Geometrical Opticsdddd

8/18/2019 Geometrical Opticsdddd

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GEOMETRICAL OPTICS

Refraction of light at plane surfaces:

Optics is a branch of physics that deals with properties and effects of light. The branch of

optics is broadly divided into two branches namely

i) Geometrical optics ii) Physical optics

i)Geometrical optics: Geometrical optics is a branch of optics that deals with properties and

effects of light assuming that light travels only in straight lines. Also the laws of reflection and

refraction will be considered.

i)Physical optics: physical optics is a branch of optics which deals with properties and effects

of light, assuming the wave nature of light.

Firstly, we shall consider and restrict only to Geometrical optics. Here, we encounter thefollowing words very frequently through out Geometrical optics.

a) Transparent bodies: Transparent bodies are those bodies which allow major portion ofthe light incident on them to pass through. Eg.Glass

b)

Translucent bodies: Translucent bodies are those bodies which allow fraction of thelight incident on them to pass through.

Eg. Oil paper, ground glass etc.

c) Opaque bodies: Opaque bodies are those bodies which do not allow light

to pass through them at all.

Eg. Wood, Wall etc.

d) Ray of light: [

] A ray of light is nothing but a straight line with an arrow to indicate

the path and direction of propagation of light respectively

e) Beam of light: A collection of rays of light is called Beam of light. There are three types

of beams: 1) Convergent 2) Divergent 3) parallel beam

f) Optical medium: Any medium that allows light to propagate through it is

called an optical medium.g) Isotropic medium: It is a medium in which the speed of light is the same in all

directions.

h) Anisotropic medium: It is an optical medium in which the speed of light has differentvalues in different directions.

i) Optical density: It is the property of a transparent material and is the measure of speedat which the light travels through that medium.

Refraction of light:Xy: Refracting surface

M N: Normal to the refracting surface

î : angle of incidencer̂ : angle of refraction

d̂ : angle of deviation

Definition: When a ray of light passes from one optical

medium to another optical medium of different opticaldensities, the velocity of light changes.

As a consequence of that, the ray bends. This

phenomenon is known as Refraction of light.

i

r

rarer

denser

M

YX

N

d

For more: www.puctime.com


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Laws of refraction of Light:I Law: “The incident ray, the refracted ray, and the normal drawn to the refracting surface at

the point of incidence, all lie in the same plane”.

II Law: (Snell’s Law): “The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a

constant for a given pair of media and colour of light(Wavelength of light)”

sini constantsinr

The constant in the above equation is called Refractive index and denoted by n or

1 2

sini

sinr

------------------ (1)

But Snell‟s law fails to explain and give the value of refractive index for normal incidence.

Then we find the refractive index in terms of velocity of light.

Refractive index in terms of velocities of light: When a ray of light is incident normally on the

ref. surface, î =0 and r̂ =0.

Let1v be the velocity of light in med 1 and 2v be the

Velocity of light in med 2 then,

1 2

vel.of light in med1

vel.of light in med2 1

1 2

2

v

v ---------------------- (2)

“This refractive index is called relative refractive index since it depends on both the media.

“Therefore Relative refractive index is defined as the ratio of velocity of light in medium1 to the velocity of light in medium2”.

Absolute Refractive Index(μ): The absolute refractive index of a medium is always defined with respect to air or vacuum.

It is defined as the ratio of velocity of light in air or vacuum to the velocity of light in the

given medium.

vel.of light in air or vacuum

vel.of light in the given medium

c

v ------------- (3)

Note: Medium with higher RI is said to be optically denser and medium with lower RI is said

to be optically rarer.

Eg.a

1 ,w

41.33

3

,g

31.5

2

,d

52.5 etc.

2

Relationship between Relative Refractive index and Absolute refractive

index

Let 1v be the velocity of light in med 1.Let 2v be the velocity of light in med 2

Let C be the velocity of light in air then,



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By definition

Relative RI 1 2vel.of light in med1

vel.of light in med2

1

1 2

2

v

v

------------------------ (4)

Also AbsoluteRI of 1

1

1

c

v

----------------------- (5)

Similarly 2

2

c

v

--------------------- (6)

2 1

1 2

21 2

1

2

1 2

1

v5

6 v

(7)

Ab.RI of med 2 Relative RI of med 2 Wrt 1 =

Ab.RI of med 1

Eg 1.g

w g

w

2. dg d

g

etc

Consider equation 7

2

1 2

1

2 1 2

1 2 1

1 2 1 1 2 2

vsin i

sin r v

sin i= sin r-----(8) v v (9)

1 2

1 1 2 2

4. sin i = sin r

5. v = v



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Lateral shift of ray of light:Consider a parallel-sided medium surrounded by

homogeneous medium. When a ray of light is incident on

such an optical medium, refraction takes place at both the

surfaces AB and CD as shown in figure, The deviation at

AB and CD are equal bet opposite in direction. Theemergent ray emerges exactly parallel to the direction of

the incident ray.

Definition: “The perpendicular distance between the emergent ray and the direction

of incident ray is known as lateral shift denoted by Ls or SL”

IO Incident rayOR Refracted ray

RE Emergent ray

IOX Direction of incident ray

T thickness of the glass slab

n or Refractive index of the glass slab

î angle of

s

incidence

r̂ angle of refraction

ê angle of emergence

L RL Lateral Shift

Applying Snell‟s Law at AB,

1 2

sin in (1)

sin r

Applying Snell‟s Law at CD

2 1

sin rn (2)

sin e

1 2

sin en (3)

sin r

Equating 1 and 3,

sin i sine

sin r sin r

i e

{Angle of incidence}={Angle of emergence}

i

e

P

d=(i r)r

A

DRC

t

r

E

N

O

B

M

X

Ls

med 2

med 1



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s

From the right angled ORP,

RPsin (i-r)=

OR

L RP OR sin (i-r)--------(4)

s

From the right angled ONR,

ONcos r=

OR

ONOR But ON=t,

cosr

tOR= (5)

cos r

U sin g 5 in 4

t L sin (i-r)cos r

,

* * Lateral Shift depends on the following factors(i) thickness of the glass slab,

(ii) Angle of incidence

(iii) Refractive index of the slab

(colour of light)

Note: Maximum Lateral Shift that the ray can suffer is equal to thickness of the slab.

i.e.s

L = t When i = 090

Normal Shift of ray of light:Consider a refracting surface xy that

separates two media, rarer from a denser

medium. Let „O‟ be luminous point object

placed at a depth „t‟ below the surface xy.

Let us consider a ray of light incident

normally on xy passes undeviated as PM.

Another ray incident obliquely at Q deviates

as QT bending away from the normal. QT

produced backwards meet at I on OP.Thus the object appears to be shifted from

O and iO . This apparent shift is called

Normal shift.

From Fig,xy refracting surface

o luminous point object

î angle of incidence

r̂ angle of refraction

I virtual image of the object

r

r

i

i

R

QX Y

O

I

M

P

T

Ns

S



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Definition:’ It is the distance through which the object appears to be shifted when it is

placed in one medium and viewed from another medium of different refractive index’.

From fig,

ˆ ˆOQR i angle of incidence

ˆ ˆSQT=r angle of refraction

ˆ ˆ ˆ

Also, OQR=POQ=i (alternate angles)ˆ ˆ ˆ SQT=PIQ=r (corresponding angles)

From right angled OPQ,

1 2

2 1

PQ sin i= (1)

OQ

PQFrom the IPQ sin r= (2)

IQ

1 sin i IQ

2 sin r OQ

IQ n =

OQ

OQ n = (3

IQ

)

If the ray strikes the surface very close to P, we can make the following approximations.

2 1

2 1

OQ OP

IQ IP

OP Real depth3 becomes n = (4)IP apparent depth

OP IP= (5)

n

But from fig,

sN = Real depth apparent depth

sN OP IP

s2 1

OPN OP

n

(Using 5 here)

s

2 1

s

2 1

1N OP 1

n

1OR N t 1

n

Normal shift depends on

1.Thickness of the medium (Real depth)2.Ref.index of the medium in which object is placed

3.Ref index of the medium from which it is viewed.



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Total Internal Reflection(TIR):

P

A B C

i > ci2

Rarer

Denser

O

Dr = 90o

A1 B

1

r 2

r 1

i1

Consider a surface xy that separates as (D) medium from a (R) med. Let us consider a

point object „O‟ in the denser medium. A ray of light such as OP incident normally on xy

passes undeviated as PM.

Consider another ray of light OA incident on xy obliquely at an angle of incidence1

i . The

ray is refracted along lAA making an angle of refraction 1r . Here the angle of refraction isgreater than angle of incidence (

1r >

1i ) because ray is passing from denser medium to rarer

medium. It is associated with weak reflection 11

AA . As the angle of incidence is increases,

angle of refraction increases. For a particular angle of incidence (Say i=c), the angle of

refraction is0

90 . The refracted ray grazes the refracting surface, this angle of incidence is

called critical anglec

i . If the angle of incidence is increases beyond critical angle, the ray

instead of refraction, gets completely reflected alongl

DD at the interface of the 2 media. This

phenomenon is called Total Internal Reflection.

Definition of critical angle

ci :

“It is a particular angle of incidence in the denser medium for which the angle of

refraction is a right angle090 ”.

Definition of TIR :“Phenomenon of light getting back denser medium by reflection at the interface when

light tend to pass from (D) med to a ® medium”.

Conditions to have TIR :1.The ray of light should pass from denser medium to a rarer medium.

2.The angle of incidence should be greater than the critical angle.

Optical Fibres:

i > c

low R

i > c

i > c



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It is a device to transmit light energy from one point to another point by large number of

TIR so that there is minimum loss of intensity of light.

It is made up of thin fibre of either glass or quartz or transparent plastic. The outer layer is

coated with material of lower RI to satisfy the TIR conditions.

Uses:1. Used in the field of computers for high speed data transmission.2. in the field of medicine – to examine the interior of heart, Stomach, intestine etc.

3. Useful tools in entertainment electronics.

Relationship between Refractive index ( ) and critical angle (c)Consider a refracting surface xy that

separates two media. Let1 be the RI

of the (D) medium, let2 be the RI of the (R) med.

From fig

Angle of incidence = critical angle (c)

Angle of refraction =090

From Snell‟s Law If

2= 1,

1 =

, i = c1 sin i =

2 sin r

sin c = 1 sin090

sin c = 1

1

sin c

Some more phenomena related to Refraction of Light(i)Twinkling of stars : The light emitted by the stars isrefracted continuously by the

different layers of the atmosphere before it reaches the

earth. Due to the repeated refractions of light, the

apparent position of the star is different from its actual

position Since the temperature and the density of the

atmosphere is changing continuously and the atmosphere

is mobile, so the apparent position of the star also

changes continuously. This continuous change in the

apparent position of a star leads to the twinkling of the

star.

(ii) Bending of an immersed object: Let a Rod AB be dipped in water. The portion OBof The rod dipped in water appears to be shortened and raised up as OC. This is due to therefraction of light. The rays of light from point B bend away from the normal after refraction at

the interface separating the air and water. These refracted rays appear to come from pointC.Thus the portion OB of the rod appears as OC. This is why an object immersed in a liquid

appears to be bent.(iii) Visibility of two images of an immersed object : Consider a fish F under

a water in the glass container. The rays of light FA and FB suffer refractionat the free surface of water and the side of the glass cotainer respectively.

After refraction, these rays appear to come from 1F and 2F which are the two virtual images ofthe fish F

iv) Sun appears oval shaped at morning and evening(i.e. flattening of the sun)

M

r = 90o

i=c

N

2= 1(air)

1= 1

X Y



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The rays of light from the lower edges of the sun are refracted more than those from the

upper edge due to passage through greater thickness of air. In other words, the rays of the

light from the upper and lower edge of the sun bend unequally. Due to this unequal bending

of light, the image of the vertical diameter of the sun appears oval and large in size.

However, at noon when the sun is overhead, the rays of light from the sun enter the

atmosphere normally and hence no bending of light takes place. Therefore, the sun appears

circular at noon.

Relevant question. Watching the sunset on a beach, one can see the sun for several

minutes after has actually set explain.

Sun is visible to an observer after the actual sunset(i.e. actual passing of the horizon by the sun)

because of refraction of light due to denser atmosphere near the earth. Due to refraction

apparent shift in the direction of the sun is about 01/ 2 leading to delay of about 2 minutes in

actual and apparent sunsets.

(v) Optic pipe and optical fibres. Optical fibre is extremely thin (radius of new microns) andlong strand of very fine quality glass or quartz coated with a thin layer of material of refractiveindex of the strand. ( If refractive index of the core is say 1.7 then refractive index of the

coating is 1.5). The coating or surrounding of optical strands is known cladding. The sleevecontaining a bundle of optical fibres is called a light pipe.

When light falls at one end of the optical fibre, it gets refracted into the fibre. The refracted

ray of light falls on the interface separating fibre and coating at an angle which is greater

than the critical angle. The total internal reflection takes place time and again as shown in

the light travels the entire length of the fibre and arrives at the other end of the fibre without

any loss in its intensity even if the fibre is rounded or curved.

Uses : (a) optical fibres are used to transmit light without any loss in its intensity over distancesof several kilometer.

(b) optical fibres are used in the manufacture of medical instruments called endoscopes. Light

pipe is inserted into the stomach of the human being. Light is sent through few optical fibres of

the light pipe. The reflected light from the stomach is taken back through the optical fibres of

the same light pipe. This helps the doctors to see deeply into the human body. Hence the doctorcan visually examine the stomach and intestines etc. of a patient.

(c) They are used in tele-communications for transmitting signals. A single fibre is able to

transmit multiple signals (say 3000) simultaneously without any interference, whereas the

electric wire can preferably transmit one signal at a time.

(d) Optical fibres are used to transmit the images of the objects.

(e) Optical fibres are used to transmit electrical signals from one place to another. Theelectrical; signals are converted into light by special devices called transducers. Then these

light signals are transmitted through optical fibres to distant places.



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REFRACTION THROUGH A PRISM

BQRC: Base of the prism

APQB: Well polished

APRC: called refracting faces.ˆBAC: Refracting angle

AP Refracting edge

ABC

PQR 2 Triangular faces

APQB:

APRC: 3 Rectangular faces

BQRC:

A prism is a transparent medium usually made of glass. It is bounded by 5 faces. 3faces are rectangular and 2 faces are triangular.

Out of 3 faces which are rectangular, 2 faces are well polished called refracting faces.

The 3rd

face which is slightly rough called Base of the prism. The edge AP common to boththe refracting faces is called the refracting edge. The angle between the two refracting faces is

called the refracting angle or Angle of the Prism.

Any section of the prism perpendicular to the refracting edge is called the principal

section of the Prism.

A Dsin

2To derive

Asin

2

A

A

Q

NP

B C

i2

S

d

r 2

r 1

i1

Z

M

Consider the principal section ABC of the Prism. Let A be the angle of the prism and

be

the Refracting index of the Prism. Consider a ray of the light PQ to be incident on the face

AB at an angle1

i and refracted as QR, with an angle of refraction1

r Let the ray QR be

incident on the face AC an angle2

r and emerge as RS making an angle of emergence2

i . In

the absence of the prism the ray PQ would have travelled straight along PQT. In the presence

of the prism, the ray PQ has undergone a total deviation equal to d.

Y

A

P

Q R

X

Z

B C



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“Angle of deviation is angle made by the emergent ray with the direction of the

incident ray”.

The angle of deviation d depends on

(i) refracting index of the glass.

(ii) Refracting index of the surrounding medium.

(iii) Angle of the prism.

(iv)

Wavelength or colour of light

(v) Angle of incidence.

From fig,

AQNR is a cyclic quadrilateral

i.e 0ˆ Â N 180 0ˆ Â 180 N (1)

01 2ˆ ˆFrom le QNR, r r 180 0

1 2ˆˆ ˆr r 180 N (2)

equating (1) and (2)

1 2ˆ ˆ Â=r r (3)

But Total deviation=(deviation at face AB)+(deviation at face AC)

1 1 2 2d=(i r ) (i r )

1 2 1 2d= i i (r r )

1 2d=i i A (4) Using 3

For a given prism and colour of light, the angle of deviation depends on the angle of

incidence. If the angle of incidence is gradually increased, the value of d, first decreases,reaches a minimum value and then increases. The lowest angle of d is called the angle of

minimum deviation. (The particular angle of incidence for which the deviation is minimum).

The variation of d with angle of incidence

is as shown in the graph.l

i is increasing

and

2i is decreasing at P.

1 2i i i

From the graph it is clear that for any value

of d, there are two values of i namely1

i

and 2i .At minimum deviation,

1 2

1 2

i i i(say)

r r r

(5)

Under this condition, the incident ray and emergent ray passes symmetrically with

respect to the refracting faces or the refracted ray passes exactly parallel to the base with inthe prism.

Using (5) in (3) and (4) we get

A r r

and D=i+i-A A=2r D=2i-A

d

D

i1

i i2

d

i



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Ar= (6)

2

2i=A+D

A+D i= (7)

2

But from Snell's Law,

sin i=

siin r

sin A+D = (8)

Asin

2

This is the expression for the refraction index of the Prism in terms of Angle of the

Prism and angle of minimum deviation.

Dispersion of Light;A beam of light consisting of a single wavelength (colour) is calleda monochromatic light. A beam of light composed of several wavelengths is called compositelight. White light is composite light consisting of seven colours (violet, indigo, blue, green,

yellow, orange and red). When a beam of white light (composite light or compound light) passes through a prism, different colours or different wavelengths get separated.

Definition: The phenomenon in which composite light splits up into its constituent colourson passing through a suitable optical medium (prism) is known as dispersion of light.

ORThe phenomenon of splitting up of white light into its constituent colours when it passes

through a prism is known as dispersion of light.OR

The splitting up of white light into its various components is known as dispersion of light.

It is clear from the fig, that red rays are deviated

Least and violet rays are deviated most. When a

ray of white light is incident at O, the angle of

incidence at O in air is the same for red and

violet rays. The angle of refraction made by red

ray OB is greater than the violet ray OC.

v R

R v

i.e. d dbut r r

Cause for dispersion: The separation of different colours present in white light is because of different refractive indices of the material of the prism for each colour, therefore,

different colours suffers different deviations.

The display of different colours on the screen is known as spectrum of white light.

The optical medium which brings about dispersion is called dispersive medium

Refrangibility: The phenomenon of different colours getting deviated to differentextents is known as refrangibility.

i

B C

r v

r r

dr d

v

A



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Impure spectrum: A spectrum in which different colours overlap and not distinctly seenis known as Impure spectrum.

Pure spectrum: A spectrum in which different colours do not overlap and are distinctlyseen is known as a pure spectrum.

To derive d=(n-1)A[deviation produced by a thin prism]

Note: A thin prism is one whose refracting angle is very small (of the order of 05 to 010 ).

I Method II Method

A+dsin

2Wkt n=

Asin

2

Consider a thin prism of refracting when the prism is set inthe minimum angle A and refractive index n.

deviation position, D=d

1 2

Wkt deviation

produced by d=i i A (1)

a prism

A Dsin

2n

Asin

2

From Snell's law at AB, for a thin prism,

1

1

sin in=

sin r

A+dn=

A

1

1

in (since prism is thin)

r

nA A d

1 1i nr (2) d nA A

Similarly2 2i nr (3) d (n 1A (4)

substitute (2) and (3) in (1)

1 2d=nr nr A

1 2d n(r r ) A

1 2but r r A

d nA A

d (n 1)A (4)

Angular Dispersion: Angular dispersion produced by a thin prism is always defined W r t two colours.

Definition: Angular dispersion produced by a thin prism is defined as the difference between

angular deviations produced for any two colours under consideration.

Eg: Angular dispersion between red and violet

v R Angular dispersion=d d (5)

V R Also Angular dispersion=(n 1)A (n 1)A

B C

r 2

d

A

i2

i1

r 1



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V R =n A A n A A

V R Angular dispersion=(n n )A

Dispersive power of the prism

:

The ability of a prism to deviate different colours of a composite light in different directions

is characterized by its dispersive power.Definition: Dispersive power is numerically equal to the ratio of the angular dispersion

between any two colours to the deviation suffered by the mean ray, it is denoted by

.

Eg: For violet and red rays,

Angular DispersionDispersive power

Mean deviation

Angular Dispersion

Mean deviation

V R

V R

d d1

d d2

V R V R

V R Y

n n n nAlso

n n n 11

2

Thus dispersive power depends only on the material of the prism and not on refracting angle.

1 2 1 2

n n n nIn general = where n=n 1 2

Dispersion without deviation:When a ray white light passes through a prism, it splits into different colours. Every colours

suffers deviation. Therefore, spectrum has to be viewed at an angle W r t direction of the

incident ray. Sometimes, it is easy to view the spectrum keeping the source directly in sight.

This is achieved by combining prisms of different dispersive powers.Eg: Direct vision spectroscope uses such a combination,

Consider two prisms (thin), one of crown glass and other

Flint glass having dispersive powers

and 1

respectively.

Let A and1

A be their refracting angles and n and 1n their

Refractive indices. If d and 1d be the deviations the mean

Ray, then

d (n 1)A(for the I prism)

To produce dispersion without deviation,



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1

1 1

1

1

1

d d 0

(n 1)A (n 1)A 0

(nh 1)A (n 1)a

n 1A A

n 1

This is the condition for zero deviation. Negative sign indicates that two prisms must be

placed with their refracting angles on opposite sides.

1 1Note: For Net dispersion =d

Spectrometer A spectrometer is an instrument in which measurement can be made on the observed

pure spectrum. It is used to measure the refractive index of the material of a prism, tomeasure the wavelength of light using diffraction grating etc.

The main parts of a spectrometer are 1) collimator 2) prism table 3) telescope 4) circular scalewith vernier.

Collimator C consists of a narrow slit S of adjustable width at one end and convex lens1

L , at

the other. The distance between the slit and1L is adjustable. When the slit S is at the focal

plane of1

L , rays of light from the source after refraction at1

L , are rendered parallel on to

the prism.

The prism table P is a circular metal plate on which a prism can be mounted. It is

capable of rotating about a vertical axis passing through its centre. There are 3 leveling

screws available at the base of the table to set it perfectly horizontal.

Telescope T is of astronomical type. The eye-piece E is provided with a pair of cross-

wires for making measurements.

The telescope T can be rotated around a circular scale(not shown in the fig) graduated

in degrees and its position can be accurately read using vernier attached to this scale.

Experimental arrangement for the production of pure spectrum

(i) Light from a source S is passed through a narrow slit so that a narrow beam is

incident on the prism.

(ii)

The beam incident on the prism is rendered parallel by placing convex lens1

L

between the slit and the prism such that the slit is at the principal focus of the lens.(iii)

The prism is set in the minimum deviation position for the mean colour.

(iv) The coloured beams emerging from the prism are brought to focus by using another

convex lens 2L .The above arrangements are available in a spectrometer.



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Refraction at spherical surfaces

1. A refracting surface which forms a part of the sphere is called a spherical refractingsurface.

2. A spherical refracting surface in which the surface is convex towards the rarermedium is called a convex spherical refracting surface.

3. A spherical refracting surface in which the surface is convex towards the densermedium is called a concave spherical refracting surface.

4. The area of the spherical surface available for refraction is called the aperture of the

spherical surface(AB).

5.

The geometrical centre of the spherical surface is called the pole of the sphericalsurface(P).

6.

The centre of the sphere of which the spherical surface forms a part is called the

centre of curvature of the spherical surface(C).7.

The radius of the sphere of which the spherical surface forms a part is called the

radius of curvature of the spherical surface(PC).

8. The straight line joining the pole and centre of curvature extendable on either sides is

called the principal axis of the spherical surface.

9. The medium or space in which the incident rays travel is called the object space.

10. The medium or space in which the refracted rays travel and form a real image is

called the image space.

Sign Convention1. All distances are measured on the principal axis from the pole of the spherical

surface.

2. Real is positive and virtual is negative.

3. The radius of curvature of a spherical surface is taken to be positive (+) if the

surface is concave towards the denser medium.

4. The radius of curvature of a spherical surface is taken to be negative ( ) if thesurface is concave towards the rarer medium.

Relation between u,v,r and n for the refraction at a spherical

surfaceLet P be the pole of spherical surface of refractive index 1n and aperture AB. Let the

surface be surrounded by a medium of refractive index 2n , where 1 2n n . Let O be aluminous point object on the principal axis in the denser medium such that PO = u (objectdistance). A ray of light along OP moves into the rarer medium, undeviated. A ray OE

refracts into the rarer medium as EF, bending away from the normal CED. Let CEO i be the angle of incidence and FED CEI r be the angle of refraction. The refractredray EF appears to originate from I on the principal axis. Hence I is the virtual image of O.

Then PI v (virtual image distance).Applying sine rule for the triangle OEC,



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CO OE COsin i sin .......(1)

sin i sin EO

Applying sine rule for the triangle IEC,

CI EI CIsin r sin .......(2)

sin r sin EI

(1) sin i CO EI ......(3)(2) sin r EO CI

For the refraction from medium (1) to (2)

2 21

1

sin i nn

sin r n

Hence from (3) 2

1

n CO EI.....(4)

n EO CI

If the ray OE is close to the principal axis (paraxial ray), by approximation

OE OP u and IE IP v

Substituting in (4), 2

1

n CO PI

n PO CI

Also from the figure, CO CP OP r u and CI CP PI r ( v) r v

2

1

n r u v

n r v u

2 1n u(r v) n v(r u)

2 2 1 1n ur n uv n vr n vu Dividing throughout by u v r,

2 2 1 1n n n n

v r u r

1 2 1 2 1 2 1 2n n n n n n n n

u v r r r u r

In general for any refracting surface, 1 2 1 2n n n n

r u v

Difference in the Refractive index Refractive index

refractive indices of the object space of the image space

Radius of curvature Object distance image distance

Power of a spherical surfaceThe capacity or ability of a spherical to cause convergence or divergence is called the

power of the spherical surface.

The power of the spherical surface is a measure of the extent or magnitude of

convergence or divergence caused in the light passing through it.

The power of a spherical surface is given by

1 2n n Differencein the refractive indicesPr Radius of curvature of the spherical surface

When „r‟ is positive (+), P is positive and represents convergence.

OC I P

i r

Rarer (n2)

F

Dr

E

denser(n1)



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18

When „r‟ is negative (-), P is negative and represents divergence.

LensesA lens is any optical medium bounded by two surfaces of which at least one is

spherical or cylindrical.

1. A lens which converges a beam of light is called a converging lens.

2. A lens which diverges a beam of light is called a diverging lens.3. By completing the two faces of a lens, two imaginary spheres are

obtained. The centres of such spheres

are called the centres of curvature of

the lens.

4. The radii of the spheres obtained by

completion of the two faces of alens are called the radii of curvature

of the lens.5.

The line joining the two centres of curvature of the lens extendable

on either sides is called the principal axis of the lens.

6. A lens whose thickness is small compared to its radii of curvature iscalled a thin lens.

7. A lens in which the radii of curvature of the two faces are equal to

one another is called an equi-convex lens(or equi-concave lens).

8. Optic centre is a point within the lens such that all

the rays passing through it will emerge parallel to

the incident ray.

9. Rays which are close and parallel to the principal axis are called

paraxial rays.

10.Rays which are away from the principal axis and incident at the

Periphery of a lens are called marginal rays or peripheral rays.

Principal focus and focal lengthWhen a narrow parallel beam of

light, parallel and close to the

principal axis is incident on one face

of a convex lens, on emergence, the

beam actually converges to a single

point on the principal axis on the

other side of the lens. Such a point is called the principal focus(F) of the

convex lens.

When a narrow parallel beam of light, parallel and close to the principal axis isincident on one face of a concave lens, on emergence, the beam actually diverges. The

diverging rays appear to originate from a single point on the principal axis on the

incidence side. Such a point is called the principal focus(F) of a concave lens.

The distance between the optic centre and the principal focus the focal length(f).

Note: A lens will have two principal foci but a single focal length.

Distance formula in lensesLet AB represent a luminous object on the principal axis of a convex lens placed

suitably to for an inverted real image 1 1A B on the other side of the lens.

From the similar triangles 1 1A B F and FDO,



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19

1 1 1A B B F ......(1)DO FO

From the similar triangles

1 1 1A B B O ......(2)AB BO

Since AB DO, from (1) and (2) 1 1B F B O .....(3)

FO BO

Let BO=u(object distance), BO=v (image distance) and FO= f(focal length) . Also

1 1B F B O FO v f . From (3) v f v

uv uf vf f u

Note: The formula can be applied to any lens, real or virtual image cases following the

sign convention.

Note: The distance formula in lenses is 1 1 1 uvor f u v f u v

Convex lens Concave lens

Converges or tends to converge light

incident on it.

Diverges or tends to diverge light incident

on it.

Has a real principal focus. Has a virtual principal focus.

Focal length is taken as positive. Focal length is taken as negative.

Can produce real and virtual image. Can produce only virtual image.

Can produce magnified , equal size and

diminished images.

Can produce only diminished image.

Can produce erect and inverted images. Can produce only erect image.Magnification

The linear magnification in a lens is defined as the ratio of the linear size(height) of

the image to the linear size(height) of the object.

If O and I are the heights of the object and the image and u and v are the object and

image distances then,

MagnificationI v

MO u

If M 1 Image is larger than the object, (enlarged image)If M 1 Image is of the same size of the object andIf M 1 Image is smaller than the object (diminished image).Magnification has no unit.

Power of a lensThe converging or delivering capacity of a lens is called the power of the

lens.power of a lens is defined as the reciprocal of its focal length expressed in metres.

1P

f .The power of a lens depends on the RI of the material of the lens with that of the

surrounding medium.The SI unit of a lens is dioptre.

The power of a lens is 1 dioptre if its focal length is 1m.

AD

B1

v u

O BF

A1

F



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20

Concave lens Converging lens ve focal length Power is v Concave lens Diverging lens ve focal length Power is v

Lens Makers FormulaLet f be the focal length of a thin

convex lens of a material of refractive

index n and radii of curvature1r and 2r .

Let the lens be placed in air (n = 1). Let

O be a Luminous point object on the

principal axis at a distance „u‟ facing the

surface ABC. Assuming the other face ADC

to be absent, let1I be the real image formed

by the sphericalsurface ABC, at a distance1v .

For the refraction at a spherical surface we have, 1 2 1 2n n n n

r u v

For ABC, Since 1n =1 (air – object space ) and 2n = n (glass – image space) and 1r r , wehave

1 1

n 1 1 n.....(1)

r u v

With presence of the other surface ADC, Let I be the final real image formed at a

distance v. For the image I,1I acts as the virtual object in dense medium. Hence the object

distance for the refraction at the ADC is 1v .Thus for the refraction at the spherical surface ADC,

2 1

n 1 n 1.....(2)

r v v

Hence (1) + (2)

2 1 1 2

n 1 n 1 1 1 1 1 1 1(n 1)

r r u v u v r r

Since for a lens

1 2

1 1 1 1 1 1(n 1)

u v f f r r

This is lens markers formula.

Note : If a lens of material of refractive index 2n is surrounded by a medium of refractive

index1n , then the lens Makers formula will be

1 2

1 2

1 1 1( n 1)

f r r

or 2

1 1 2

1 n 1 11

f n r r

Note : The focal length of a lens depends upon

(1) the refractive index of the material of the lens and

(2) The refractive index of he surrounding medium and the radii of

curvature of the lens.

Combination of thin lensesA single lens which can effectively replace a combination of lenses is called the

equivalent lens of the combination.

A

B

C

O I I'

uv'

D

r 1 r 2



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21

Expression for the focal length of the equivalent lens for a

combination of thin lenses placed in contactLet O be a luminous point on the

Axis of a thin convex lens of focal length

1f at a distance „u‟. Let 1I be the real image

Formed by the lens at a distance 1v . Then byDistance formula for lenses,

1 1

1 1 1....(1)

u v f

Let another lens of focal length2f be placed coaxially in

contact with first lens in its image side. Let I be the final image formed. Here1

I acts as the

virtual object for I and hence1v is the object distance for the second lens.

1 2

1 1 1.....(2)

v v f

(1) + (2)

1 2

1 1 1 1...(3)

u v f f

If F is the focal length of the equivalent lens of the said combination,

1 1 1....(4)

u v F

From (3) and (4)

1 2

1 1 1

F f f Also 1 2

1 2

f f F

f f

For „n‟ lenses in contact,1 2 n

1 1 1 1......F f f f .

1 2 n

1 2 n 1

f f ....f F f f ...f

The reciprocal of the focal length of the equivalent lens of combination of thin lenses

in contact is equal to the algebraic sum of the reciprocals of the focal length of the individual

lenses.

Note: If F (effective focal length) is positive Converging combinationIf F (effective focal length ) is negative Diverging combinationThis formula is applicable ONLY for thin lenses.

Consider

1 2 n

1 1 1 1......

F f f f

Since1

Pf (Power) 1 2 nP P P .........P

The effective power of a combination of lenses in contact is the algebraic sum of the

powers of the individual lenses.

Expression for the focal length of a combination of two thin lenses

separated by a finite distanceLet two thin lenses 1L and 2L be two thin lenses of focal length 1f and 2f

respectively be placed coaxially with a separation „d‟ in air.

Let F be the focal length of the combination of lenses. It can be shown

that

O(u)

O(u)

f 1 f 2

F

I (v)

I (v) I1(v1)



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1 2 1 2

1 1 1 d

F f f f f or 1 2

1 2

f f F

f f d

dv1

u v

O

L1

L2

I'I

If 1 2P ,P and P are the powers of the two lenses forming the combination and the

equivalent lens respectively, then

1 2 1 2P P P dP P

Conjugate fociIn the case of a convex lens, if an object is placed suitably so to form

a real image, then the positions of the object and the image are

interchangeable. Such positions are called conjugate foci.A real image can be got on a screen

with a convex lens iff the distance between

the object and the screen D is equal or more

than 4 time the focal length of the lens.

If D 4f , two real images are formed.If D=4f, only real image is possible.If D

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