can therefore see a rainbow only when his back is towards

the Sun.

In order to understand the formation of rainbows, consider

Fig. 9.25(a). Sunlight is first refracted as it enters a raindrop, which causes

the different wavelengths (colours) of white light to separate. Longer

wangelength of light (red) are bent the least while the shorter wavelength

(violet) are bent the most. Next, these component rays strike the inner

surface of the water drop and get internally reflected if the angle between

the refracted ray and normal to the drop surface is greater then the critical

angle (48

in this case). The reflected light is refracted again when it comes

out of the drop, as shown in the figure. It is found that the violet light

emerges at an angle of 40

related to the incoming sunlight and red light

emerges at an angle of 42

. For other colours, angles lie in between these

two values.

Figure 9.25(b) explains the formation of primary rainbow. We see

that red light from drop 1 and violet light from drop 2 reach the observer’s

eye. The violet from drop 1 and red light from drop 2 are directed at level

above or below the observer. Thus the observer sees a rainbow with

red colour on the top and violet on the bottom. The primary

rainbow is a result of three-step process, that is, refraction, reflection

and refraction.

Formation of rainbows

http://www.eo.ucar.edu/rainbows

http://www.atoptics.co.uk/bows.htm

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When light rays undergoes two internal reflections inside a raindrop,

instead of one as in the primary rainbow, a secondary rainbow is formed

as shown in Fig. 9.25(c). It is due to four-step process. The intensity of

light is reduced at the second reflection and hence the secondary rainbow

is fainter than the primary rainbow. Further, the order of the colours is

reversed in it as is clear from Fig. 9.25(c).

9.7.2 Scattering of light

As sunlight travels through the earth’s atmosphere, it gets scattered

(changes its direction) by the atmospheric particles. Light of shorter

wavelengths is scattered much more than light of longer wavelengths.

(The amount of scattering is inversely proportional to the fourth power

of the wavelength. This is known as Rayleigh scattering). Hence, the bluish

colour predominates in a clear sky, since blue has a shorter wave-

length than red and is scattered much more strongly. In fact, violet

gets scattered even more than blue, having a shorter wavelength.

But since our eyes are more sensitive to blue than violet, we see the

sky blue.

Large particles like dust and water droplets present in the atmosphere

behave differently. The relevant quantity here is the relative size of the

wavelength of light

, and the scatterer (of typical size, say, a). For a <<

one has Rayleigh scattering which is proportional to 1/

. For a >>

i.e., large scattering objects (for example, raindrops, large dust or ice

particles) this is not true; all wavelengths are scattered nearly equally.

Thus, clouds which have droplets of water with a >>

are generally white.

FIGURE 9.25 Rainbow: (a) The sun rays incident on a water drop get

refracted twice and reflected internally by a drop; (b) Enlarged view of

internal reflection and refraction of a ray of light inside a drop forming

primary rainbow; and (c) secondary rainbow is formed by rays undergoing

internal reflection twice inside the drop.

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At sunset or sunrise, the sun’s rays have to pass through a larger

distance in the atmosphere (Fig. 9.26). Most of the blue and other shorter

wavelengths are removed by scattering. The least scattered light reaching

our eyes, therefore, the sun looks reddish. This explains the reddish

appearance of the sun and full moon near the horizon.

9.8 OPTICAL INSTRUMENTS

A number of optical devices and instruments have been designed utilising

reflecting and refracting properties of mirrors, lenses and prisms.

Periscope, kaleidoscope, binoculars, telescopes, microscopes are some

examples of optical devices and instruments that are in common use.

Our eye is, of course, one of the most important optical device the nature

has endowed us with. We have already studied about the human eye in

Class X. We now go on to describe the principles of working of the

microscope and the telescope.

9.8.1 The microscope

A simple magnifier or microscope is a converging lens of small focal length

(Fig. 9.27). In order to use such a lens as a microscope, the lens is held

FIGURE 9.26 Sunlight travels through a longer distance in the

atmosphere at sunset and sunrise.

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near the object, one focal length away or less, and the eye is positioned

close to the lens on the other side. The idea is to get an erect, magnified

and virtual image of the object at a distance so that it can be viewed

comfortably, i.e., at 25 cm or more. If the object is at a distance f, the

image is at infinity. However, if the object is at a distance slightly less

than the focal length of the lens, the image is virtual and closer than

infinity. Although the closest comfortable distance for viewing the image

is when it is at the near point (distance D ≅ 25 cm), it causes some strain

on the eye. Therefore, the image formed at infinity is often considered

most suitable for viewing by the relaxed eye. We show both cases, the

first in Fig. 9.27(a), and the second in Fig. 9.27(b) and (c).

The linear magnification m, for the image formed at the near point D,

by a simple microscope can be obtained by using the relation

v f

= =

























1 1

1– –

Now according to our sign convention, v is negative, and is equal in

magnitude to D. Thus, the magnification is

= +













(9.39)

FIGURE 9.27 A simple microscope; (a) the magnifying lens is located

such that the image is at the near point, (b) the angle subtanded by the

object, is the same as that at the near point, and (c) the object near the

focal point of the lens; the image is far off but closer than infinity.

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Since D is about 25 cm, to have a magnification of six, one needs a convex

lens of focal length, f

= 5 cm.

Note that m = h

′

/h where h

is the size of the object and h′ the size of

the image. This is also the ratio of the angle subtended by the image

to that subtended by the object, if placed at D for comfortable viewing.

(Note that this is not the angle actually subtended by the object at the

eye, which is h/u.) What a single-lens simple magnifier achieves is that it

allows the object to be brought closer to the eye than D.

We will now find the magnification when the image is at infinity. In

this case we will have to obtained the angular magnification. Suppose

the object has a height h. The maximum angle it can subtend, and be

clearly visible (without a lens), is when it is at the near point, i.e., a distance

D. The angle subtended is then given by

tan













≈

(9.40)

We now find the angle subtended at the eye by the image when the

object is at u. From the relations

h v

h u

′

= =

we have the angle subtended by the image

tan

h h v h

v v u u

′

= = ⋅ =

− − −

≈

. The angle subtended by the object, when it

is at u = –f.













(9.41)

as is clear from Fig. 9.27(c). The angular magnification is, therefore













(9.42)

This is one less than the magnification when the image is at the near

point, Eq. (9.39), but the viewing is more comfortable and the difference

in magnification is usually small. In subsequent discussions of optical

instruments (microscope and telescope) we shall assume the image to be

at infinity.

A simple microscope has a limited maximum magnification (≤ 9) for

realistic focal lengths. For much larger magnifications, one uses two lenses,

one compounding the effect of the other. This is known as a compound

microscope. A schematic diagram of a compound microscope is shown

in Fig. 9.28. The lens nearest the object, called the objective, forms a

real, inverted, magnified image of the object. This serves as the object for

the second lens, the eyepiece, which functions essentially like a simple

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microscope or magnifier, produces the final image, which is enlarged

and virtual. The first inverted image is thus near (at or within) the focal

plane of the eyepiece, at a distance appropriate for final image formation

at infinity, or a little closer for image formation at the near point. Clearly,

the final image is inverted with respect to the original object.

We now obtain the magnification due to a compound microscope.

The ray diagram of Fig. 9.28 shows that the (linear) magnification due to

the objective, namely h′/h, equals

h L

h f

′

= =

(9.43)

where we have used the result

tan













′













Here h′ is the size of the first image, the object size being h and f

being the focal length of the objective. The first image is formed near the

focal point of the eyepiece. The distance L, i.e., the distance between the

second focal point of the objective and the first focal point of the eyepiece

(focal length f

) is called the tube length of the compound microscope.

As the first inverted image is near the focal point of the eyepiece, we

use the result from the discussion above for the simple microscope to

obtain the (angular) magnification m

due to it [Eq. (9.39)], when the

final image is formed at the near point, is

= +1













[9.44(a)]

FIGURE 9.28 Ray diagram for the formation of image by a

compound microscope.

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When the final image is formed at infinity, the angular magnification

due to the eyepiece [Eq. (9.42)] is

= (D/f

) [9.44(b)]

Thus, the total magnification [(according to Eq. (9.33)], when the

image is formed at infinity, is

m m m

o e

= =

























(9.45)

Clearly, to achieve a large magnification of a small object (hence the

name microscope), the objective and eyepiece should have small focal

lengths. In practice, it is difficult to make the focal length much smaller

than 1 cm. Also large lenses are required to make L large.

For example, with an objective with f

= 1.0 cm, and an eyepiece with

focal length f

= 2.0 cm, and a tube length of 20 cm, the magnification is

m m m

o e

= =

























20 25

250

1 2

  

Various other factors such as illumination of the object, contribute to

the quality and visibility of the image. In modern microscopes, multi-

component lenses are used for both the objective and the eyepiece to

improve image quality by minimising various optical aberrations (defects)

in lenses.

9.8.2 Telescope

The telescope is used to provide angular magnification of distant objects

(Fig. 9.29). It also has an objective and an eyepiece. But here, the objective

has a large focal length and a much larger aperture than the eyepiece.

Light from a distant object enters the objective and a real image is formed

in the tube at its second focal point. The eyepiece magnifies this image

producing a final inverted image. The magnifying power m is the ratio of

the angle

subtended at the eye by the final image to the angle

which

the object subtends at the lens or the eye. Hence

o o

e e

f f

f h f

  





(9.46)

In this case, the length of the telescope tube is f

+ f

Terrestrial telescopes have, in addition, a pair of inverting lenses to

make the final image erect. Refracting telescopes can be used both for

terrestrial and astronomical observations. For example, consider

a telescope whose objective has a focal length of 100 cm and the eyepiece

a focal length of 1 cm. The magnifying power of this telescope is

m = 100/1 = 100.

The world’s largest optical telescopes

http://astro.nineplanets.org/bigeyes.html

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Let us consider a pair of stars of actual separation 1′ (one minute of

arc). The stars appear as though they are separated by an angle of 100 ×

1′ = 100′ =1.67°.

FIGURE 9.29 A refracting telescope.

The main considerations with an astronomical telescope are its light

gathering power and its resolution or resolving power. The former clearly

depends on the area of the objective. With larger diameters, fainter objects

can be observed. The resolving power, or the ability to observe two objects

distinctly, which are in very nearly the same direction, also depends on

the diameter of the objective. So, the desirable aim in optical telescopes is

to make them with objective of large diameter. The largest lens objective

in use has a diameter of 40 inch (~1.02 m). It is at the Yerkes Observatory

in Wisconsin, USA. Such big lenses tend to be very heavy and therefore,

difficult to make and support by their edges. Further, it is rather difficult

and expensive to make such large sized lenses which form images that

are free from any kind of chromatic aberration and distortions.

For these reasons, modern telescopes use a concave mirror rather

than a lens for the objective. Telescopes with mirror objectives are called

reflecting telescopes. There is no chromatic aberration in a mirror.

Mechanical support is much less of a problem since a mirror weighs

much less than a lens of equivalent optical quality, and can be supported

over its entire back surface, not just over its rim. One obvious problem

with a reflecting telescope is that the objective mirror focusses light inside

the telescope tube. One must have an eyepiece and the observer right

there, obstructing some light (depending on the size of the observer cage).

This is what is done in the very large 200 inch (~5.08 m) diameters, Mt.

Palomar telescope, California. The viewer sits near the focal point of the

mirror, in a small cage. Another solution to the problem is to deflect the

light being focussed by another mirror. One such arrangement using a

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SUMMARY

1. Reflection is governed by the equation ∠i = ∠r

′

and refraction by the

Snell’s law, sini/sinr = n, where the incident ray, reflected ray, refracted

ray and normal lie in the same plane. Angles of incidence, reflection

and refraction are i, r ′ and r, respectively.

2. The critical angle of incidence i

for a ray incident from a denser to rarer

medium, is that angle for which the angle of refraction is 90°. For

i > i

, total internal reflection occurs. Multiple internal reflections in

diamond (i

≅ 24.4°), totally reflecting prisms and mirage, are some

examples of total internal reflection. Optical fibres consist of glass

fibres coated with a thin layer of material of lower refractive index.

Light incident at an angle at one end comes out at the other, after

multiple internal reflections, even if the fibre is bent.

3. Cartesian sign convention: Distances measured in the same direction

as the incident light are positive; those measured in the opposite

direction are negative. All distances are measured from the pole/optic

centre of the mirror/lens on the principal axis. The heights measured

upwards above x-axis and normal to the principal axis of the mirror/

lens are taken as positive. The heights measured downwards are taken

as negative.

FIGURE 9.30 Schematic diagram of a reflecting telescope (Cassegrain).

convex secondary mirror to focus the incident light, which now passes

through a hole in the objective primary mirror, is shown in Fig. 9.30.

This is known as a Cassegrain telescope, after its inventor. It has the

advantages of a large focal length in a short telescope. The largest telescope

in India is in Kavalur, Tamil Nadu. It is a 2.34 m diameter reflecting

telescope (Cassegrain). It was ground, polished, set up, and is being used

by the Indian Institute of Astrophysics, Bangalore. The largest reflecting

telescopes in the world are the pair of Keck telescopes in Hawaii, USA,

with a reflector of 10 metre in diameter.

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4. Mirror equation:

1 1 1

v u f

+ =

where u and v are object and image distances, respectively and f is the

focal length of the mirror. f is (approximately) half the radius of

curvature R. f is negative for concave mirror; f is positive for a convex

mirror.

5. For a prism of the angle A, of refractive index n

placed in a medium

of refractive index n

A D

= =

( )









( )

sin /

where D

is the angle of minimum deviation.

6. For refraction through a spherical interface (from medium 1 to 2 of

refractive index n

and n

, respectively)

2 1 2 1

n n n n

v u R

−

− =

Thin lens formula

1 1 1

v u f

− =

Lens maker’s formula

1 1 1

2 1

1 1 2

n n

n R R

−

(

)

−













and R

are the radii of curvature of the lens surfaces. f is positive

for a converging lens; f is negative for a diverging lens. The power of a

lens P = 1/f.

The SI unit for power of a lens is dioptre (D): 1 D = 1 m

–1

If several thin lenses of focal length f

, f

,.. are in contact, the

effective focal length of their combination, is given by

1 2 3

1 1 1 1

f f f f

= + + +

…

The total power of a combination of several lenses is

P = P

+ P

+ …

7. Dispersion is the splitting of light into its constituent colour.

8. Magnifying power m of a simple microscope is given by m = 1 + (D/f),

where D = 25 cm is the least distance of distinct vision and f is the

focal length of the convex lens. If the image is at infinity, m = D/f. For

a compound microscope, the magnifying power is given by

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m = m

× m

where m

= 1 + (D/f

), is the magnification due to the

eyepiece and m

is the magnification produced by the objective.

Approximately,

o e

L D

f f

= ×

where f

and f

are the focal lengths of the objective and eyepiece,

respectively, and L is the distance between their focal points.

9. Magnifying power m of a telescope is the ratio of the angle

subtended

at the eye by the image to the angle

subtended at the eye by the

object.

= =

where f

and f

are the focal lengths of the objective and eyepiece,

respectively.

POINTS TO PONDER

1. The laws of reflection and refraction are true for all surfaces and

pairs of media at the point of the incidence.

2. The real image of an object placed between f and 2f from a convex lens

can be seen on a screen placed at the image location. If the screen is

removed, is the image still there? This question puzzles many, because

it is difficult to reconcile ourselves with an image suspended in air

without a screen. But the image does exist. Rays from a given point

on the object are converging to an image point in space and diverging

away. The screen simply diffuses these rays, some of which reach our

eye and we see the image. This can be seen by the images formed in

air during a laser show.

3. Image formation needs regular reflection/refraction. In principle, all

rays from a given point should reach the same image point. This is

why you do not see your image by an irregular reflecting object, say

the page of a book.

4. Thick lenses give coloured images due to dispersion. The variety in

colour of objects we see around us is due to the constituent colours

of the light incident on them. A monochromatic light may produce an

entirely different perception about the colours on an object as seen in

white light.

5. For a simple microscope, the angular size of the object equals the

angular size of the image. Yet it offers magnification because we can

keep the small object much closer to the eye than 25 cm and hence

have it subtend a large angle. The image is at 25 cm which we can see.

Without the microscope, you would need to keep the small object at

25 cm which would subtend a very small angle.

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EXERCISES

9.1 A small candle, 2.5 cm in size is placed at 27 cm in front of a concave

mirror of radius of curvature 36 cm. At what distance from the mirror

should a screen be placed in order to obtain a sharp image? Describe

the nature and size of the image. If the candle is moved closer to the

mirror, how would the screen have to be moved?

9.2 A 4.5 cm needle is placed 12 cm away from a convex mirror of focal

length 15 cm. Give the location of the image and the magnification.

Describe what happens as the needle is moved farther from the mirror.

9.3 A tank is filled with water to a height of 12.5 cm. The apparent

depth of a needle lying at the bottom of the tank is measured by a

microscope to be 9.4 cm. What is the refractive index of water? If

water is replaced by a liquid of refractive index 1.63 up to the same

height, by what distance would the microscope have to be moved to

focus on the needle again?

9.4 Figures 9.31(a) and (b) show refraction of a ray in air incident at 60°

with the normal to a glass-air and water-air interface, respectively.

Predict the angle of refraction in glass when the angle of incidence

in water is 45° with the normal to a water-glass interface [Fig. 9.31(c)].

FIGURE 9.31

9.5 A small bulb is placed at the bottom of a tank containing water to a

depth of 80cm. What is the area of the surface of water through

which light from the bulb can emerge out? Refractive index of water

is 1.33. (Consider the bulb to be a point source.)

9.6 A prism is made of glass of unknown refractive index. A parallel

beam of light is incident on a face of the prism. The angle of minimum

deviation is measured to be 40°. What is the refractive index of the

material of the prism? The refracting angle of the prism is 60°. If the

prism is placed in water (refractive index 1.33), predict the new

angle of minimum deviation of a parallel beam of light.

9.7 Double-convex lenses are to be manufactured from a glass of

refractive index 1.55, with both faces of the same radius of

curvature. What is the radius of curvature required if the focal length

is to be 20cm?

9.8 A beam of light converges at a point P. Now a lens is placed in the

path of the convergent beam 12cm from P. At what point does the

beam converge if the lens is (a) a convex lens of focal length 20cm,

and (b) a concave lens of focal length 16cm?

9.9 An object of size 3.0cm is placed 14cm in front of a concave lens of

focal length 21cm. Describe the image produced by the lens. What

happens if the object is moved further away from the lens?

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9.10 What is the focal length of a convex lens of focal length 30cm in

contact with a concave lens of focal length 20cm? Is the system a

converging or a diverging lens? Ignore thickness of the lenses.

9.11 A compound microscope consists of an objective lens of focal length

2.0cm and an eyepiece of focal length 6.25cm separated by a

distance of 15cm. How far from the objective should an object be

placed in order to obtain the final image at (a) the least distance of

distinct vision (25cm), and (b) at infinity? What is the magnifying

power of the microscope in each case?

9.12 A person with a normal near point (25cm) using a compound

microscope with objective of focal length 8.0 mm and an eyepiece of

focal length 2.5cm can bring an object placed at 9.0mm from the

objective in sharp focus. What is the separation between the two

lenses? Calculate the magnifying power of the microscope,

9.13 A small telescope has an objective lens of focal length 144cm and

an eyepiece of focal length 6.0cm. What is the magnifying power of

the telescope? What is the separation between the objective and

the eyepiece?

9.14 (a) A giant refracting telescope at an observatory has an objective

lens of focal length 15m. If an eyepiece of focal length 1.0cm is

used, what is the angular magnification of the telescope?

(b) If this telescope is used to view the moon, what is the diameter

of the image of the moon formed by the objective lens? The

diameter of the moon is 3.48 × 10

m, and the radius of lunar

orbit is 3.8 × 10

9.15 Use the mirror equation to deduce that:

(a) an object placed between f and 2f of a concave mirror produces

a real image beyond 2f.

(b) a convex mirror always produces a virtual image independent

of the location of the object.

diminished in size and is located between the focus and

the pole.

(d) an object placed between the pole and focus of a concave mirror

produces a virtual and enlarged image.

[Note: This exercise helps you deduce algebraically properties of

images that one obtains from explicit ray diagrams.]

9.16 A small pin fixed on a table top is viewed from above from a distance

of 50cm. By what distance would the pin appear to be raised if it is

viewed from the same point through a 15cm thick glass slab held

parallel to the table? Refractive index of glass = 1.5. Does the answer

depend on the location of the slab?

9.17 (a) Figure 9.32 shows a cross-section of a ‘light pipe’ made of a

glass fibre of refractive index 1.68. The outer covering of the

pipe is made of a material of refractive index 1.44. What is the

range of the angles of the incident rays with the axis of the pipe

for which total reflections inside the pipe take place, as shown

in the figure.

FIGURE 9.32

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(b) What is the answer if there is no outer covering of the pipe?

9.18 Answer the following questions:

(a) You have learnt that plane and convex mirrors produce virtual

images of objects. Can they produce real images under some

circumstances? Explain.

(b) A virtual image, we always say, cannot be caught on a screen.

Yet when we ‘see’ a virtual image, we are obviously bringing it

on to the ‘screen’ (i.e., the retina) of our eye. Is there a

contradiction?

the bank of a lake. Would the fisherman look taller or shorter to

the diver than what he actually is?

(d) Does the apparent depth of a tank of water change if viewed

obliquely? If so, does the apparent depth increase or decrease?

(e) The refractive index of diamond is much greater than that of

ordinary glass. Is this fact of some use to a diamond cutter?

9.19 The image of a small electric bulb fixed on the wall of a room is to be

obtained on the opposite wall 3m away by means of a large convex

lens. What is the maximum possible focal length of the lens required

for the purpose?

9.20 A screen is placed 90cm from an object. The image of the object on

the screen is formed by a convex lens at two different locations

separated by 20cm. Determine the focal length of the lens.

9.21 (a) Determine the ‘effective focal length’ of the combination of the

two lenses in Exercise 9.10, if they are placed 8.0cm apart with

their principal axes coincident. Does the answer depend on

which side of the combination a beam of parallel light is incident?

Is the notion of effective focal length of this system useful at all?

(b) An object 1.5 cm in size is placed on the side of the convex lens

in the arrangement (a) above. The distance between the object

and the convex lens is 40cm. Determine the magnification

produced by the two-lens system, and the size of the image.

9.22 At what angle should a ray of light be incident on the face of a prism

of refracting angle 60° so that it just suffers total internal reflection

at the other face? The refractive index of the material of the prism is

1.524.

9.23 A card sheet divided into squares each of size 1 mm

is being viewed

at a distance of 9 cm through a magnifying glass (a converging lens

of focal length 9 cm) held close to the eye.

(a) What is the magnification produced by the lens? How much is

the area of each square in the virtual image?

(b) What is the angular magnification (magnifying power) of the

lens?

Explain.

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9.24 (a) At what distance should the lens be held from the figure in

Exercise 9.29 in order to view the squares distinctly with the

maximum possible magnifying power?

(b) What is the magnification in this case?

Explain.

9.25 What should be the distance between the object in Exercise 9.24

and the magnifying glass if the virtual image of each square in the

figure is to have an area of 6.25 mm

. Would you be able to see the

squares distinctly with your eyes very close to the magnifier?

[Note: Exercises 9.23 to 9.25 will help you clearly understand the

difference between magnification in absolute size and the angular

magnification (or magnifying power) of an instrument.]

9.26 Answer the following questions:

(a) The angle subtended at the eye by an object is equal to the

angle subtended at the eye by the virtual image produced by a

magnifying glass. In what sense then does a magnifying glass

provide angular magnification?

(b) In viewing through a magnifying glass, one usually positions

one’s eyes very close to the lens. Does angular magnification

change if the eye is moved back?

to the focal length of the lens. What then stops us from using a

convex lens of smaller and smaller focal length and achieving

greater and greater magnifying power?

(d) Why must both the objective and the eyepiece of a compound

microscope have short focal lengths?

(e) When viewing through a compound microscope, our eyes should

be positioned not on the eyepiece but a short distance away

from it for best viewing. Why? How much should be that short

distance between the eye and eyepiece?

9.27 An angular magnification (magnifying power) of 30X is desired using

an objective of focal length 1.25cm and an eyepiece of focal length

5cm. How will you set up the compound microscope?

9.28 A small telescope has an objective lens of focal length 140cm and

an eyepiece of focal length 5.0cm. What is the magnifying power of

the telescope for viewing distant objects when

(a) the telescope is in normal adjustment (i.e., when the final image

is at infinity)?

(b) the final image is formed at the least distance of distinct vision

(25cm)?

9.29 (a) For the telescope described in Exercise 9.28 (a), what is the

separation between the objective lens and the eyepiece?

(b) If this telescope is used to view a 100 m tall tower 3 km away,

what is the height of the image of the tower formed by the objective

lens?

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25cm?

9.30 A Cassegrain telescope uses two mirrors as shown in Fig. 9.30. Such

a telescope is built with the mirrors 20mm apart. If the radius of

curvature of the large mirror is 22

0mm and the small mirror is

140mm, where will the final image of an object at infinity be?

9.31 Light incident normally on a plane mirror attached to a galvanometer

coil retraces backwards as shown in Fig. 9.33. A current in the coil

produces a deflection of 3.5

of the mirror. What is the displacement

of the reflected spot of light on a screen placed 1.5 m away?

FIGURE 9.33

9.32 Figure 9.34 shows an equiconvex lens (of refractive index 1.50) in

contact with a liquid layer on top of a plane mirror. A small needle

with its tip on the principal axis is moved along the axis until its

inverted image is found at the position of the needle. The distance of

the needle from the lens is measured to be 45.0cm. The liquid is

removed and the experiment is repeated. The new distance is

measured to be 30.0cm. What is the refractive index of the liquid?

FIGURE 9.34

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Notes

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Physics

350

Notes

2020-21