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Chapter 4 : Optical Instruments
M A M EL-Morsy
Optical Instruments
1. PHOTOGRAPHIC CAMERA
A photographic camera consists essentially of a convex lens, a
light sensitive film or plate at the back and a focussing arrangement.
There is an adjustable aperture known as diaphragm that permits
different amounts of light to enter the camera.
Fig. 4-1
The image is focussed on the plate or film by altering the
distance of the lens from it. There is a shutter placed in the path of
light and the time for which it is opened depends upon the brightness
of the object, the sensitivity of the photographic emulsion, the focal
length of the lens and the size of the stop (Fig. 4. 1).
The diameter d of the stop is usually given as a fraction of the
focal length f of the lens system and the ratio
called the f-ratio or f-number.
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focal length
diameter of the stop
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Suppose the focal length of the lens is 11 cm and the diameter of the
f
11
stop is 1 cm. Then the f-ratio or f-number is 11 and d  .
The diameters of the stop of a camera are usually marked so
that the exposure time is doubled between each position e.g.
f
,
5.6
f
,
8
f
,
11
f
,
16
f
22
Therefore, the f-ratios are 5.6, 8, 11, 16, 22 etc. and their
squares are in the ratio of 1 : 2 : 4 : 8 : 16.
Therefore, if the correct exposure for
for
2.
f
is one second, then,
5 .6
f
f
it is 2 seconds and for
it is 4 seconds and so on.
11
8
DEPTH OF FOCUS
Depth of focus is the maximum distance by which the film in a
camera may buckle (move forward or backward along the axis)
without causing the blurring of the image.
Fig. 4-2
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Let r be the radius of the exit pupil of the coaxial optical
system (camera lens system) and P the point image (Fig, 4.2). When
the film is moved forward by a distance xl, the ray passing through the
point A on the circumference of the exit pupil E, cuts the plane
passing through P1, and normal to the axis at the point B. When the
film is moved backward by a distance x2, the ray passing through the
point A cuts the plane passing through P2, and normal to the axis at
the point C. The location of the paint P2 is such that x1= x2.
P1B=P2C=R
s PP1B and POA are similar

P1 B PP1

OA OP
or
R x1

r
x

x1 
But,
Rx
r
x2=x1.
Depth of focus,
P1P2 = x1 + x2 =
2 Rx
r
Determination of the depth of focus in a camera is of the great
importance. It helps in estimating the amount by which the film in the
camera may buckle without disturbing the image beyond the tolerable
limit.
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3. TELEPHOTO LENS
To obtain an image of the object situated at a very long
distance, a convex lens of large focal length should be used. In this
case, to obtain a magnified image the distance between the lens and
the photographic plate should be large. Thus a magnified image
requires a long camera which may be unmanageable and
inconvenient. To avoid this difficulty a telephoto lens is used. It
consists of two lenses mounted in a tube. The tube is fixed in the
camera in place of a single lens. One of the lenses is convex and the
other is concave. Each lens is made achromatic (Fig. 4.3).
Fig. 4-3
The concave lens L2 is placed in the original position of the
lens in the camera, whereas the convex lens L1, is placed in the front
of the lens L2. The combination of these two lenses acts as a single
lens of large focal length. The combination acts as a convex lens.
If fl and f2 are the focal lengths of the lenses L1 and L2 and is the
distance between them, then
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1
1
1
d



f
f1
f2
f1 f 2
Consider that the rays from the distant object after passing
through the lens L1 meet at F1, but the lens L2 placed in the path
diverges the rays and finally the rays meet at F2.
The combination of lenses, L1 and L2 has formed the final
image at F2 which can be formed by a single convex lens of focal
length DF2 placed at D. Thus we find that the telephoto lens acts as a
converging lens of large focal length. It also avoids the use of very
long camera box and can be used for taking highly magnified images
even with a small camera.
4. MICROSCOPES
In the seventeenth century convex lenses were used as
magnifying glasses. Afterwards two or more lenses were used to form
powerful microscopes.
In 1648, Hooke made use of microscopes in the study of
existence of cells in animal and vegetable tissues.
The angular magnification of a microscope in normal use is:
M
angle subtended at the eye by the image
angle subtended at the unaided eye by the object placed at the near po int
M


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5. SIMPLE MICROSCOPE (MAGNIFYING GLASS)
Suppose an object of length h is placed at the near point A and
viewed by the eye. Then the visual angle is .
tan 
h
D
(i)
Fig. 4.4
Now suppose that a convex lens L of focal length f is used as a
magnifying glass and the object AB is placed between O and F, such
that the magnified erect image PQ is formed. If the observer's eye is
close to the lens, then the distance OP is equal to the least distance of
distinct vision. Here the visual angle  is given by:
tan  
PQ
D
Dividing (ii) by (i),
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(ii)
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tan 
PQ

tan
h

PQ


h
or
(for small angles)
PQ
v
D


h
u
u
But,
If M is the magnifying power
M
 D

 u
M
D
u
1 1
1
 
v u
f
Here,
v = -D, and u is also negative

1 1
1
 
D u
f
1
i.e.
D D

u
f
D
D
 1
u
f
M  1
D
f
6. VIRTUAL IMAGE AT INFINITY
When an object is placed at the focus of the lens the image is,
formed at infinity (Fig. 4.5).
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Fig. 4.5
Then, linear magnification

The visual angle

PQ
h
is infinite,
h
f
h

f
M

 h
D
Example 4.1
D
M


f
A convex lens of focal length 10 cm is used as a
magnifying glass. Find the magnifying power when (i) the image is
formed at infinity and (ii) the image is formed at the least distance of
distinct vision (25 cm from the lens).
The Answer:
(i) When the image in formed at infinity
M 
D
25

 2.5
f
10
(ii) When the image is formed at the near point
M 
D
25
 1  1
f
10
= 3.5
Example 4.2 A person has his near point at 15 cm and a range of
distinct vision of 85 cm. What is his range of distinct vision when he
wears close fitting spectacles having a power of -0.8 diopter ?
The Answer:
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Power = -0,8 diopter
f = metre = -125 cm
In the first case, let the near point using spectacles be at a
distance of ul cm from the eye.
Here,
v = -15 cm
1
1
1


v u1
f

u1 =-17 cm
1
1
1


15 u1
125
( approximately)
In the second case, let the far point using spectacles be at a
distance of u2 cm from the eye.
Here,
v = -(15+85) =-100 cm
1
1
1


v u2
f
u2 = -500 cm
Hence the range of vision
= - 500-17 cm.
= - 483 cm.
Example 4.3 The ratio of the magnifying power of a converging lens
when it is used to throw an image at infinity is 1.20.
(a)
What is the distance of distinct vision for the
eye if the focal length of the lens is 4.80 cm ?.
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(b) At what distance from the lens must an object be placed so
that the image falls at the distance of distinct vision calculated
in (a) ?.
The Answer:
D
f
1 
(a)
Here
D
f
 1.20
f
 1  1.20
D
But f = 4.80 cm
4 .8
D
 0.20
D = = 24 cm
The least distance of distinct vision is 24 cm.
(b) Here, f = 4.80 cm. u = ?
v = -24 cm.
1
1
1


v
u2
f

1
1
1


24
u
4.8
u = -4 cm.
The object is placed at a distance of 4 cm from the lens.
7. COMPOUND MCROSCOPE
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The magnifying power of a simple microscope can be increased
D
M

1

by decreasing the focal length of the lens as
f .
But due to constructional difficulties, the focal length of a lens
cannot be decreased beyond a certain limit. Moreover, the lens of
small focal length has a small diameter because the curvature of the
surface is large and the field of view is small.
Therefore, to increase the magnifying power, two separate
lenses are used. The lens near the object is called the objective and
the other which is nearer the eye is known as the eyepiece. The
objective and the eyepiece are both convex lenses. The objective is of
small diameter and small focal length ( high power ) whereas the
eyepiece is of large focal length than the objective.
The object AB is placed at a distance slightly greater than the
focal length of the objective (Fig. 4.6). An inverted image AB is
formed at A. The eyepiece is adjusted so that the distance of A' from
it is less than its focal length. As the eyepiece acts as a simple
magnifying glass, the final image PQ is formed at P which is
magnified and virtual.
Magnifying Power
When the microscope is in normal use, the final image PQ is
formed at the ‘near point’ at a distance D from the eye.
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Fig. 4.6
M 
Angle subtended at the eye by the final image at D
Angle subtended at the eye by the object at the same distance D
PQ
PQ
M D 
AB
AB
D
M
PQ
A B

A B
AB
PQ
 M o  Magnification of the eyepiece
A B
and
A B
 M e  Magnification of the objective
AB
M = Me x Mo
As the eyepiece acts as a simple magnifying glass, its magnifying
power
= 1
D
f

Me  1 
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D
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where fo is the focal length of the eyepiece.
If the distance of the image AB from the objective = v and the
distance of the image A B from the objective = v, and the distance of
the object AB from the objective = u. Then,
Mo 
Also
1 1
1


v u
f
v
u
(According to the sign convention u is –ve)
1 1
1


,
v u
fo
Mo =
v
v
(
1)
u
fo
v
v
(
1)
u
fo
M = Mo x Me
M (
v
D
 1 )( 1 
)
fo
fe
If the case of most of the microscopes the distance between the
objective and the eyepiece is fixed. The lenses are fixed at the two
ends of a tube. Then the microscope is focussed by moving it bodily
either towards the object or away from the object.
Therefore, in the relation for the magnifying power,
M (
v
D
 1 )( 1  ) , it is necessary to find v in terms of fo, fe and the
fo
fe
distance between the objective and the eyepiece.
Suppose, the distance between the objective and the eyepiece
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= OE = L,
OA = v
 v= OE - A E
v = L- A E
As A' acts as a point object for the eyepiece whose image is
formed at P,
1
1
1


D A E
fe
(According to the sign convention D and
A E are –ve)

1
1
1


D
A E
fe
D  fe
1
1
1



A E
D
fe
D. f e
or

AE =
D . fe
D  fe
v ( L
Dfe
)
D  fe
Substituting the value of v in the formula for magnifying
power,
M  (1
(
D
)(
fe
( L
D  fe
)(
fe
L
103
Dfe
)
D  fe
1)
fo
Dfe
)  fo
D  fe
)
fo
Chapter 4 : Optical Instruments

( D  fe )
fe
M A M EL-Morsy
( L  fo 
Dfe
)
D  fe
fo
 DL  D f o  Lfe  f o f e  Df e 
M 

fe . fo


 D( L  f o  f e )  f e ( L  f o ) 


fe . fo


From the last equation, knowing fe, fo, D and L, the magnifying
power of the microscope can be calculated.
Note. Actually the objective is not a single lens but it consists of one
or more achromatic lenses and the eyepiece also consists of two
lenses which give an optically true image. The above discussion only
gives the simplest case where O represents the centre of the lens
which in equivalent to the lenses present in the objective while E
represents the centre of the lens which is equivalent to the lenses
present in the eyepiece.
8. SPECTROMETER
A spectrometer consists mainly of three parts:
1.
Collimator,
2.
Prism table and,
3.
Telescope.
Collimator
It consists of an achromatic lens L1, such that the slit S is at its
focus. The slit is placed in front of a source of light and the width of
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the slit can be adjusted (Fig. 4.7). The slit acts as a source of light and
the rays coming out of the lens L1 are parallel.
Fig.(4.7)
Table
The table can be adjusted and its position can be read with the
help of the verniers V1, and V2. The table can be rotated about a
vertical axis and its axis coincides with the axis of rotation of the
telescope. A prism is placed on the table. A parallel beam of light is
incident on the prism and the emergent beam is also parallel.
Telescope
It is an astronomical telescope fitted with a Ramsden's eyepiece
and cross wires. When a parallel beam of light coming out of the
prism falls on the objective, the spectrum produced is viewed through
the eyepiece. If a photograph is to be taken, a photographic plate
replaces the eyepiece.
1.
Adjustments
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The eye-piece of the telescope is adjusted such that the cross wires
are clearly visible. The telescope is focussed on a distant object. The
parallax between the image and the cross-wires is removed. Thus, the
telescope is set for parallel rays. Then, the spectrometer is placed in
front of a monochromatic light source (sodium lamp) and the slit is
adjusted. The position of the lens of the collimator is adjusted such
that a sharp and well-defined image is seen through the telescope.
2.
Setting the table
The table can he levelled with the help of a spirit level. For
accurate work three levelling screws A, B and C provided with the
prism table are adjusted. Parallel lines are marked on the table, which
are parallel to the line AB. The prism is placed with its edge ab
perpendicular to the line AB (Fig. 4.31).
Fig. (4.8)
The table is adjusted so that both the faces ab and ac of the prism
receive light coming out of the collimator. The telescope is brought
into the position to receive light from the face ab. The screws A and
B are adjusted till the image of the slit is in the centre of field. Then,
the telescope is turned to receive light from the face ac. Again by
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adjusting the screw C, the image of the slit is brought to the centre of
the field of view.
3.
Schuster's method
The prism is placed on the table and the approximate position for
minimum deviation is observed through the telescope. The telescope
is rotated for a position making an angle more than the minimum
deviation. In this position of the telescope, there will be two positions
of the prism when the image of the slit can be viewed through the
telescope. Suppose, in the first case, the prism is in the position A
(Fig. 4.9). The incident rays from the collimator are falling more
obliquely on the prism. The telescope is focussed so that the image is
sharp and there is no parallax between the image and the cross-wires.
In this case the rays coming out of the prism become more parallel.
Keeping the position of the telescope fixed, the prism table is
rotated such that the image of the slit is again viewed through the
telescope. Suppose the prism in now in the position B. Here less
oblique rays pass through the prism. The image in this case will be
blurred. It in focussed by adjusting the distance between the slit and
the collimating lens. The telescope in not adjusted in this case.
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Fig.(4.10)
The prism is again brought back to the position A and the
process is repeated. When the image of the slit is sharp and there is no
parallax between the image and the cross-wires, the spectrometer is
adjusted. The collimator produces parallel rays, which are focussed on
the cross-wires of the telescope after passing through the prism.
9.  OF THE MATERIAL OF A PRISM
Angle of the prism
After setting the spectrometer, the prism is placed on the table
as shown in Fig. (4.11). The slit is just close to the source of
monochromatic light i.e., to light from the sodium lamp.
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Fig. (4.11)
Both the faces ab and ac receive light from the collimator. The telescope is rotated and the image formed by reflection at the face a b is
observed through the telescope in the position P. The readings of the
verniers are noted. The telescope is rotated to receive light reflected
by the face ac and again the readings are taken. The angle between
the positions P and Q is twice the angle of the prism.
Fig. (4.12).
Use Fig.(4.12) to proof that the angle ( = 1+2) is equal to 2A .
Where A is the angle of the prism.
Angle of minimum deviation
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The prism is placed on the prism table and the image of the slit
after refraction through the prism is observed
through
the
telescope.
The prism table is rotated so that the angle of deviation decreases.
The telescope is also rotated so as to follow the image. Finally, a
position is reached when the image is stationary and with the rotation
of the prism table in the same direction, the image recedes and the
angle of the deviation increases. The cross-wires of the telescope are
made to coincide with the image of the slit when it is stationary.
This is the position of minimum deviation (Fig. 4.13). The
position of the telescope is observed. The telescope is brought to view
the image of the slit directly and its position is noted. The angle
through which the telescope has been rotated from the position of
minimum deviation, gives the angle of minimum deviation.
Fig.(4.13)
Suppose the angle of the prism = A and the angle of minimum
deviation = 
The refractive index of the material of the prism for the given
monochromatic light in given by
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
M A M EL-Morsy
A
2
A
sin
2
sin
In a good spectrometer two lines are observed for sodium light.
The two angles of minimum deviation are observed and the refractive
indices for the two lines of sodium (D1 and D2) are calculated.
The spectrometer is also used to determine:
1.
The dispersive power of the material of a given prism.
2.
The wavelength of a given source of light by using a diffraction
grating.
10. PULFRICH REFRACTOMETER
It is used to measure the refractive indices of solids and liquids.
It consists of a right-angled prism A having its two faces perfectly
plane. One of the faces is horizontal and the other is vertical (Fig.
4.14).
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Fig.(4.14)
The solid B whose refractive index is to be determined is taken
having two faces cut perpendicular to one another. The faces are
made perfectly plane. The solid B is placed on the prism A. To bring
perfect contact between the horizontal faces of A and B, a few drops
of a liquiq whose refractive index is greater than the solid is used. In
the case of a liquid, whose refractive index is to be determined, a
glass cell ia used.
Light is incident in a direction parallel to the horizontal surface
so that light entering the prism A is incident at the critical angle c with
the normal. Finally it emerges from the prism at an angle i.
Let the refractive index of the solid be  and that of the
material of the prism A be o.
Here o > and c is the critical angle.
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sin c =
M A M EL-Morsy

o
...
(i)
sin i

sin r
Also
r + c = /2 , r = /2 -c
sin i
sin (

2

c)
sin i
 o
cosc
…
(ii)
Squaring and adding equations (i) and (ii)
 2 sin 2 i
sin c  cos c  2 
o
 o2
2
2
   o2  sin2 i
(iii)
Knowing  o and i, the value of  can be calculated.
To measure i, the apparatus has a collimating telescope with a
circular scale attached to it. The telescope is adjusted so that the
cross-wires lie on the dark edge of the field of view. This gives the
position of the minimum angle of emergence i. The material of the
prism A has a refractive index of 1.74. In modem instruments, the
circular scale is calibrated in terms of the refractive index and
readings can be read directly from the scale. In some cases a table is
provided with the instruments, which gives the value of 
corresponding to i.
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11. ABBE REFRACTOMETER
Abbe refractometer is used to measure the refractive index of
liquids. It consists of two prisms A and B. The prisms have angles 30o
, 60o and 90o. The liquid whose refractive index is to be determined
is placed between the two prisms (Fig. 4.15).
Light is reflected from the glass plate or a ground glass surface
P into the prism A. The emergent light is received by the telescope.
The rays of light are incident on the hypotenuse face of the prism at
various angles of incidence.
Fig.(4.15)
At a particular angle of incidence, the ray passes along the
plane of the liquid and just passes into the second prism. These are the
rays, which have an angle of incidence equal to the critical angle. The
theory is just similar to Pulfrich refractometer. The telescope is set to
a position where the cross-wires lie on the dark edge of the field of
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view. This reading is noted. The scale attached to the telescope is
calibrated to give the refractive index of the liquid directly.
12. PRISM BINOCULARS
Prism binoculars consists of two astronomical telescopes each
containing two totally reflecting right angled prisms. In this way the
distance between the objectives of the telescope is made Larger than
the distance between the two eyes, thus increasing the field of view
and the stereoscopic effect pertaining to distance. Moreover, the
effective length is made half of the length of the telescope to be used
for the same magnification and the final image is also erect.
The Prism A is placed with its refracting edge horizontal while
the prism B is placed with its refracting edge vertical. In an
astronomical telescope the final image formed is inverted.
The objective and the eyepiece are convex lenses. The prism A
turns the image in the vertical direction upside down ( Fig.4.14). The
prism B turns the image in the horizontal direction. The image
produced after reflection is erect and magnified. The optical path is
about three times the length of the binocular. Therefore, it works as
astronomical telescope equal to three times its length.
13. DISPERSION BY A PRISM
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A beam of white light, when it passes through a prism is split
up into its constituent colours and this is called dispersion of light.
The image thus formed on a screen is called a spectrum.
Fig. (4.16)
The spectrum consists of visible and invisible regions. In the
visible region the order of the colours is from violet to red. The
principal colours are given by the word VIBGYOR (Violet, Indigo,
Blue, Green, Yellow, Orange and Red). The deviation produced for
the violet rays of light is maximum and for red rays of light it is
minimum. Fig. (4.16) represents the dispersion of a white ray of light
by a prism in the visible region. The region of the spectrum, of wavelengths shorter than violet is called ultra-violet and the region of
wavelengths longer than red in called infra-red. In the present
chapter, the discussion relates only to the visible region of the
spectrum.
The refractive index for the material of a prism (or a lens) is
different for different wavelengths (or colours). The deviation and
hence the refractive index is more for blue rays of light than the
corresponding values for red rays of light. The deviation and the
refractive index of the yellow constituent are taken as the mean
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values. If the dispersion through a prism does not follow the order
given by VIBGYOR, it is said to be anomalous dispersion.
14. REFRACTION THROUGH A PRISM
The refractive index of the material of a prism is given by
Where A is the angle of
the prism and D is the angle of minimum
deviation. For a small angled prism
 

2


 

2
where  and  refer to the angle of the prism and the angle of
minimum deviation (for small values of , the angle  is also small
and the sines of the angles are taken equal to the angles).
  (   1 )
Fig.(4.17)
Fig. (4.17) represents the angles of deviation b, and r,
produced in the blue, mean yellow and red rays of light. The
deviations b,  and r can be written as :
  (   1 )
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for mean yellow light ... (i)
Chapter 4 : Optical Instruments
M A M EL-Morsy
 b  (  b  1 )
for blue light
 r  (  r  1 )
...(ii)
for red light ... (iii)
The difference in deviation between any two colours is called
angular dispersion. From (ii) and (iii)
 b   r  (  b  1 )  (  r  1 )
 (  b   r )
(iv)
 b   r (  b   r )  b   r



(   1 )
 1
d

 1
... (v)
Dividing (iv) by (i)
where b and r are the refractive indices for the blue and red
rays of light and  is the refractive index for the mean yellow rays of
light. The expression
b  r
d
(
)
 1
 1
is called the dispersive pews;
of the material of the prism. It is constant for two colours (or
wavelengths) chosen and is represented by .

b  r
 1
The reciprocal of  is called the constringence. It is also
customary to represent b,  and r as F , D and C where F, D and
C correspond to the Fraunhoffer lines (dark lines) in the solar
spectrum. The F, D and C lines lie in the blue, yellow and red regions
of the spectrum and their wavelengths are 4861Å, 5893 Å and 6663 Å,
respectively. (1 Å =1 Angstrom unit=10-8 cm).
15. CAUCHY'S DISPERSION FORMULA
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M A M EL-Morsy
When an electromagnetic wave is incident on an atom or a
molecule, the periodic electric force of the wave sets the bound
charges into vibratory motion. The frequency with which these
charges are forced to vibrate is equal to the frequency of the wave.
The phase of this motion as compared to the impressed electric force
will depend on the impressed frequency. It will vary with the
difference between the impressed frequency and the natural frequency
of the charges.
Dispersion can be explained with the concept of secondary
waves that are produced by the induced oscillations of pound charges.
When a beam of light propagates through a transparent medium (solid
or liquid), the amount of lateral scattering is extremely small. The
scattered waves travelling in a lateral direction produce destructive
interference.
However, the secondary waves travelling in the same direction
as the incident beam superimpose on one another. The resultant
vibration will depend on the phase difference between the primary
and the secondary waves. These superimposition changes the phase of
the primary waves and this is equivalent to a change in the wave
velocity. Wave velocity is defined as the speed at which a condition
of equal phases is propagated. Hence the variation in phase due to
interference, changes the velocity of the wave through the medium.
The phase of the oscillations and hence that of the secondary waves
depends upon the impressed frequency. It is clear, therefore, that the
velocity of light in the medium varies with the frequency of light.
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Also refractive index depends upon the velocity of light in the
medium. Therefore the refractive index of the medium varies with the
frequency (wavelength) of light.
The relative permittivity of the medium in the case of dynamic
polarizability is used to obtain the Cauchy’s dispassion formula which
can be written as:
  A 
B
2
This equation represents Cauchy's dispersion formula. A and B
are called Cauchy's constant. The values of A and B depend on the
medium.
It is evident that the refractive index of the medium
decreases with increase in wavelength of light.
Example 4.1. Calculate the values of Cauchy's constants A and B for
crown glass. Given
c = 1.514,
F = 1.524
c = 6563Å
F = 4862Å
and
According to Cauchy's formula
  A 
B
2
1.514  A 
B
( 6563 x 10 8 )2
1.524  A 
B
( 6862 x 10 8 )2
0.1  B{
1
1

}
8 2
( 6862 x 10 )
( 6563 x 10 8 )2
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(i)
Chapter 4 : Optical Instruments
M A M EL-Morsy
B= 5.236 X 10-11 cm2
Substituting the value of B in equation (i) and solving
A = 1.502
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