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Chapter Nine

RAY OPTICS

AND OPTICALINSTRUMENTS

9 . 1 INTRODUCTION

Natu re h as endowed th e hu ma n eye (retina ) with th e sen sitivity to detect

e lectromagnetic waves within a small range of the e lectromagneticspectrum. Electromagnetic radiation belonging to this region of the

sp ectru m (wavelen gth of ab ou t 40 0 n m t o 750 n m) is called light. I t is

ma inly through light an d th e sens e of vision tha t we know an d interpret

the wor ld a roun d u s .

There a re two things t ha t we can int u itively ment ion abou t light from

common experience. First, th at it travels with enormou s s peed an d s econd ,th at it tra vels in a str aigh t lin e. It took s ome time for people to realise th at

th e speed of light is finite and m eas u rab le. Its p resen tly accepted value

in vacuu m is c = 2.99792458 × 108

m s–1

. For man y purp oses, it su ffices

to take c = 3 × 108

m s–1

. The s peed of l ight in vacuu m is the h ighest

speed at ta inable in n atu re .

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

c o n t r a d i c t w h a t w e h a v e l e a r n t i n C h a p t e r 8 , t h a t l i g h t i s a nelectroma gnetic wave of wavelength belongin g to th e visible par t of the

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

wavelen gth of ligh t is very sma ll comp ar ed to th e size of ordina ry objects

th at we en cou n ter com m only (genera lly of th e order of a few cm or larger).In this s itu at ion, a s you wil l learn in Chap ter 10, a l ight wave can becons idered to tra vel from on e poin t to an oth er, along a st raight line join ing

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them. The path is cal led a ra y of l ight , and a bundle of such rayscons t itu tes a beam of light .

In th is cha pter, we consider th e ph enom ena of reflection , refractionand dispersion of light, using the ray picture of light. Using the basiclaws of reflection an d refract ion , we sha ll stu dy th e ima ge forma tion byplan e an d sp h erical reflecting and r efractin g su r faces. We then go on todescr ibe the cons t ruc t ion and working of some impor tant opt ica l

ins t ru ments , inc lud ing the hu man eye .

PARTICLE MODEL OF LIGHT

Newton ’s fun dam enta l contribu tions to ma th ematics, mecha nics, an d gravitation often b lindu s to h is d eep experimenta l and t heoretical stu dy of light. He mad e pion eering contribu tionsin the field of optics. He further developed the corpuscular model of light proposed byDescartes. I t presu mes th at light en ergy is concen trated in tiny particles called corpuscles .He fur ther assumed that corpuscles of l ight were massless e last ic par t ic les . With hisu nd erstan ding of mecha nics, he cou ld come u p with a s imple model of reflect ion a ndrefract ion. I t is a common observation tha t a bal l boun cing from a sm ooth plane s u rfaceobeys th e laws of reflection. When th is is a n elas tic collision, th e ma gnitu de of th e velocityrema ins th e sam e. As th e su rface is s mooth, th ere is n o force acting para llel to the s u rface,so the componen t of mom entu m in th is direction a lso remains the sa me. Only th e comp onentperpendicular to th e su rface, i.e ., th e norm al compon ent of the mom entu m, gets reversedin reflection . Newton argu ed th at s mooth su rfaces like mirrors r eflect th e corpu scles in asimilar man ner .

In order to explain the p hen omena of refract ion, Newton p ostu la ted tha t th e speed of th e corpus cles was great er in water or glass tha n in air. However, later on it was discover edth at th e speed of light is less in wat er or glass t h an in air.

In th e field of optics, Newton – the experimen ter, was greater th an Newton – the th eorist.He himself observed m an y phen omena , which were difficult to u nd erstan d in terms of pa rticle na tu r e of light. For exam ple, the colou rs obs erved du e to a th in film of oil on wat er.Property of pa rtial reflection of light is yet an oth er su ch exa mp le. Everyone wh o ha s lookedinto th e water in a p ond s ees ima ge of the face in it , bu t a lso sees th e bottom of the p ond.Newton argu ed th at some of the corpu scles, which fall on th e water, get reflected an d s omeget tran sm itted. Bu t wha t property cou ld distingu ish th ese two kinds of corpu scles? Newtonha d to postu late some kind of u np redictable, cha nce ph enomen on, which decided wheth eran individua l corpu scle would be reflected or not. In explaining other p hen omen a, h owever,

th e corpu scles were presu med to beh ave as if th ey are identical. Su ch a d ilemm a does n otoccur in th e wave pictu re of light . An incom ing wave can be divided in to two weaker wavesa t th e boun dary be tween a i r and water .

9 . 2 REFLECTION OF LIGHT BY S PHERICAL MIRRORS

We are fam iliar with th e laws of r eflection . The a n gle of reflection (i.e., th e

an gle between reflected ra y and th e norm al to the reflectin g su rface or

th e mirror) equ als th e an gle of incidence (an gle between incident r ay an d

the n ormal). Also th at t he incident ra y, reflected ra y and th e norm al to

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

(Fig. 9.1). These laws a re valid a t ea ch point on an y reflecting su rfacewheth er plane or cu rved. However, we sh all rest rict ou r discu ss ion t o the

sp ecial case of cu rved su rfaces, th at is, sph erical su rfaces. The norm al in

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th is case is to be taken a s n ormal to the tan gentto su rface at th e poin t of incidence. Th at is, th e

norm al is a long the ra dius, th e line joining th e

centr e of cu rvatu re of th e mirror to the p oin t of incidence.

We ha ve already stu died tha t th e geometric

centr e of a s ph erical m irror is ca lled its p ole while

th at of a s ph erical lens is called its optical cent re.

Th e line joinin g the pole an d th e centre of cur vatu reof th e sph er ical mirror is kn own as th e principal

axis . In the case of sph erical lenses , th e principal

axis is th e lin e join ing th e optical centr e with its

pr incipa l focu s a s you will see later.

9 .2 .1 S ig n co n v en t io n

To derive th e relevan t form u lae for reflection by sp h erical m irrors a n d

refraction by sp herical lens es, we mu st first ad opt a s ign convention for

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

c o n v e n t i o n . A c c o r d i n g t o t h i sconvention, all distan ces are meas u red

from t he pole of th e mirror or th e optical

c e n t r e o f t h e l e n s . T h e d i s t a n c e s

measu red in th e same direct ion as th e

inciden t light a re taken as p ositive an dt h o s e m e a s u r e d i n t h e d i r e c t i o n

opposite to the direction of incident

light are tak en as n egative (Fig. 9.2).

The heights measured upwards with

respect to x -axis and normal to theprincipal axis ( x -axis) of the mirror/

len s are t ak en as positive (Fig. 9.2). Th e

h e i g h t s m e a s u r e d d o w n w a r d s a r e

tak en as n egative.

With a comm on accepted convention, it turn s ou t th at a single formu lafor sph erical mirrors an d a s ingle formu la for sph erical lens es can ha nd le

all differen t cas es.

9.2 .2 Focal length of spherical mirrors

Figure 9.3 s hows wha t h app ens when a para llel beam of light is incident

on (a) a con cave mirror, and (b) a con vex mirror. We as su m e th at t h e rays

are paraxial, i.e., they are inciden t a t points close to th e pole P of th e mirror

an d m ake s ma ll an gles with th e 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, th e r eflected ra ys a ppea r to d iverge from a p oin t F

on its p rincipa l axis [Fig. 9.3 (b)]. The point F is called th e principa l focus

of th e mirror. If th e par allel pa ra xial beam of light were inciden t, m ak ing

som e an gle with th e prin cipal axis, th e reflected rays wou ld converge (orapp ear to diverge) from a point in a p lan e th rough F n ormal to the p rincipal

axis. This is called th e focal plane of th e m irror [Fig. 9.3 (c)].

FIGURE 9.1 The incident ray, reflected ray

and the normal to the reflecting surface l ie

in the same plane .

FIGURE 9 .2 The Cartesian Sign Convention.

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Th e distan ce between th e focus F an d th e pole P of th e mirror is called

th e focal length of th e mirror, denoted b y f . We now sh ow tha t f = R / 2 ,

where R is the ra diu s of cu rvatu re of th e mirror. The geometryof reflection of an incident ray is s h own in Fig. 9.4.

Let C be the cen tr e of cu rvatu re of th e mirr or. Consider a

ray pa rallel to the principal axis str iking th e mirror at M. Th en

CM will be perp end icu lar to t he mirror a t M. Let θ be the a ngle

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

principal axis. Then ,∠MCP = θ a n d ∠MFP = 2θ

Now,

ta n θ =MD

CDa n d t a n 2θ =

MD

FD(9.1)

For small θ , which is true for paraxial rays, tan θ ≈

θ ,

t a n 2θ ≈ 2θ. Ther efore, Eq. (9.1 ) gives

MD

F D= 2

MD

CD

or, FD =CD

2(9.2)

Now, for s m all θ , th e point D is very close to th e point P.Therefore, FD = f an d CD = R. Equ ation (9.2) then gives

f = R/ 2 (9 .3 )

9.2 .3 The mirror equation

I f rays emanating f rom a point actually meet a t another point af terreflect ion a nd / or refract ion, tha t point is cal led th e image of th e first

point. The ima ge is real if th e rays actu ally converge to th e 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 i f the ra ys do not actu ally meet but app eart o d i v e r g e f r o m t h e p o i n t w h e n p r o d u c e d

backwards . An image i s thus a poin t - to-poin t

c o r r e s p o n d e n c e w i t h t h e o b j e c t e s t a b l i s h e dth rough reflection an d/ or refraction.

I n p r i n c i p l e , w e c a n t a k e a n y t w o r a y s

eman ating from a point on a n object , t race their

paths, f ind their point of intersection and thus,

obtain th e ima ge of th e point du e to reflection at asphe r i c a l m i r r o r . I n p r a c t i c e , howe ve r , i t i s

convenient to ch oose an y two of th e followin g rays:

(i) The ray from th e point wh ich is pa ral le l to the

principal axis. The reflected ra y goes th rou ghth e focu s of th e mirror.

(ii) Th e r a y p a s s i n g t h r o u g h t h e c en t r e o f

curvatu re of a concave mirror or app ear ing to pa ss th rough i t for a

convex mirror. The reflected ra y simply retr aces th e pa th .

(iii) The ray pa ss ing th rough (or directed towards) the focus of th e concave

mirror or app ear ing to pas s throu gh (or directed towards) the focu sof a con vex m irror. Th e reflected ra y is p ar allel to th e prin cipal axis.

(iv) Th e ra y inciden t a t a n y an gle at th e pole. The reflected r ay follows

laws of reflection.

Figure 9.5 sh ows th e ray diagram cons ider ing th ree rays. It s howsth e ima ge A′B ′ (in th is cas e, real) of an object AB form ed by a con cave

mirror. It does n ot mean th at on ly three rays ema na te from th e point A.An infin ite n u mb er of rays em an ate from a n y source, in all directions .

Thu s, point A′ is ima ge point of A if every ray origina tin g at p oint A an d

fallin g on th e con cave mirror after reflection p as ses t h rou gh th e poin t A′.We now derive th e mirror equ ation or t h e relation b etween th e object

distan ce (u ), image dista nce (v ) an d th e focal length ( f ).From Fig. 9.5, the two right-angled triangles A′B ′F and MPF are

similar. (For pa raxial rays, MP can be cons idered to be a s traight lin e

perp en dicu lar to CP.) Therefore,

B A B F

PM FP

orB A B F

BA FP

(Q PM = AB) (9 .4 )

Since ∠ APB = ∠ A′PB ′, the righ t an gled trian gles A′B ′P an d ABP are

als o similar. Therefore,

B A B P

B A B P

(9.5)

Compa ring Eqs . (9.4) an d (9.5), we get

B P – FPB F B P

FP FP BP

(9.6)

Equ ation (9.6) is a relation in volving ma gnitu de of dista n ces. We nowap ply the sign convention. We note th at light tra vels from t h e object to

th e mirror MPN. Hence th is is ta ken as th e positive direction. To reach

FIGURE 9.5 Ray diagram for image

formation by a concave mirror.

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th e object AB, image A′B′ as well as t h e focus F from t h e pole P, we ha veto tra vel opposite to th e direction of in ciden t light. Hen ce, all th e th ree

will h ave n egative sign s. Thu s,

B′ P = –v , FP = – f , BP = –u

Using th ese in Eq. (9.6 ), we get

– –

–

v f v

f u

–

or–v f v

f u

1 1 1

v u f

(9.7)

This re la t ion is kn own a s th e m irror equ ation .

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

import an t qu an tity to cons ider. We define lin ear magnification (m ) as th e

rat io of th e height of th e image (h ′) to th e h eight of th e object (h ):

m =h

h

(9.8)

h a n d h′ will be ta ken positive or n egative in accordan ce with th e accepted

sign convention. In trian gles A′B ′P an d ABP, we h ave,

B A B P

BA BP

With th e sign convent ion , th is becomes– –h v

h u

–

so tha t

m = –h v

h u

(9.9)

We ha ve der ived h ere t he m ir ror equ a t ion , Eq. (9 .7), an d th em agn ification form u la, Eq. (9.9 ), for th e case of rea l, inverted im age form ed

by a concave mirror. With th e proper u se of sign convention, th ese are,

in fact , valid for a ll th e cas es of reflection b y a sp h erical mirror (con cave

or convex) wheth er th e image formed is real or virtu al. Figure 9 .6 s hows

th e ray diagram s for virtu al im age form ed by a conca ve an d convex mirror.You sh ou ld verify tha t Eqs . (9.7 ) an d (9.9 ) ar e valid for th ese ca ses

as well.

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

P an d F, a nd (b) a convex mirror.

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E XAMP L

E 9 . 3

E XAMP L E

9 .2

E XAMP L E

9 .1

Example 9 .1 Su ppos e tha t th e lower ha l f of th e concave mir ror’sreflecting 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?

Solut ion You ma y think th at th e image will now sh ow only ha lf of th e

object , bu t ta king th e laws of reflection to be tr u e for al l point s of th e

r ema ining part of th e mirror, th e image will be th at of th e 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

mirr or, as s hown in Fig. 9.7. Show by suitable diagram, th e for ma tion

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

distort ion of image depend on the location of the phone with respect

to the mir ror?

FIGURE 9 .7

S o l u t i o n

The r ay diagram for the formation of the ima ge of th e ph one is sh own

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

to principal axis wil l be on the sa me p lane. I t wil l be of the s am e size,

i.e., B ′C = BC. You ca n you rs elf r ealise why the ima ge is dist orted.

Example 9 .3 An object is placed at (i) 10 cm, (ii) 5 cm in front of a

concave mirror of rad ius of cur vatur e 15 cm. Find the posit ion, n atu re,

and magnification of the image in each case.

S o l u t i o n

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

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

– – .

1 1 1

1 0 7 5v

or.

.

1 0 7 5

2 5v

= = – 30 cm

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

Also, magnification m =( 3 0 )

– – – 3

( 1 0 )

v

u

The image is magnified, real and inverted.

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E X A M P L E

9 . 4

E X A M P L E

9 . 3

(i i) The object d is tan ce u = –5 cm. Then from Eq. (9.7),

1 1 1

5 7 .5v

or .

. –

5 7 51 5 cm

7 5 5v

=

This ima ge is form ed at 15 cm behind th e mirror. It is a virtu al ima ge.

Magnification m =1 5

– – 3( 5 )

v

u

The image is magnified, vir tual and erect .

Example 9 .4 Suppose whi le s i t t ing in a parked car , you not ice a

jogger a pproa ching towards you in th e 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.

S o l u t i o n

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

fuv

u f

For convex mirro r, s ince R = 2 m, f = 1 m. Then

for u = –39 m ,

( 3 9 ) 1 39

m3 9 1 4 0v

Since the jogger moves at a constant speed of 5 m s –1, a f ter 1 s the

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

The shift in the posit ion of image in 1 s is

1 3 6 5 1 3 6 03 9 3 4 5 1m

4 0 3 5 1 40 0 1 4 0 0 2 8 0

Therefore, th e average speed of th e image when th e jogger is between

39 m an d 34 m from the m ir ror, is (1 / 280) m s –1

Similarly, i t can be seen that for u = –29 m, –19 m an d –9 m, the

speed with which the image appears to move is

–1 –1 –11 1 1

m s , m s a n d m s ,

1 5 0 6 0 1 0respectively.

Although t he jogger has b een moving with a const an t speed, th e speed

of h is / her image appears to increase su bs tan t ia l ly as he/ sh e moves

closer to the mirr or. This ph enomen on can be noticed by any person

si t t ing in a s ta t ionary car or a bus . In case of moving vehic les , 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 encou nters an other trans parent m ediu m, a pa r t of

light gets reflected ba ck into th e first m edium while the res t ent ers th eother. A ray of light rep resen ts a b eam . Th e direction of propagation of

an obliqu ely in ciden t ray of light th at en ters th e oth er mediu m, ch an ges

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a t t h e i n t e r f a c e o f t h e t w o m e d i a . T h i sphenomenon is cal led refraction of light . Snell

experimentally obtained the following laws of

refraction:

(i) The inc ident ray , the refrac ted ray and the

n o r m a l t o t h e i n t e r f a c e a t t h e p o i n t o f

inciden ce, all lie in th e sa me p lane.

(ii) Th e rat io of th e sine of th e an gle of incidence

to th e sine of angle of refraction is con st an t.

Remem ber th at th e an gles of inciden ce (i ) an d

refra ction (r ) are th e an gles th at th e incident

an d i ts refracted ray ma ke with th e normal,

resp ectively. We ha ve

21

sin

s in

in

r (9.10)

where n 21 is a cons tan t, called the refractive inde x of the s econd medium

with respect to th e first mediu m. Equ at ion (9.1 0) is th e well-kn own Sn ell’s

law of refra ction. We n ote th at n 21 is a cha racteristic of th e pair of media

(an d also depen ds on th e wavelength of light), bu t is in depen den t of th e

an gle of inciden ce.

From Eq. (9.1 0), if n 21 > 1 , r < i , i.e. , th e refracted r ay bend s toward s

the n ormal. In su ch a case mediu m 2 is sa id to be optically den se r (or

d e n s e r , in s hor t ) than medium 1 . On the o ther han d, if n 21 <1 , r > i, t herefracted ray bend s a way from th e norma l. This is th e case when incident

ray in a d enser m ediu m refracts into a rarer medium .

Note: Optical dens ity s hould n ot be confuse d w ith ma ss dens ity ,

w hich is ma ss per unit volum e. It is possible that m as s dens ity o f

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

tw o med ia). For exam ple, turpentine a nd w ater. Mas s d ens ity of

turpentine is les s than that of w ater but i ts optical dens ity is higher.

If n21

is th e refractive index of m edium 2

with resp ect to mediu m 1 an d n12

the refractive

ind ex of medium 1 with respect to mediu m 2 ,

th en i t sh ould be clear tha t

12

21

1n

n (9.11)

It also follows that if n32

is th e refra ctive

ind ex of medium 3 with respect to medium 2

then n32 = n

31 × n12 , where n

31 is th e refra ctiveind ex of medium 3 with resp ect to mediu m 1.

Some elementa ry resu lts ba sed on th e laws

o f r e f r a c t i o n f o l l o w i m m e d i a t e l y . F o r a

rectangular s lab, refract ion ta kes p lace a t twointerfaces (air -glass an d glass -air). It is eas ily seen from Fig. 9.9 th atr

2= i

1, i.e. , the emer gent ra y is p ara llel to th e in ciden t ra y—th ere is n o

FIGURE 9.8 Refra ction a n d r eflection of ligh t.

FIGURE 9.9 Lateral shift of a ray refracted

through a para l le l -s ided s lab .

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9 . 5

deviation, but it does suffer lateral displacement/ sh ift with respect to the inciden t ra y. Another familiar

observat ion is t h at t h e bottom of a tan k filled with

water a pp ears to be r aised (Fig. 9.10 ). For viewin gnear th e normal direction, it can be sh own th at th e

apparent depth, (h1) is real depth (h

2) divided by

th e refra ctive in dex of th e med ium (water ).

Th e refraction of light th rough th e atm osph ere

is resp onsible for man y interesting ph enomen a. Forexamp le, th e su n is visible a little before the act u al

sunrise and unti l a l i t t le af ter the actual sunset

du e to refract ion of l ight th rough the a tmosp here

(Fig. 9.11). By actu al su nr ise we mean th e actu alcrossing of the hor izon b y the s u n. Figu re 9.11sh ows th e actual and a pparen t positions of the su n

with respect to the horizon. The figure is highly

exaggerated to sh ow the effect. Th e refra ctive index

of a ir with respect to vacu u m is 1.00 029. Du e to

this , th e appa rent s hift in th e direct ion of the s u nis by abou t h alf a degree an d th e correspond ing

time difference between actu al su ns et and app arent

su ns et is a bout 2 minu tes (see Example 9.5). The

appa rent flatten ing (oval sha pe) of the s u n at s u ns et

and su nr ise is a l so due to the sam e phenomenon.

FIGURE 9.10 Apparent depth for

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

FIGURE 9.11 Advance sunr ise and delayed sunset due to

a tm ospher i c r e fr ac t ion .

Example 9 .5 The ear th takes 2 4 h to ro ta te once about it s axis . How

m u c h t i m e d o e s t h e s u n t a k e t o s h i f t b y 1 º w h e n v i e w e d f r o m

t h e e a r t h ?

S o l u t i o n

Time taken for 360° shift = 24 hTime tak en for 1° sh ift = 24/ 360 h = 4 min.

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9 . 4 TOTAL INTERNAL REFLECTION

When light t ra vels f rom an optical ly denser medium to a r arer m ediu m

at the interface, it is pa r t ly reflected ba ck into th e sam e medium an d

pa rtly refract ed to th e second m edium . Th is reflection is called the internal

reflection.

When a ra y of light enters from a denser m ediu m to a rar er medium ,

it bend s a way from t h e norm al, for examp le, th e ray AO1

B in Fig. 9.12 .

Th e incident ray AO1

is p ar tially reflected (O1C) an d p artially tran sm itted

(O1B) or refracted , th e an gle of refraction (r ) being larger th an th e an gle of

incidence (i). As th e an gle of inciden ce increas es, s o does t h e an gle of refract ion, t ill for th e ra y AO

3, th e a n gle of refract ion is π / 2 .Th e refracted

ray is ben t so mu ch away from th e norma l th at i t grazes the su rface a t

th e int erface between th e two m edia. Th is is s hown by the ra y AO3 D inFig. 9.1 2. If th e an gle of inciden ce is increas ed s till fu rth er (e.g., th e ra y

AO4 ), refraction is n ot poss ible, an d t h e inciden t ra y is t otally reflected.

THE DROWNING CHILD, LIFEGUARD AND SNELL’S LAW

Cons ider a recta n gular s wimm ing pool PQSR; see figu re h ere. A lifegua rd s ittin g at Gouts ide the pool notices a ch ild drowning at a p oint C. The gua rd wan ts to reach the

child in the shortest possible time. Let SR be the

side of th e pool between G an d C. Should he/ sh e

take a s tra ight l ine p ath GAC between G a nd C or

GBC in which th e path BC in water wou ld be th esh ortest, or some oth er path GXC? The gua rd kn ows

tha t h i s / he r r unn ing spee d v1

on grou nd is h igher

than h is / he r swimm ing speed v2.

Su ppose th e gu ard en ters water at X. Let GX =l1

an d XC =l2. Then the t ime taken to reach from G to

C wou ld be

1 2

1 2

l lt

v v

T o m a k e t h i s t i m e m i n i m u m , o n e h a s t o

different iate it (with resp ect to th e coordinat e of X) and find th e point X when t is aminimu m. On doin g all th is algebra (which we skip here), we find th at th e gu ard s hou ld

enter water a t a point where Snell’s law is s a t isfied. To un dersta n d th is , dra w a

perp en dicu lar LM to side SR at X. Let ∠GXM = i a n d ∠CXL = r . Then it can be seen tha t t

is m inimu m when

1

2

s in

sin

vi

r v

In the case of light v1 / v

2, th e rat io of th e velocity of light in vacu u m t o tha t in th e

med ium , is th e refractive index n of the m ediu m.

In s h ort, wheth er it is a wave or a particle or a hu ma n b ein g, when ever two mediu ms

an d two velocities ar e involved, on e m u st follow Sn ell’s law if one wa n ts to ta ke t h esh ortest time.

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This is ca lled total interna l reflection. Whenlight gets reflected b y a s u rface, n orma lly

some fraction of it gets transmitted. The

reflected ra y, th erefore, is always less int ensetha n th e incident ray, howsoever smooth th e

reflecting su rface ma y be. In t otal in tern al

r e f l e c t i o n , o n t h e o t h e r h a n d , n o

tra ns miss ion of light ta kes place.

Th e an gle of incidence corresp ond ing toan angle of refraction 90º, say ∠AO

3N, is

called th e critical an gle (ic

) for th e given p a ir

of m edia . We see from Sn ell’s law [Eq. (9.1 0)]

that if the relative refractive index is lessthan one then , s ince the maximu m va lueof sin r is unity, there is an upper l imit

to the value of sin i for which the law can be satisfied, that is, i = ic

suc h tha t

s in ic = n 21 (9.12)

For values of i larger than ic, Sn ell’s law of refraction ca n n ot be

sa tisfied, an d h ence n o refraction is poss ible.

Th e refractive ind ex of den ser m edium 2 with respect to ra rer med ium

1 will be n12

= 1/ s in ic. Some typical critical an gles a re list ed 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 d em o ns t r a t i o n f o r t o ta l in t e r na l re f l ec t i o n

All optical phen omen a can be dem ons trat ed very easily with th e u se of a

laser torch or pointer, wh ich is eas ily available nowada ys. Tak e a glass

beak er with clear wa ter in it . Stir th e water a few times with a p iece of soap, s o tha t it becomes a little turb id. Take a las er point er an d sh ine its

beam th rough th e tu rbid water. You wil l find th at the p ath of th e beam

in side the water sh in es brigh tly.

Shine the beam from below the beaker such that i t s tr ikes a t the

u pper water su rface at the other end . Do you fin d th at it un dergoes part ialreflection (which is s een a s a sp ot on th e tab le below) an d p art ial refra ction

[which comes out in th e air an d is s een a s a spot on th e roof; Fig. 9.13(a)]?

Now direct the laser beam from one s ide of the b eaker su ch th at it strikesthe upper surface of water more obliquely [Fig. 9.13(b)]. Adjust the

direction of las er bea m u n til you find t h e an gle for which t h e refraction

TABLE 9 . 1 CRITICAL ANGLE OF SOME TRANSPARENT MEDIA

Subs tanc e m e dium Re frac t ive inde x Crit ic al angle

Wa ter 1.33 48.75°

Crown gla s s 1.52 41.14°

Den s e flin t gla s s 1.62 37.31°

Dia m on d 2.42 24.41°

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above the water s u rface is totally absen t a nd the beam is totally reflectedba ck to water. This is t otal int erna l reflection at its simp lest.

Pou r th is water in a long test tu be an d s hine th e laser l ight from top,

as sh own in Fig. 9.13(c). Adjus t th e direction of th e laser beam su ch t ha tit is tot ally inter n ally reflected every time it strikes th e walls of th e tu be.

Th is is sim ilar to wha t ha ppen s in optical fibres.

Tak e care not to look in to th e las er beam directly and n ot to poin t it

at a n ybody’s face.

9 .4 .1 To ta l in tern a l re f lec t io n in n atu re a n d

its t ec hnological applications

(i) Mirage: On h ot sum mer days, the a ir near th e ground becomes hotter

th an th e air at h igh er levels. Th e refract ive index of air increas es withits dens ity. Hotter air is less d ens e, and ha s sm aller refractive index

tha n th e cooler air. If th e air curren ts are sm all, tha t is, the a ir is s till,

th e optical dens ity at different layers of air in creases with h eight. As a

resu lt, ligh t from a tall object such as a tree, pas ses th rough a m ediu m

whose refractive in dex decreases toward s th e grou n d. Thu s, a ra y of light from s u ch a n object su ccess ively bends away from th e norm al

an d u nd ergoes t otal in tern al reflection, if th e an gle of incidence for

the a ir n ear th e grou nd exceeds th e cr it ical an gle . This is s hown in

Fig. 9.14(b). To a d istan t obs erver, th e light a ppea rs to be com ing

from s omewhere below the groun d. The observer na tu ral ly ass u mes

th at light is b ein g reflected from th e groun d, s ay, by a p ool of watern ear th e tall object. Such inverted ima ges of dista n t tall objects cau se

an optical illu sion to th e observer. Th is ph enom enon is called mirage .

Th is type of mirage is especially common in h ot deserts . Some of you

might ha ve noticed th at while moving in a bu s or a car du r ing a hotsu mm er day, a dista nt p atch of road, especially on a h ighway, appears

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

reach th at s pot . This is a lso du e to mirage.

FIGURE 9.13

Observing total

internal reflection inwater 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 i ts place when the air above the ground is

a t uni form temperature , (b) When the layers of air close to the ground have varying

temperature wi th hot tes t layers near the ground, l ight f rom a dis tant t ree mayundergo total internal reflection, and the apparent image of the tree may create

an i l lusion to the observer that the tree is near a pool of water.

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(ii) Diamond : Diamond s ar e kn own for th eirsp ectacu lar brilliance. Th eir brillian ce

i s m a i n l y d u e t o t h e t o t a l i n t e r n a l

reflection of light ins ide th em. The criticalan gle for diam ond -air in terface (≅ 24.4°)

is very sm all, th erefore once light ent ers

a d iamon d, it is very likely to un dergo

t o t a l i n t e r n a l r e f l e c t i o n i n s i d e i t .

Diam onds foun d in n atu re rarely exhibitth e brillian ce for which th ey are kn own.

It is the technical skill of a diamond

c u t t e r w h i c h m a k e s d i a m o n d s t o

sp ark le so brillian tly. By cu tting th ed i a m o n d s u i t a b l y , m u l t i p l e t o t a li n t e r n a l r e f l e c t i o n s c a n b e m a d e

to occur.

(iii) Prism : Prism s d esign ed to bend light by

90º or by 180º ma ke u se of total intern al

reflection [Fig. 9 .15 (a) an d (b)]. Su ch apr ism is a lso used to inver t images

withou t ch an gin g their size [Fig. 9.1 5(c)].

In th e first two cas es, th e critical angle ic for th e ma terial of th e prism

mu st b e less t ha n 45º . We see from Table 9.1 th at th is is t ru e for both

crown glass an d d ense f lint glass .(iv) Optical fibres: Now-a-days optical fibres are extensively used for

tran sm itting au dio an d video signals th rough long dista nces. Optical

fibres too ma ke u se of th e ph enom enon of total in tern al reflection .

Optical fibres are fab ricated with h igh qu ality compos ite glas s/ qu art z

fibres. Ea ch fibre consist s of a core an d cladd ing. Th e refractive in dexof th e ma terial of th e core is h igh er th an th at of th e clad ding.

When a s igna l in the form of l ight i s

directed at one en d of th e fibre at a s u itable

angle , i t undergoes repeated tota l internal

reflection s a lon g the length of th e fibre an d

finally comes ou t a t th e other end (Fig. 9.16 ).

Since light u n dergoes tota l in tern al reflectionat each sta ge, there is n o appr eciable loss in

th e inten sity of th e ligh t s ign al. Optical fibres

are fab ricated s u ch th at light reflected at on e

side of inn er su rface str ikes th e other a t an

an gle larger th an th e critical angle. Even if th e

fibre is bent, light can easily travel along its

length. Thu s, an optical fibre can be u sed to act a s a n optical pipe.

A bu n dle of optical fibres can be pu t to s everal u ses . Optical fibres

are exten sively u sed for tra n sm itting an d receiving electrical signa ls which

are converted to light b y su itable tran sd u cers. Obvious ly, optical fibres

can a lso be u sed for t ra ns mission of optical s ignals . For exam ple, th esear e u sed a s a ‘light p ipe’ to facilita te visu al exam ina tion of inter n al organ s

like esoph agus , stoma ch an d intestines. You m ight h ave seen a common ly

FIGURE 9.15 Prisms des igned to bend rays by

90º and 180º or to inver t image wi thout changing

its size make use of total internal reflection.

FIGURE 9.16 Light undergoes successive total

internal reflections as i t moves through a n

optical fibre.

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available decorative lamp with fine plastic fibres with their free endsformin g a fou n ta in like st ru ctu re. Th e other en d of th e fibres is fixed over

an electric lamp . Wh en th e lam p is switch ed on, th e ligh t tra vels from th e

bottom of each fibre an d ap pea rs a t th e tip of its free end a s a d ot of light.The fibres in su ch d ecorative lamps are opt ical fibres.

The m ain requ iremen t in fabr icating optical fibres is th at th ere shou ld

be very little ab sorpt ion of light as it tra vels for lon g dista nces inside

th em. Th is has been a chieved by pur ification an d sp ecial prepara tion of

ma ter ia ls su ch a s qu ar tz . In s i lica glass f ibres, i t is p ossible to tran sm itmore th an 95% of th e light over a fibre length of 1 km . (Compa re with

wha t you expect for a block of ordina ry window glas s 1 k m t h ick.)

9 .5 REFRACTION AT SPHERICAL SURFACES

AND BY LENSES

We h ave so far cons idered refraction a t a plan e inter face. We sh all now

cons ider refraction at a s ph erical in terface between two tran sp aren t media.

An inf ini tesima l par t of a s ph er ical su rface can be regarded as plana r

an d th e sam e laws of refract ion can be app lied a t every point on th e

su rface. J u st a s for reflect ion by a s ph er ical mirror, the n orm al a t th e

point of incidence is p erpendicu lar to th e tan gent plane to th e sph er ical

su rface at th at point an d, therefore, pass es thr ough its centre of curvatu re.

We first con sider r efra ction b y a sin gle sp h erical su rface a n d follow it by

thin lenses. A thin lens is a t ra ns parent optical mediu m b oun ded by twosu rfaces; at leas t one of which s h ou ld be sph erical. App lying the formu la

for image forma tion by a single sp herical su rface s u ccessively at th e two

su rfaces of a lens, we sha ll obtain t he lens m aker’s formu la a nd th en th e

lens formu la.

9.5 .1 Refraction at a sphe rical surface

Figu re 9.17 sh ows th e geometr y of form at ion of image I of an object O onth e principal axis of a s ph erical su rface with centr e of cu rvatu re C, an d

radius of curvatu re R . Th e rays are incident from a m edium of refractive

index n1, to a n other of refractive index n

2. As b efore, we take t he a pertu re

(or the lateral size) of the surface to be smallcompared t o other distan ces involved, so tha t sm allan gle app roximation can be m ade. In pa r t icu lar ,

NM will be ta ken to be n early equ al to th e length of

th e perpendicular from th e point N on the principal

axis. We h ave, for s m all an gles,

ta n ∠NOM =MN

OM

ta n ∠NCM =MN

MC

ta n ∠NIM =MN

MI

FIGURE 9.17 Refraction at a spherical

surface separa t ing two media .

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Now, for ΔNOC, i is th e exterior an gle. Th erefore, 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 Sn ell’s law

n1 s in i = n 2 s in r

or for s ma ll an gles

n1i = n

2r

LIGHT SOURCES AND PHOTOMETRY

It is kn own tha t a body above absolut e zero temp eratu re emits electromagn etic rad iation.

The wavelength region in wh ich t he b ody emits th e radiat ion depends on i ts a bsolute

temperatu re . Radiat ion emitted by a h ot body, for exam ple, a t u ngsten filament lam ph avin g tempera tu re 285 0 K are p art ly invisible and mos tly in infrar ed (or hea t) region.

As t he t emp eratu re of th e body in creases rad iation em itted by it is in visible region. The

su n with temp eratu re of abou t 550 0 K emits rad iation whose energy versu s wavelength

graph p eaks a pproxima tely a t 550 nm corresponding to green l ight an d is a lmost in th e

midd le of th e visible region. The en ergy versu s wa velength distr ibu tion grap h for a givenbody peaks a t some wavelength, which is inversely proport ional to the absolute

temperatu re of th at body.

The m easu remen t of light as perceived by h u ma n eye is called photometry . Photom etry

is meas u remen t of a ph ysiological phen omen on, being the st imu lus of light as received

by the hu ma n eye, t ran sm itted by the optic nerves an d an alysed by the brain. The m ainphys ic a l qua n t i t i e s in pho tom e t r y a r e ( i ) t he luminous in tens i ty of the sou r c e ,

(ii) th e lum inous flux or flow of light from th e sou rce, a n d (iii) illuminance of th e su rface.

Th e SI u nit of lum inous intens ity ( I ) is can dela (cd). Th e can dela is th e lu min ous inten sity,

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

540 × 1012

Hz and that has a radiant intensity in that direction of 1/ 683 watt per steradian.If a light sou rce emits one can dela of lu minou s int ens ity int o a s olid a n gle of one s teradian ,

the tota l luminous f lux emitted into that sol id angle is one lumen (lm). A standard

100 watt incadescent l ight bu lb emits app roxima tely 1700 lumen s.In ph otometry, th e only para meter, which can be m easu red d irectly is illuminance . It

is defined as lum inous flu x in cident p er un it area on a s u rface (lm/ m2

or lux ). Most lightmeters m easu re this quan ti ty. The i llu mina nce E , produ ced by a source of lu minou s

intensi ty I , is given by E = I / r 2, where r is th e norm al dista nce of the s u rface from th e

source. A qua ntity nam ed luminance ( L ), is u sed t o chara cterise th e brigh tn ess of emitting

or reflecting flat su rfaces. I ts u n it is cd/ m2

(som etimes called ‘n it’ in ind u st ry) . A good

LCD compu ter monitor has a br ightn ess of abou t 250 n its .

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E XAMP L E

9 . 6

Subst i tu t ing i a n d r from Eqs . (9.1 3) an d (9.14 ), we get

1 2 2 1

O M MI MC

n n n n (9.15)

Here, OM, MI an d MC represen t m agnitu des of dista n ces. Applyin g the

Cartesian sign convent ion,

OM = –u, MI = +v , MC = + R

Su bs titu ting th ese in Eq. (9.15 ), we get

2 1 2 1n n n n

v u R

(9.16)

Equ ation (9.16 ) gives u s a r elation between object an d image distan ce

i n t e r m s o f r e f r a c t i v e i n d e x o f t h e m e d i u m a n d t h e r a d i u s o f curvature of the curved spher ical surface. I t holds for any curvedsph erical su rface.

Example 9 .6 Light f rom a point source in a i r fa l l s on a spher ica l

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

of th e light sou rce from th e glas s s u rface is 100 cm. At wha t posit ion

the image is formed?

S o l u t i o n

We u se th e r elation given by Eq. (9.16). Here

u = – 100 cm, v = ? , R = +20 cm , n1

= 1 , and n2

= 1.5.

We th en h ave

1 .5 1 0.5

1 0 0 2 0v

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 t he geometr y of image forma tion by a dou ble convex

l e n s . T h e i m a g e f o r m a t i o n c a n b e s e e n i n t e r m s o f t w o s t e p s :

( i) The f irs t refract ing surface forms the image I1 of the object O[Fig. 9.1 8(b)]. The ima ge I1

acts as a virtua l object for the s econ d su rface

th at form s t h e ima ge at I [Fig. 9.1 8(c)]. App lying Eq. (9.15 ) to th e first

int erface ABC, we get

1 2 2 1

1 1OB BI BC

n n n n (9.17)

A similar procedu re ap plied to th e second in terface*ADC gives ,

2 1 2 1

1 2DI DI DC

n n n n (9.18)

* Note th at now th e refract ive index of the m edium on th e r ight s ide of ADC is n 1

while on its left it is n2. Fur ther DI

1i s nega t ive as the d i s t an ce is m easu red

agains t the di rect ion of inciden t l ight .

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F or a th in l e ns , B I 1 = DI1 . AddingEqs . (9.1 7) an d (9.18 ), we get

1 12 1

1 2

1 1( )

OB DI BC DC

n nn n

(9.19)

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

OB → ∞ an d DI = f , Eq . (9.1 9) gives

12 1

1 2

1 1( )

BC DC

nn n

f

(9.20)

The point where image of an object

placed at infinity is formed is called the focus F, of th e lens a nd th e dista nce f gives

it s focal len gth . A len s h as two foci, F an d

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

sign convent ion,

BC1 = + R1 ,

DC2

= – R2

So Eq. (9.20 ) can be written a s

22 1 2 1

1 2 1

1 1 11

nn n

f R R n

Q (9.21)

Equa tion (9.21) is k nown a s th e lens

maker ’s formula. I t is useful to design

len ses of desired focal length u sing su rfaces

of su itable rad ii of cu rvatu re . Note that th eformu la is t ru e for a con cave lens also. In

tha t c a se R1is negative, R

2positive and

therefore, f is nega tive.

From Eqs . (9.1 9) an d (9.2 0), we get

1 1 1

OB DI

n n n

f (9.22)

Again, in t he th in lens app roxima tion, B a nd D are both c lose to theoptical cen tre of the lens . Applyin g th e s ign convention ,

BO = – u, DI = +v, we get

1 1 1

v u f (9.23)

Equ ation (9.23) is th e fam iliar thin lens form ula . Thou gh we d erived

it for a real image formed by a convex lens , th e formu la is valid for both

convex as well as concave lens es an d for both rea l an d virtu al ima ges.

It is worth men tion ing tha t th e two foci, F and F ′, of a d ou ble convexor con cave len s a re equidistan t from th e optical cen tre. Th e focus on th e

side of th e (origina l) sou rce of ligh t is ca lled t h e first focal point , wh ereas

the oth er is cal led th e s econd focal point .To find t h e ima ge of an object by a lens , we can , in p rinciple, tak e an y

two rays ema na ting from a p oint on an object ; t race their path s u sing

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

image formed by a double convex lens,

(b) Refraction at the f irst spherical surface and

(c) Refraction at the second spherical surface.

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E XAMP L E

9 .7

th e laws of refraction a n d find th e poin t whereth e refracted rays meet (or ap pear to meet). In

pra ctice, however, it is convenient t o choose an y

two of th e followin g ra ys:(i) A ray eman ating from th e object paral le l to

th e principal axis of the lens after refraction

passes through the second pr incipal focus

F′ (in a con vex lens ) or a pp ear s t o diverge (in

a con cave lens) from th e first p rincipa l focus F.(ii) A ray of light, pa ss ing throu gh th e optical

cent re of the lens , emerges wi thout any

deviation after refraction .

(iii) A ra y of l ight p as sin g th rou gh t h e firs tp r i n c i p a l f o c u s ( f o r a c o n v e x l e n s ) o rap pear ing to meet at it (for a con cave lens )

emerges parallel to the principal axis after

refraction.

Figures 9.19(a) an d (b) illus tra te th ese ru les

for a convex an d a conca ve lens , respectively.Yo u s h o u l d p r a c t i c e d r a w i n g s i m i la r r a y

diagra m s for differen t pos itions of th e object with

respect to th e lens an d a lso ver ify tha t th e lens

form u la, Eq. (9.2 3), h olds good for all cas es.

Here again it mu st be rememb ered that eachpoint on a n object gives ou t infinite n u mb er of

rays. All th ese rays will pas s t h rough th e sa me ima ge point after refraction

at th e lens.

Magnification (m ) produ ced by a lens is defined, like th at for a m irror,

as th e rat io of th e size of th e ima ge to th at of the object. Proceeding in th esa me way as for sph erical mirrors, it is ea sily seen th at for a lens

m =h

h

=

v

u(9.24)

Wh en we ap ply the sign convention, we see th at, for erect (an d virtu al)

image formed by a con vex or concave lens , m is p ositive, wh ile for a n

inverted (an d rea l) ima ge, m is n egative.

E x a m p l e 9 . 7 A m agic i an dur ing a show m akes a g l a s s l ens w i th

n = 1.47 disa ppea r in a tr ough of l iqu id. Wha t is th e refractive index

of the l iquid? Could the l iquid be water?

S o l u t i o n

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

make the lens d isappear . This means n1

= n2 .

. This gives 1/ f =0 o r

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 m eas u re of th e convergence or divergence, which a

len s introdu ces in th e light falling on it . Clearly, a lens of sh orter focal

FIGURE 9 . 1 9 Tracing rays through (a)

convex lens (b) conca ve lens .

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E X A M P L E

9 . 8

length b ends th e in cident light m ore, while converging itin case of a convex lens and diverging it in case of a

concave lens. The pow er P of a lens is defined as the

tan gen t of the a n gle by which it con verges or diverges abeam of light falling a t u nit dista nt from t he optical centre

(Fig. 9. 20 ).

1tan ; if 1 ta n

hh

f f or

1

f f o r s m a l l

value of δ . Thu s ,

P =1

f (9.25)

Th e SI u n it for power of a lens is dioptr e (D): 1D = 1m–1

. Th e power of

a len s of focal len gth of 1 m etre is on e dioptre. Power of a len s is p ositivefor a con vergin g lens an d n egative for a d ivergin g lens . Th u s, wh en a n

optician pres cribes a corrective lens of power + 2.5 D, the r equired lens is

a con vex lens of focal len gth + 40 cm . A lens of power of – 4.0 D m ean s a

conca ve lens of focal len gth – 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. I ts 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 wat er? (Refra ctive ind ex of air-water = 1.33 , refra ctive

index for air-glass = 1.5.)

S o l u t i o n

(i) Pow er = +2 d iop t r e .

(i i) Here, we have f = +12 cm, R1

= +10 cm, R2

= –15 cm.

Refractive index of air is taken as unity.

We u se th e lens formu la of Eq. (9.22). The sign convention ha s t o

be applied for f , R1

an d R2.

Subst i tu t ing the values , we have

1 1 1( 1)

1 2 1 0 1 5n

This gives n = 1.5.

(iii) For a glas s lens in a ir, n2

= 1.5, n1

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

formula gives

1 2

1 1 10.5

2 0 R R

For the same glass lens in water, n2

= 1.5, n1

= 1.33. Therefore,

1 2

1 .3 3 1 1(1 .5 1 .3 3 )

f R R

(9.26)

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

9 .5 .4 Combinat ion of th in lens es in contact

Cons ider two len ses A an d B of focal len gth f 1 a n d f

2 placed in cont act

with each other. Let th e object be placed at a poin t O beyon d th e focus of

FIGURE 9.20 Power of a lens.

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th e first len s A (Fig. 9.2 1). Th e first len s p rodu cesan image a t I

1. Since image I

1is real, it s erves a s a

virtu al object for th e second lens B, produ cin g the

fina l im age at I. It m u st , however, be born e in m indth at forma tion of ima ge by the first lens is p resu med

only to facilita te deter min at ion of th e position of th e

final ima ge. In fact, t he direction of rays emerging

from th e first lens gets m odified in a ccordan ce with

th e an gle at which th ey strike the second lens. Sinceth e lenses are th in, we assu me th e optical centres of the lenses to be

coin ciden t. Let th is centra l point be den oted by P.

For th e image formed by th e first lens A, we get

1 1

1 1 1

v u f (9.27)

For th e ima ge formed by th e second len s B, we get

1 2

1 1 1

v v f (9.28)

Add ing Eqs . (9.2 7) an d (9.2 8), 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 lengt h f , we have1 1 1

v u f

so th at we get

1 2

1 1 1

f f f (9.30)

Th e derivation is valid for an y nu mb er of th in lenses in conta ct. If severa l thin lens es of focal len gth f

1, f

2, f

3,... are in con ta ct, th e effective

focal length of th eir comb inat ion is given by

1 2 3

1 1 1 1

f f f f … (9.31)

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

P = P1

+ P2

+ P3

+ … (9.32 )

where P is th e net power of the lens combina tion. Note tha t th e su m in

Eq. (9.32) is an algebraic su m of in dividua l powers, s o some of th e term s

on t h e righ t s ide ma y be positive (for convex lens es) an d s ome n egative

(for conca ve lens es). Comb inat ion of len ses h elps to obta in d ivergin g orconverging len ses of desired ma gnification. It also en h an ces sh arp n ess

of th e ima ge. Since th e ima ge formed by th e first lens b ecomes t h e object

for th e second , Eq. (9.25 ) implies t h at th e total m agnification m of th e

combin ation is a pr odu ct of ma gnification (m1

, m2

, m3

,...) of in dividu al

lenses

m = m1

m2 m

3... (9 .33)

FIGURE 9.21 Image formation by a

combination of two thin lenses in contact .

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E X A M P L E 9 . 9

Su ch a system of combination of lens es is common ly u sed in designinglenses for cameras , microscopes, telescopes a nd other optical ins tru men ts.

E x a m p l e 9 . 9 Find the pos i t i on o f t he im age fo rm ed by the l ens

combination given in the Fig. 9.22.

FIGURE 9 .2 2

Solut ion Image formed by the f irst lens

1 1 1

1 1 1

v u f

1

1 1 1

3 0 1 0v

or v1

= 15 cm

The ima ge formed b y the f irst lens serves as the object for th e second .This is a t a distan ce of (15 – 5) cm = 10 cm to th e r ight of th e second

lens. Though the image is real , i t serves as a vir tual object for the

second lens , which means that the rays appear to come f rom i t for

the second lens .

2

1 1 1

10 10v

or v2

= ∞

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

second lens . This ac ts as an object for the th i rd lens .

3 3 3

1 1 1

v u f

or3

1 1 1

3 0v

or v3

= 30 cm

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

9 . 6 REFRACTION THROUGH A PRISM

Figure 9.23 s hows th e pas sa ge of light th rou gh a trian gular prism ABC.Th e an gles of incidence an d r efraction a t th e first face AB are i a n d r

1,

while the a n gle of inciden ce (from glas s t o air) at th e secon d face AC is r 2

and the angle of refraction or emergence e . The angle between theemergent r ay RS and th e direction of th e in ciden t ray PQ is called th e

an gle of d eviation , δ .

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In th e qu ad rilat eral AQNR, two of th e an gles(a t the ver t ices Q and R) a re r ight angles .

Therefore, the sum of the other angles of the

qua dri la tera l is 1 80º .

∠ A + ∠QNR = 180 º

From t h e trian gle QNR,

r 1 + r

2+∠QNR = 180 º

Compa ring thes e two equ ations , we get

r 1 + r

2 = A (9.34)

The tota l deviation δ is the s u m of deviations

at t he two faces,δ = (i – r

1 ) + (e – r

2 )

tha t is ,

δ = i + e – A (9.35)

Th u s, th e an gle of deviat ion depen ds on t h e an gle of incidence. A plot

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

Fig. 9.24. You can see th at, in general, an y given valu e of δ , except for

i = e, correspond s to two values i an d hen ce of e . Th is, in fact, is expected

from th e symm etry of i a n d e in Eq. (9.35 ), i.e., δ remains the sam e if ia n d e are interch an ged. Ph ysically, th is is related

to the fact th at th e path of ray in Fig. 9.23 can be

t r a c e d ba c k , r e su l t ing in the sa m e a ng le o f deviat ion. At the minimum deviat ion D

m , the

refracted ra y ins ide th e pr ism becomes para lle lto its ba se. We have

δ = Dm

, i = e which imp lies r 1

= r 2.

Equ at ion (9.3 4) gives

2r = A or r =2

A(9.36)

In th e sa m e way, Eq. (9.3 5) gives

Dm = 2 i – A , or i = ( A + Dm )/ 2 (9 .37 )

Th e refra ctive index of th e prism is

22 1

1

sin [( )/ 2]

sin [ / 2]m

A Dnn

n A

(9.38)

The a ng le s A a n d Dm

c a n b e m e a s u r e d

experimen tally. Equ ation (9.38 ) th u s provides a

meth od of determ ining refractive index of th e m aterial of th e prism .

For a sm all an gle prism, i.e. , a thin p rism, Dm is also very sm all, an dwe get

2 1

/ 2sin [( )/ 2]

sin [ / 2 ] / 2

mm A D A D

n A A

;

Dm

= (n21

–1) A

It implies th at , th in pr isms do n ot deviate l ight m u ch.

FIGURE 9.23 A ray of l ight passing through

a t r iangular g lass pr ism.

FIGURE 9.24 Plot of angle of deviation (δ )

versus angle of incidence ( i) for a

t r i angu la r p r i sm .

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9 . 7 DISPERSION BY A PRISM

It h as been kn own for a long t ime tha t when a n arrow beam of su nlight ,usually called white light, is incident on a glass prism, the emergent

light is seen to be consisting of several colours. There is actually a

continuous variation of colour, but broadly, the different component

c o lou r s tha t a ppe a r in s e que nc e a r e :

v iolet, indigo, b lue, green, y ellow, orangea n d re d ( g i v e n b y t h e a c r o n y m

VIBGYOR) . The red l ight bends the

leas t, while th e violet light b end s th e most

(Fig. 9 .25 ).

Th e ph enom enon of sp litting of ligh tinto i ts component colou rs is kn own a s

d i s p e r s i o n . T h e p a t t e r n o f c o l o u r

compon ent s of light is called th e spectru m

of ligh t. Th e word spectrum is n ow u sed

i n a m u c h m o r e g e n e r a l s e n s e : w ed i s c u s s e d i n C h a p t e r 8 t h e e l e c t r o -

ma gnetic spectru m over the large range

of wavelengths, f rom γ - r a ys to r a d io

waves, of which the spectrum of light

(visible spectru m) is only a s ma ll pa rt.

Thou gh th e reason for appearan ce of spectrum is n ow common kn owledge, it was a m atter of mu ch d ebate in

th e h istory of ph ysics. Does th e prism itself create colour in some way or

does it only separa te th e colou rs a lready presen t in white ligh t?

In a class ic experimen t k nown for its s implicity bu t great sign ifican ce,

Isaa c Newton s ettled th e issu e once for all. He pu t a noth er similar p rism,

bu t in an inver ted posit ion, a nd le t the em ergent b eam from the f irs t

prism fall on th e second prism (Fig. 9.26). Th e resu ltin g emergent bea m

was fou n d to be wh ite ligh t. Th e explan ation was clear— the first pr ism

sp lits t h e white ligh t into its compon ent colou rs, wh ile the inverted prism

recombin es th em to give white ligh t. Th u s, wh ite

light itself consists of light of different colours,

which are sepa rated by the pr ism .It mu st b e un derstood here tha t a ra y of light ,

as def ined mathematical ly, does not exist . An

actu al ray is really a b eam of ma n y rays of light.

Each ray splits into componen t colours when i t

enters th e glass pr ism. When those coloured ra ys

come ou t on th e other s ide , th ey again p rodu ce a

white beam .

We now kn ow tha t colour is as sociated with

wavelength of light. In t h e visible spectru m, red

light is at t h e long wavelen gth en d (~700 n m ) while

the violet light is at the short wavelength end(~ 400 n m ). Disp ersion ta kes p lace becau se th e refract ive in dex of m edium

for differen t wavelength s (colou rs ) is differen t. For exam ple, th e ben ding

FIGURE 9.25 Dispersion of sunlight or white light

on passing through a glass prism. The relat ive

deviation of different colours shown is highly

exaggera t ed .

FIGURE 9.26 Schematic diagram of

Newton’s classic experimen t on

dispersion of white light.

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of red component of white light is least while it is most for the violet.Equ ivalently, red ligh t tra vels faster th an violet light in a glass prism .

Table 9.2 gives th e refractive in dices for differen t wa velen gth for crown

glass an d fl int glass . Thick lenses cou ld be as su med a s m ade of ma nypr i sm s , t he r e f o r e , t h i c k l e nse s show chromatic aberration d u e t o

disp ers ion of ligh t.

TABLE 9 .2 REFRACTIVE INDICES FOR DIFFERENT WAVELENGTHS

Colour Wave le n gt h (n m ) Cro wn glas s Flin t glas s

Violet 3 96 .9 1 .533 1 .6 63

Blu e 4 86 .1 1 .5 23 1 .6 39

Yellow 5 89 .3 1 .517 1 .6 27

Red 6 56 .3 1 .515 1 .6 22

The variation of refractive index with wavelength may be more

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

speed of l ight is independent of wavelength. Thus, vacuum (or a ir

ap proximately) is a n on-disp ersive mediu m in wh ich a ll colou rs t ravelwith th e sam e speed. Th is also follows from th e fact th at s u n light rea ches

u s in th e form of white l ight an d n ot as i ts component s . On th e other

ha nd , glass is a d ispers ive medium .

9 . 8 SOME NATURAL PHENOMENA DUE TO SUNLIGHT

Th e in terplay of light with th ings ar oun d u s gives rise to several beau tifu l

ph enomen a. The s pectacle of colour t ha t we see aroun d u s a ll the time is

poss ible only du e to su n ligh t. The b lue of th e sky, white clou ds , the red-

hue a t sunr ise and sunset , the ra inbow, the br i l l iant colours of some

pear ls, sh ells, and win gs of birds, a re ju st a few of th e na tu ral won ders

we are u sed t o. We describe som e of th em h ere from t h e point of view

of ph ysics.

9.8 .1 The rainbowThe rainbow is an example of the dispersion of sunlight by the water

drops in th e a tm osph ere . This is a p hen omenon du e to comb ined effect

of dispersion, refraction and reflection of sunlight by spherical water

droplets of rain. The conditions for observing a ra inbow are th at th e su n

sh ould be sh ining in one pa rt of th e sky (sa y near western horizon) while

i t is ra ining in the opposite par t of the sky (say eastern hor izon) .

An obs erver can th erefore see a rainbow only when h is back is towards

the sun .

I n o r d e r t o u n d e r s t a n d t h e f o r m a t i o n o f r a i n b o w s , c o n s i d e r

Fig. (9.27(a). Sunlight is f irst 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 b ent th e most . Next, these com ponen t ra ys str ike

F or m a t i on of r a i n b o w s

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r ai n b ow s

h t t p: / / www. a t o p t i c s . c o. u

k / b ow s .h t m

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th e in n er su rface of th e water drop a n d get int erna lly reflected if th e an gle

between th e refracted ra y and n ormal to the drop su rface is greater th en

th e critical an gle (48 º, in th is cas e). Th e reflected light is refra cted a gain

as it comes ou t of th e drop as sh own in the f igure . It is foun d th at th e

violet light em erges at a n a ngle of 40º related to th e in coming su n lightan d red ligh t emerges a t an an gle of 42 º. For other colou rs, a n gles lie in

between th ese two values .

FIGURE 9.27 Rainbow: (a) The sun rays incident on a water drop get refracted twice

and reflected internally by a drop; (b) Enlarge view of internal reflection and

refraction of a ray of l ight inside a drop form primary rainbow; and

(c) secondary rainbow is formed by rays

undergoing internal reflection twice

ins ide the drop.

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Figure 9 .27(b) explains th e for ma tion of prima ry rainb ow. We seeth at red light from d rop 1 a n d violet light from d rop 2 rea ch th e observers

eye. The violet from drop 1 a n d red light from dr op 2 a re directed a t level

above or below the observer. Thus the observer sees a rainbow withred colour on the top and viole t on the bottom. Thus, the pr imary

rain bow is a resu lt of th ree-step p rocess, th at is, refraction, reflection

an d refraction.

When light rays u nd ergoes tw o int erna l reflections inside a raindrop,

ins tead of on e as in t he pr imary rainbow, a secondary rainb ow is formedas sh own in Fig. 9.27 (c). It is du e to fou r-step process . Th e in ten sity of

light is redu ced at th e secon d reflection a n d h ence the second ary rainbow

is fa inter th an th e pr imary ra inbow. Furth er, th e order of the colours is

reversed in it a s is clear from Fig. 9.27(c).

9.8 .2 Scatt ering of l ight

As s u nlight t ra vels th rough the ear t h’s a tmosp here , it gets scattered

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

wavelengths is scattered much more than light of longer wavelengths.

(Th e am oun t of sca ttering is in versely proportiona l to th e fou rth powerof th e wavelength . Th is is k n own as Rayleigh s catter ing). Hence, the b luish

colour predominates in a c lear sky, s ince blue has a shor ter wave-

length than red and is scat tered much more strongly. In fact , viole t

ge ts sca t te red even more than b lue , having a shor te r wave length .

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

Large particles like dust and water

d r o p l e t s p r e s e n t i n t h e a t m o s p h e r e

beh ave different ly. The r elevan t qu an tity

h ere is th e relative size of th e wavelen gthof light λ , an d t h e sca tterer (of typical size,

say , a ). For a << λ , one has Rayleigh

scat tering wh ich is proportional to 1/ λ 4.

For a >> λ , i.e., large scattering objects

(for examp le, raind rops , large du st or icepa rticles) this is n ot tru e; all wavelen gths

are sca ttered n early equ ally. Thu s, cloud swhich h ave droplets of water with a >> λ

ar e gen erally white.At su ns et or sun r ise , the su n’s rays

ha ve to pas s t hrou gh a larger dista nce in th e a tm osph ere (Fig. 9.28).

Most of th e blue an d oth er sh orter wavelength s a re removed by scatt ering.The least scattered light reaching our eyes, therefore, the sun looks

reddish. This explains th e reddish appeara nce of the s u n an d fu ll moon

near the h or izon.

9 . 9 OPTICAL INSTRUMENTS

A n u mb er of optical devices an d instr u men ts h ave been des igned u tilisingref lect ing and refract ing proper t ies of mirrors , lenses and pr isms.

Periscope, kaleidoscope, binocu lars, telescopes, microscopes are s ome

FIGURE 9.28 Sunl ight t ravels through a longer

d i s t ance in the a tm osphere a t sunse t and sunr i se .

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exam ples of optical devices an d instru men ts th at ar e in comm on u se.Our eye is, of course, on e of th e most importan t optical device the n atu re

ha s en dowed u s with. Star t ing with the eye, we then go on to descr ibe

th e principles of working of th e microscope an d t he t elescope.

9 .9 .1 Th e ey e

Figure 9.29 (a) shows the eye. Light enters the eye through a curved

front su rface, the cornea. I t passes throu gh the pu pil which is th e centra l

hole in th e iris. Th e size of th e pu pil can cha nge u n der contr ol of mu scles.

The light is fu rth er focus sed b y the eye lens on th e retin a. The retina is a

film of nerve fibres covering th e cu rved ba ck s u rface of th e eye. The retin a

c on ta ins r ods a nd c one s wh ic h se nse l i gh t i n t e ns i ty a nd c o lou r ,

resp ectively, an d tra n sm it electrical signa ls via the optic nerve to th e brain

which finally processes this information. The shape (curvature) and

th erefore the focal length of th e lens can be m odified s omewha t by th e

ciliary m u scles. For example, when th e mu scle is r elaxed, th e focal length

is ab out 2 .5 cm an d objects a t inf ini ty are in sh arp focus on th e re t ina .

When the object is brou ght c loser to the eye, in order to mainta in the

same image-lens distance (≅ 2.5 cm), the focal length of the eye lens

becomes s h orter by th e action of th e ciliary mu scles. Th is property of th e

eye is ca lled accommodation . If th e object is t oo close to th e eye, th e len s

cann ot cu rve enough to focus th e ima ge on to the re t ina , and the image

is b lurred. The closest dista n ce for which t he lens ca n focu s light on t he

retina is called the least distance of distinct vision , or the near point .The s tan dard value for norma l vision is taken as 25 cm . (Often th e near

point is given t he s ymbol D.) Th is distan ce in creas es with age, becau se

of the decreasing effectiveness of the ciliary muscle and the loss of

flexibility of th e lens . Th e nea r point m ay be as close as ab ou t 7 t o 8 cm

in a child ten years of age, and may increase to as m u ch as 200 cm at 60

years of age. Thu s, if an elderly person tries to read a book at a bou t 25 cm

from t h e eye, th e image ap pears blur red. This cond ition (defect of th e eye)

is ca lled presbyopia. It is corrected by u sin g a convergin g len s for rea ding.

Thus, our eyes are marvellous organs that have the capabil i ty to

interpret incoming electromagn etic waves as images th rough a comp lex

process. These are our greatest ass ets an d we mu st tak e proper care toprotect th em. Ima gin e th e world with ou t a pa ir of fu nct ion al eyes. Yet

m an y am ongst u s bra vely face th is ch allenge by effectively overcoming

th eir limitat ion s t o lead a n orma l life. Th ey deserve ou r a pp reciation for

th eir coura ge an d conviction .

In s pite of all preca u tions a n d pr oactive action, ou r eyes m ay develop

som e defects du e to various reason s. We sha ll restrict our discu ss ion to

som e comm on opt ical defects of the eye. For exam ple, the light from a

dista n t object ar riving at th e eye-lens m ay get converged a t a p oin t in

fron t of the retina . This type of defect is ca lled nearsightedness or myopia .

This m eans tha t th e eye is p roducing too mu ch convergence in th e incident

beam . To compen sa te this, we interpose a conca ve lens between th e eyean d th e object, with th e divergin g effect desired to get th e image focus sed

on th e retin a [Fig. 9.29 (b)].

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9 . 1 2

E X A M P L E

9 . 1 1

Example 9 .11

(a ) The far point of a m yopic person is 80 cm in front of th e eye. Wha t

is the power of the lens required to enable him to see very distant

objects clearly?

(b) In what way does the correct ive lens help the a bove person? Does

the lens magnify very distant objects? Explain carefully.

(c) The a bove person prefers to remove his spectacles whi le reading

a book. Explain why?

S o l u t i o n

(a) Solving as in th e previous examp le , we find tha t the person sh ould

u se a conca ve len s of focal length = – 80 cm, i.e., of power = – 1.25

d iop t r e s .

(b) No. The conca ve lens , in fact , redu ces th e size of th e object , bu tthe angle subtended by the dis tant object a t the eye i s the same

as the angle subtended by the image (at the far point) at the eye.

The eye is able to see distant objects not because the corrective

lens magnifies the object , but because i t brings the object ( i .e. , i t

produces vir tual image of the object) at the far point of the eye

which then can be focussed by the eye- lens on the re t ina .

(c ) The m yopic pe r son m ay have a norm al nea r po in t , i .e . , abou t

25 cm (or even less) . In order to read a book with the spectacles,

s u c h a p e r s o n m u s t k e e p t h e b o o k a t a d i s t a n c e g r e a t e r t h a n

25 cm s o tha t the ima ge of th e book by th e conca ve lens is produ ced

not closer than 25 cm. The angular size of the book (or i ts image)

a t t he g rea t e r d i s t ance i s ev iden t ly l e s s t han the angu la r s i ze

when the book is p laced a t 25 cm and no spectacles are needed .

Hence, the person prefers to remove the spectacles while reading.

Example 9 .1 2 (a) The n ear p oint of a h ypermetropic person is 75 cm

from the eye. What is the power of the lens required to enable the

person to read clearly a b ook held at 2 5 cm from th e eye? (b) In wh at

way does the corrective lens help the above person? Does the lens

magnify objects held near the eye? (c) The above person prefers to

remove the spectacles while looking at the sky. Explain why?

S o l u t i o n

(a ) u = – 25 cm, v = – 75 cm

1 / f = 1 / 25 – 1 / 75 , i. e. , f = 37.5 cm.

The correct ive lens needs to have a converging power of +2.67

d iop t r e s .

(b) The corrective lens produ ces a vir tu al ima ge (at 7 5 cm ) of an object

a t 25 cm. The a ngular s ize of th is image is th e sam e as tha t of the

object . In th is sens e the lens d oes not m agnify the object bu t merely

brings the object to the near point of the hypermetric eye, which

then gets focussed on the re t ina . However , t he angu la r s i ze i s

g rea t e r t han tha t o f t he sam e ob jec t a t t he nea r po in t (75 cm )

viewed without the spectacles.

(c) A hyperm etropic eye ma y have norma l far point i .e . , i t ma y have

enough converging power to focus parallel rays from infinity on

th e ret ina of th e sh ortened eyeball . Wearing sp ectacles of converging

lens es (us ed for nea r vision) will am oun t to more converging power

tha n needed for para l le l rays . Hence the person p refers n ot to use

the spectacles for far objects.

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9 .9 .2 The mic rosc opeA sim ple ma gnifier or m icroscope is a convergin g len s of sm all focal len gth

(Fig. 9.30). In ord er to us e su ch a len s a s a microscope, the lens is h eldnea r th e object, one focal length away or less, a nd

th e eye is p ositioned close to th e lens on th e other

side. The idea is to get an erect, magnified and

virtu al ima ge of th e object at a dista n ce so tha t it

can be viewed comforta bly, i.e., at 2 5 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 th an th e focal length of th e lens , th e

image is virtu al an d closer th an infinity. Althou ghth e closest comfortab le dista n ce for viewin g theimage is when i t is a t the near point (distance

D ≅ 25 cm), i t causes some stra in on the eye.

Therefore, the image formed at infinity is often

cons idered m ost s u itable for viewin g by the relaxed

eye. We sh ow both cas es, th e first in F ig. 9.30 (a),an d th e second in Fig. 9.30 (b) an d (c).

The linear magnification m , for the image

formed at th e near point D, by a simple microscope

can be obtained by u sing the re la t ion

1 1– 1 –v vm vu v f f

Now according to our sign convention, v is

negative , and is equal in ma gnitude to D. T h u s ,

th e ma gnification is

1 D

m f

(9.39)

Since D is about 25 cm , to have a ma gnification of

s ix, one needs a convex lens of focal length,

f = 5 cm.Note tha t m = h ′ / h where h is th e size of th e

object an d h ′ th e size of th e im age. Th is is also th er a t i o o f t h e a n g l e s u b t e n d e d b y t h e i m a g eto tha t su btend ed by the object , if placed a t D forcomfortab le viewing. (Note th at th is is n ot th e an gleactu ally su btend ed by the object at the eye, which

is h / u .) What a s ingle- lens simple magnif ierach ieves is th at it allows the object to be brou ght closer to th e eye tha n D.

We will n ow find th e ma gnification wh en t h e image is at infin ity. Inth is case we will ha ve to obtained t h e angular magnification. Supposeth e object has a h eight h. The ma ximu m a ngle it can s u btend, and be

clearly visible (with ou t a len s), is when it is a t th e nea r p oint , i.e., a dista n ce D. The a ngle su btend ed is then given b y

ta no

h

D ≈ θ

o(9.40)

FIGURE 9.30 A simple microscope; (a) the

magnifying lens is located such that the

image is at th e near point , (b) th e angle

su btan ded by the object , is the sam e as

tha t a t the near point , an d (c) the object

near the focal point of the lens; the image

is far off but closer than infinity.

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We now find the a ngle su btend ed at th e eye by the image when theobject is a t u. From th e re la t ions

h v

mh u

we ha ve the an gle su btend ed by the ima ge

ta ni

h h v h

v v u u ≈θ . The a n gle su bten ded by th e object, when it

is a t u = –f .

i

h

f (9.41)

as is clear from Fig. 9.2 9(c). Th e an gular m agn ification is, th erefore

i

o

Dm

f

(9.42)

Th is is one less th an th e magnification when th e ima ge is at th e near

point, Eq. (9.39), bu t t he viewin g is more com fortable an d t he difference

in ma gnificat ion is u su ally sma ll. In su bsequ ent discuss ions of optical

ins tru men ts (microscope and telescope) we shall as su me th e ima ge to be

at infin ity.

A simp le microscope ha s a limited maximu m m agn ification (≤ 9) for

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

lenses, one compounding the effect of the other. This is known as acompound microscope . A sch ema tic diagram of

a compou nd microscope is s hown in Fig. 9.31.

Th e lens nea rest th e object, called th e objective,

forms a real, inverted, m agn ified ima ge of th e

object. Th is serves as th e object for th e secondlens, th e eyepiece , which fun ctions essen tially

like a s imple micr oscope or ma gnifier, produ ces

th e final ima ge, which is enlarged an d virtu al.

The first inverted image is thus near (at or

within) the focal plane of the eyepiece, at a

dista n ce app ropriate for fin al image forma tionat infin ity, or a little closer for ima ge form at ion

at the near point. Clearly, the final image is

inverted with respect to th e original object.

We now obtain t h e ma gnification d u e to acompound microscope . The ray d iagram of

Fig. 9.31 sh ows th at th e (lin ear) magn ification

du e to the objective, na mely h ′ / h , equa ls

O

o

h Lm

h f

(9.43)

where we have used th e result

ta no

h h

f L

FIGURE 9.31 Ray diagram for the

formation of image by a compound

microscope.

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Terres t r ia l te lescopes have , inad dition, a pa ir of inverting len ses to

ma ke t he final image erect. Refracting

t e l e s c o p e s c a n b e u s e d b o t h f o rt e r r e s t r i a l a n d a s t r o n o m i c a l

observations . For examp le, con sider

a t elescope whos e objective h as a focal

len gth of 10 0 cm a nd th e eyepiece a

focal len gth of 1 cm. The m agn ifyin gp o w e r o f t h i s t e l e s c o p e i s

m = 100 / 1 = 100 .

Let u s cons ider a pair of sta rs of

ac tua l separa t ion 1 ′ (one minute of arc). The sta rs ap pear as though theyare sepa rated b y an an gle of 100 × 1 ′

= 100 ′ =1.67º .

Th e ma in considerations with an as tronom ical telescope are its lightgath ering power an d its r esolution or r esolving power. Th e form er clearly

depen ds on th e area of th e objective. With larger diam eters , faint er objectscan be obs erved. Th e res olving power, or th e ab ility to obs erve two objects

distinctly, which ar e in very nearly th e sam e direction, also depen ds onth e diam eter of th e objective. So, the d esirable aim in optical telescopes

is to ma ke th em with objective of large diam eter. Th e lar gest lens objectivein u se h as a d iameter of 40 inch (~1.02 m). It is at th e Yerkes Ob servatory

in Wiscons in, USA. Su ch b ig len ses tend to be very hea vy an d th erefore,difficult to m ake an d s u pport b y their edges. Furth er, it is rath er difficult

an d expensive to ma ke su ch large sized lens es which form images th atare free from an y kind of chr oma tic ab erration an d distortions.

For these reasons , modern te lescopes u se a concave mirror ra th erth an a lens for th e objective. Telescopes with mirror ob jectives a re called

reflecting telescopes. They have several advantages. First, there is no

ch roma tic aberra tion in a mirror. Second, if a p ara bolic reflecting su rfaceis chosen, s ph er ical aberrat ion is a lso removed. Mechan ical su pport is

mu ch less of a problem since a mirror weighs m u ch less tha n a lens of e q u i v a l e n t o p t i c a l q u a l i t y , a n d c a n b e

su pported over its en tire back s u rface, not jus t over its rim. One obvious prob lem with areflecting telescope is th at th e objective mirror

focus ses light inside the telescope tub e. Onemu st h ave an eyepiece and the observer right

th ere, obs tru cting some ligh t (depen ding on

th e size of th e obs erver cage). Th is is wha t isdone in the very large 200 inch (~5.08 m)

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

mirr or, in a s ma ll cage. An oth er solution to

the problem is to de f lec t the l ight be ing

f o c u s s e d b y a n o t h e r m i r r o r . O n e s u c harran gemen t u sing a convex secondary m irror to focus th e incident light,which now pas ses th rou gh a h ole in th e objective primary mirror, is s hown

FIGURE 9.32 A refracting telescope.

FIGURE 9.33 Schematic diagram of a reflecting

te lescope (Cassegra in) .

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SUMMARY

1 . Reflec t ion is governed by the equa t ion ∠i =∠r ′ an d refract ion by the

Sn ell’s law, s ini / s inr = n, where th e incident ra y, reflected ra y, refracted

ray a nd norm al l ie in th e sa me plane. Angles of incidence, ref lect ion

an d refract ion are i, r ′ a n d r , respectively.

2. Th e critical an gle of incide nce ic for a ra y incident f rom a d enser to rarer

medium, is that angle for which the angle of ref ract ion is 90°. For

i > ic , total internal reflection occurs. Multiple internal reflections in

d iamond (ic ≅ 24.4°), total ly reflect ing pr isms an d m irage, are som e

examples of total internal reflection. Optical f ibres 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 , af ter

multiple internal reflections, even if the fibre is bent.

3. Ca rtes ia n sign convention: Dis tan ces measu red in the s ame d i rec t ion

as the incident l ight are pos i t ive; those measured in the oppos i te

direct ion ar e negat ive. Al l dis tan ces are m easu red from th e pole/ opt ic

centre of the m irror / lens on th e pr incipal axis . The h eights m easu red

u pwards above x -axis an d n orma l to the p r incipal axis of the m irror /

lens a re taken a s pos i t ive. The h eights m easu red downwards are ta ken

as n egat ive.

4. Mirror equ a tion:

1 1 1

v u f

where u a n d v are object and image dis tances , respect ively an d f is th e

focal length of the mirror. f i s (ap proxim ately) ha lf th e rad ius of

c u r v a t u r e R. f is negative for concave mirror; f is pos itive for a convex

mirror.

5. F or a p r is m o f t h e an gle A , of refra ctive in dex n2 placed in a medium

of refractive index n1 ,

2

21

1

sin / 2

s in / 2

m A Dn

nn A

where Dm

i s the a ngle of minimu m deviat ion.

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

refractive ind ex n1 a n d n

2, respectively)

2 1 2 1n n n n

v u R

Thin lens form ula

1 1 1v u f

in Fig. 9.33 . Th is is known a s a Cassegrain telescope, after its inventor.It has the advantages of a large focal length in a short telescope. The

largest t elescope in India is in Kavalur, Tam il Nad u . It is a 2.3 4 m d iamet er

reflectin g telescope (Cass egrain). It was groun d, p olish ed, set u p, a nd isbeing u sed b y the Indian In stitu te of Astroph ysics, Ban galore. The largest

reflecting telescopes in th e world a re th e pa ir of Keck telescopes in Ha waii,

USA, with a reflector of 10 m etre in diam eter.

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EXERCISES

9 . 1 A sm all can dle, 2.5 cm in s ize is p laced at 2 7 cm in front of a concavemirror of rad ius of cur vatu re 36 cm . At what distan ce from th e mirror

sh ould a s creen be placed in order to obta in a sh arp image? Descr ibe

th e na tu re an d size of th e ima ge. If th e can dle is m oved closer to th e

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 ha ppen s a s th e n eedle is moved farth er from th e mirr or.

9 . 3 A tank i s f i l led wi th 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. Wha t is the refractive index of water? If

water is replaced by a l iquid 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 Figu res 9 .34(a) and (b) sh ow refraction of a ray in a ir incident at 6 0°

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 45º with th e n orma l to a water -glas s int erface [Fig. 9.34 (c)].

withou t a screen. Bu t th e image does exis t . Rays from a given point

on th e object are converging to an image point in spa ce an d diverging

away. The s creen s imp ly diffu ses th ese rays , som e of which reach our

eye and we see th e ima ge. This can be seen b y the images formed in a ir

du r ing a laser sh ow.

3 . Ima ge fo rmat ion needs r egu lar r e flec t ion / r e fr ac t ion . In p r inc ip le , a ll

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

why you do n ot see your ima ge by an i r regu lar reflect ing object , say

th e page of a b ook.

4 . Thick lenses g ive co loured images du e to d i sper s ion . The var ie ty in

colour of objects we see arou nd u s is du e to the cons t i tu ent colou rs of

the l ight incident on them. A monochromatic l ight may produce an

ent i rely different p ercept ion a bou t th e colours on a n object as s een in

white l ight.5 . For a s imple microscope , the an gular s i ze of the ob jec t equa l s the

an gular size of th e ima ge. Yet i t offers m agn ification bec au se we can

keep the sm al l ob jec t much c loser to the eye tha n 25 cm an d hen ce

ha ve i t subten d a large an gle. The ima ge is at 25 cm which we can s ee.

Without t he m icroscope, you would n eed to keep th e sm al l object at

25 cm which would su b tend a very smal l an g le .

FIGURE 9.3 4

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9 . 5 A small bulb is placed at the bottom of a tank containing water to adepth of 80 cm. Wha t i s the area of the s u rface of water throu gh

which l ight from th e bu lb can em erge ou t? Refractive ind ex 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 l ight is incident on a face of th e prism. The a ngle of minimu m

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

ma terial of th e prism? The refracting an gle of th e 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 D o u b l e - c o n v e x l e n s e s a r e t o b e m a n u f a c t u r e d f r o m a g l a s s o f

r e f r a c t i v e i n d e x 1 . 5 5 , w i t h b o t h f a c e s o f t h e s a m e r a d i u s o f

cu rvatu re. Wha t is the ra dius of cu rvatu re requ ired if th e focal length

is to be 20 cm?9 . 8 A beam of light converges at a point P. Now a lens is placed in the

pa th of the convergent beam 1 2 cm from P. At what p oint does th e

bea m con verge if th e len s is (a) a convex len s of focal length 20 cm ,

an d (b) a con cave lens of focal length 16 cm?

9 . 9 An object of size 3.0 cm is p laced 14 cm in front of a con cave lens of

focal length 21 cm. Describe the image produ ced by the lens. What

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

9 . 1 0 Wh at is th e focal length of a con vex lens of focal length 30 cm in

conta ct with a concave lens of focal length 20 cm? Is th e system a

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

9 . 1 1 A compound microscope consists of an objective lens of focal length

2 . 0 c m a n d a n e ye p i ec e of fo c a l l en g t h 6 . 2 5 c m s e p a r a t e d b y a

distan ce of 15 cm. How far from th e objective sh ould an object be

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

dist inct vision (25 cm), and (b) at infinity? What is th e m agnifying

power of the microscope in each case?

9 . 1 2 A p e r s o n w it h a n o r m a l n e a r p o i n t (2 5 c m ) u s in g a c om p o u n d

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

focal length 2 .5 cm can br ing an object p laced a t 9 . 0 mm from th e

object ive in sharp focus . What i s the separa t ion between the two

lenses? Calculate the magnifying power of the microscope,

9 . 1 3 A sm all telescope h as an objective lens of focal length 144 cm a nd

an eyepiece of focal length 6.0 cm. Wha t is th e ma gnifying p ower of

the te lescope? What i s the separa t ion between the object ive and

the eyepiece?9 . 1 4 (a ) A gian t refracting telescope at an obs ervatory ha s an objective

lens of focal length 15 m. If an eyepiece of focal length 1. 0 cm is

used, what is the angular magnification of the telescope?

(b) If th is te lescope is u sed to v iew the moon, what is th e diameter

of the image of the moon formed by the object ive lens? The

diameter of the moon is 3.48 × 10 6 m, and the radius of lunar

orbit is 3 .8 × 10 8 m .

9 . 1 5 Use the mir ror equat ion to deduce that :

(a ) an ob ject p l aced be tw een f a n d 2 f of a concave mirror produces

a real image beyond 2 f.

(b) a convex mirror a lways produ ces a vir tu al ima ge independ ent

of the location of the object.

(c ) t h e vi r t u a l i m a g e p r o d u c e d b y a c o n v e x m i r r o r is a l w a y sd i m i n i s h e d i n s i z e a n d i s l o c a t e d b e t w e e n t h e f o c u s a n d

the pole.

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(d) an object p laced between the p ole an d focus of a concave mir rorproduces a v i r tual and enlarged image.

[ Note: This exerc ise helps you deduce a lgebra ica l ly proper t ies of

images that one obtains from explici t ray diagrams.]

9 . 1 6 A small pin fixed on a table top is viewed from above from a distance

of 50 cm. By what distan ce would th e pin app ear to be ra ised if i t is

viewed from th e sam e point throu gh a 15 cm th ick glass s lab h eld

par allel to the t able? Refractive index of glas s = 1.5. Does the a ns wer

depend on the location of the slab?

9 . 1 7 (a) Figure 9 .35 s hows a cross-s ect ion of a ‘l ight p ipe’ ma de of a

glass f ibre 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 shownin the f igure.

(b) Wha t is the an swer if there is no out er covering of the pipe?

FIGURE 9.3 5

9 . 1 8 Answer the following questions:

(a) You ha ve lear nt th a t p lane and convex mirrors pr oduce vir tua l

images of objects . Can they produce rea l images under some

c i r cum s tances? Exp la in .

(b) A vir tu al image, we a lways say, cann ot be caught on a screen.

Yet wh en we ‘see’ a virtu al ima ge, we are obviou sly brin gin g it

o n t o t h e ‘s c r e e n ’ (i . e . , t h e r e t i n a ) o f o u r e y e . Is t h e r e a

con t r ad ic t ion?

(c) A diver u nder water, looks obliquely a t a fish erm an s tan ding on

the b an k of a lake. Would th e fisher ma n look tal ler or sh orter to

the diver than what he ac tual ly i s?

(d ) D oes the a ppa ren t dep th o f a t a nk o f w a te r chan ge if viewed

obliquely? If so, does the app aren t depth increas e or decrease?

(e ) The r e fr ac t ive index o f d i am ond i s m u ch g rea t e r t han tha t o f

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

9 . 1 9 The ima ge of a s ma ll electric bu lb fixed on th e wall of a r oom is to b e

obtained on the opp osite wall 3 m a way by mean s of a large convex

lens . Wha t is the m aximu m p ossible focal length of th e lens required

for the purpose?

9 . 2 0 A screen is p laced 90 cm from an object . The ima ge of the object on

the screen i s formed by a convex lens a t two di f ferent locat ions

sepa rated by 20 cm. Determine th e focal length of th e lens .

9 . 2 1 (a ) Determine t he ‘effective focal length ’ of th e combina tion of th e

two lens es in Exercise 9.10, if th ey are placed 8.0 cm a par t with

the i r p r inc ipa l axes co inc iden t . D oes the answ er depend on

which side of the combina tion a b eam of par allel light is inciden t?

Is the notion of effective focal length of this system useful at all?(b) An object 1.5 cm in size is placed on th e side of the con vex lens

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

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a n d t h e c o n v e x l e n s i s 4 0 c m . D e t e r m i n e t h e m a g n i f i c a t i o nproduced by the two-lens system, and the size of the image.

9 . 2 2 At wha t an gle sh ould a ray of l ight be incident on th e face of a p rism

of refracting a ngle 60° so th at i t just su ffers total intern al reflection

at th e other face? The refractive ind ex of th e ma terial of th e prism is

1 .524 .

9 . 2 3 You ar e given pr isms ma de of crown glas s a n d flint glass with a

wide variety of angles. Suggest a combination of prisms which will

(a) devia te a penci l of white l ight withou t m uch dispers ion,

(b ) d i spe r se (an d d i sp l ace ) a p enc i l o f w h i t e ligh t w ithou t m uch

devia t ion.

9 . 2 4 For a normal eye, the far point is at infinity and the near point of

dist inct vision is ab out 25 cm in front of the eye. The corn ea of th e

eye provides a converging power of about 40 dioptres, and the leastconverging power of the eye- lens behind the cornea i s about 20

diopt res . From th is rou gh da ta es t imate the ra nge of accommodat ion

(i.e., the range of converging power of the eye-lens) of a normal eye.

9 . 2 5 D o e s s h o r t - s i g h t e d n e s s ( m y o p i a ) o r l o n g - s i g h t e d n e s s ( h y p e r -

metropia) imply necessari ly that the eye has part ial ly lost i ts abil i ty

of accomm odation? If not , wh at might ca u se th ese defects of vision ?

9 . 2 6 A myopic person has been using spectacles of power –1.0 dioptre

fo r d i s t an t v i s ion . D ur ing o ld age he a l so needs to use sepa ra t e

r e a d i n g g l a s s o f p o w e r + 2 . 0 d i o p t r e s . E x p l a i n w h a t m a y h a v e

h a p p e n e d .

9 . 2 7 A p e r s o n l o o k i n g a t a p e r s o n w e a r i n g a s h i r t w i t h a p a t t e r n

comprising vert ical and horizontal l ines is able to see the vert ical

l ines more dist inctly than the horizontal ones. What is this defectdu e to? How is s u ch a d efect of vision corrected ?

9 . 2 8 A man with normal near point (25 cm) reads a book with small print

using a magnifying glass: a thin convex lens of focal length 5 cm.

(a) Wha t is the c loses t and th e far th es t d is tan ce a t which he sh ould

keep the lens f rom the page so that he can read the book when

viewin g throu gh th e ma gnifyin g glass ?

(b ) Wha t is t h e m ax im u m and the m in im u m angu la r m agn ifica t ion

(magnifying power) possible using the above simple microscope?

9 . 2 9 A card s heet d ivided int o squ ares each of size 1 m m 2 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) Wha t is the magnifica t ion produced by the lens? How mu ch isthe area of each square in the vi r tual image?

(b ) Wha t i s t he an gu la r m agn i fi ca t ion (m a gn i fy ing p ower ) o f t he

l e n s ?

(c) Is th e ma gnification in (a) equa l to the m agnifying power in (b)?

Exp la in .

9 . 3 0 (a ) At w ha t d i s t an ce shou ld the lens b e he ld from the f igu re in

Exercise 9 .29 in order to v iew the squares d is t inct ly wi th the

maximum possible magnifying power?

(b) Wha t is the ma gnifica t ion in th is case ?

(c) Is th e magnification equa l to the ma gnifying power in this case?

Exp la in .

9 . 3 1 What should be the dis tance between the object in Exerc ise 9 .30

and the magnifying glass if the vir tual image of each square in thefigure is to have an area of 6.25 mm 2 . Would you b e ab le to see the

squ ares dist inctly with you r eyes very close to th e ma gnifier ?

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Optical Instruments

349

[ Note: Exercises 9.29 to 9.31 will help you clearly understand thedifference between magnification in absolute size and the angular

magnification (or magnifying power) of an instrument.]

9 . 3 2 Answer the following questions:

(a ) Th e a n g le s u b t e n d e d a t t h e e ye b y a n o b j ec t is e q u a l t o t h e

angle subtended at the eye by the vir tual image produced by a

magnifying glass. In what sense then does a magnifying glassprovide angular magnif ica t ion?

(b) In viewing thr ough a m agnifying glas s , one us u al ly pos i t ions

one’s eyes very close to the lens . Does an gular m agn ification

change if the eye is moved back?

(c) Magnifying power of a s imple microscope is inversely proportiona lto 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 mu st both th e object ive and th e eyepiece of a compoun d

microscope have short focal lengths?(e) When viewing throu gh a compoun d microscope, our eyes sh ould

be pos i t i oned no t on the eyep iece bu t a shor t d i s t ance aw ay

from it for best viewing. Why ? How mu ch sh ould be tha t sh ort

distance between the eye and eyepiece?

9 . 3 3 An angular magnification (magnifying power) of 30X is desired using

an objective of focal len gth 1 .25 cm a n d a n eyepiece of focal len gth

5 cm. How will you set u p th e compoun d microscope?

9 . 3 4 A sm all telescope h as an objective lens of focal length 140 cm a nd

an eyepiece of focal length 5.0 cm. Wha t is th e m agnifying p ower of the telescope for viewing distant objects when

(a) the te lescope is in norma l adjus tment (i.e ., when th e fina l imageis at infinity)?

(b) the f ina l image is form ed at th e least d istan ce of dist inct vision

(25 cm )?

9 . 3 5 (a ) For the t e l escope desc r ibed in Exerc ise 9 .34 ( a ), w ha t i s t h e

separation between the objective lens and the eyepiece?

(b) If this telescope is u sed to view a 10 0 m t al l tower 3 km away,

what is th e h eight of the image of th e tower formed by th e objectivel e n s ?

(c) Wha t is th e height of th e final ima ge of th e tower if i t is form ed at

2 5 c m ?

9 . 3 6 A Cass egrain telescope us es two mirrors as s hown in Fig. 9.33. Su ch

a te lescope is bu ilt with the m ir rors 20 mm apa r t . If the ra dius of

cu rva tu re o f t he l a rge m i r ro r i s 220m m and the sm a l l m i r ro r i s140 mm , where will the f ina l ima ge of an object at infinity be?

9 . 3 7 Light incident n ormally on a plane m irror at t ached to a galvan ometer

coil retraces backwards as shown in Fig. 9.36. A current in the coil

produces a deflection of 3.5o

of the mirror. What is the displacement

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

FIGURE 9.3 6

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Physics

9 . 3 8 Figure 9.37 shows an equiconvex lens (of refractive index 1.50) incontact with a liquid layer on top of a plane mirror. A small needle

with i ts t ip on the principal axis is moved along the axis unti l i ts

inverted image is foun d a t th e posit ion of the n eedle. The distan ce of

the n eedle from th e lens is m easu red to be 45. 0 cm. The l iquid i s

r e m o v e d a n d t h e e x p e r i m e n t i s r e p e a t e d . T h e n e w d i s t a n c e i s

meas u red to be 3 0.0 cm. What is th e refractive index of the liqu id?

FIGURE 9.3 7

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