09.Ray optics.pdf

09.Ray Optics & Optical Instruments Xplore Learning centre
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Originally, the term optics was used only in relation to the eye and vision. Later, as
lenses and other devices for aiding vision began to be developed.
Optics is a branch of physics that deals with the study of behavior and the
properties of light, along with its interactions with the matter and also with the
instruments which are used to detect it.
Properties of light:
1)Light is a form of energy which is in the form of an electromagnetic wave.
2) The visible light has wavelengths measuring between 400–700 nanometers.
3) Light travels in straight line.
4)Light has speed 3 × 108
ms-1.
Optics is divided into two main branches:
1)GEOMETRICAL ( RAY) OPTICS : In geometrical optics, light is considered to travel
in straight lines(as ray).
It explain the phenomena such as reflection, refraction, Total internal reflection etc.
2)PHYSICAL (WAVE) OPTICS: While in physical optics, light is considered as an
electromagnetic wave.
It explains the phenomena such as interference , diffraction, polarization of light etc.
Note: Read this and try to understand--In this chapter, we consider the
phenomena of reflection, refraction and dispersion of light, using the ray picture of light.
Using the basic laws of reflection and refraction, we study how the image
formation by plane and spherical reflecting(mirror) and refracting surfaces(lens).
Also explain the construction and working of some important optical
instruments like microscope and telescope.

Reflection of light: The phenomenon in which change in the direction of light when it
is incident on plane well polished surface and bounce back in the same medium.
Laws of reflection:
1) The angle of incident and the angle of reflection are equal. ∠𝑖 = ∠𝑟
2) The incident ray, normal and reflected ray all are in same plane
Two kind of reflection: Depending on the surface of the object where the light is
incident and laws of reflections are valid in both the reflection.
 Regular reflection/ Specular reflection: When a beam of light is perfectly
reflected(parallel to each other) back from shiny, polished surface.
Exa: reflection from Mirror, glass, steel etc.
 Irregular/diffused reflection: When a beam of light is reflected irregular(in
different direction) from an rough surface.
Exa: Reflection from Sheet of paper, table top etc.

PLANE MIRROR : A plane mirror is a mirror with a flat reflective surface.
 In plane mirror the reflection is of regular reflection and image formed in plane
mirror is always virtual and laterally inverted.
 Image size and object size also same. The image is formed at the same distance as
that of the object from the mirror.
 In plane mirror the image of an object cannot be magnified(Large in size) or
diminished(small in size). Also field of view is smaller.
Image formation in plane mirror is shown below

Applications of plane mirror:
 They are used as looking mirrors.
 They are used to make periscope.
 They are used in various scientific instruments.
But some times we need image must be larger or smaller in size compare to object.
This cannot be achieved by using simply plane mirror. Hence to over come this problem
we take help of spherical(Curved) mirrors.
SPHERICAL MIRRORS
Spherical mirrors are the parts of glass sphere whose inner or outer surface
is polished like mirror. Depending on the which surface is behave like mirror, there are
two types of spherical mirror.
1) CONCAVE MIRROR: If the mirror coating is outside of the spherical surfaces,
the mirror is called as concave mirror or converging mirror.
2) CONVEX MIRROR: If the mirror coating is inside of the spherical surfaces, the
mirror is called as convex mirror or diverging mirror.
Image Formation by Spherical mirrors.
The image formed can be real as well as virtual depending on the positions of
the object from mirror. The image is either magnified, reduced or has the same size,
depending on the position of the object.

 There are a few basic terminologies that one needs to know while studying
spherical mirrors, and they are
1. Center of Curvature(C)
Center of curvature is the centre of the sphere of which the spherical mirror is a part.
2. Radius of Curvature(R)
It’s the linear distance between Pole and the Center of curvature.
3. Principal axis: The imaginary line passing through the optical center and the center
of curvature of any lens or a spherical mirror.
4.Pole(P):The midpoint of the spherical mirror.
5.Aperture : An aperture of a mirror or lens is a point from which the reflection of light
actually happens. It also gives the size of the mirror.
6.Principal Focus : Principal Focus can also be called Focal Point. It’s on the axis of a
mirror or lens wherein rays of light parallel to the axis converge or
appear to converge after reflection or refraction.
7.Focus(F): It’s any given point, where light rays parallel to the principal axis, will
converge after getting reflected from the mirror.

Reflection in spherical mirrors: In plane mirror reflection of beam of light is parallel to
each other after reflection. But in case of spherical mirrors the reflection take place at
curved mirror surface is different.
Image formation In spherical mirrors: In both concave or convex mirror image of an
object can be enlarged, diminished, real, virtual, erect and inverted depending on the
position of object from the mirror.
 IMAGE FORMATION IN CONCAVE MIRROR
CONCAVE MIROR: A parallel beam of light
incident on the mirror and reflected back and
meet at one common point at Focus(f). Hence
this mirror is called as converging mirror.
Here the rays are actually meeting at focus
after reflection. Hence it forms an real image
of object.
CONVEX MIROR: A parallel beam of light
incident on the mirror and reflected back and
appears to meet at one common point at Focus
(f). Hence this mirror is called as diverging
mirror.
Here the rays are appears to be meeting at
focus after reflection. Hence it forms an virtual
image of object.

Above two figure shows how the image formed in concave mirror. Here the image
formed is real, inverted and diminished type. The nature of the image can be changed as
we change the position of object from the mirror.
 IMAGE FORMATION IN CONVEX MIRROR
The above two figures shows how the images are formed in convex mirror. Here
the image is virtual, erect and diminished.
Mechanism or Rules for formation of image in spherical mirrors
The Image formation in mirrors can be traced by using the following simple rules. These
rules tells us how image is formed ,at what place and nature of image.
For Concave mirror rays are actually passes through the focus after reflection, while for
convex the rays appears to be passes through the focus after reflection

Rule 3-
Ray passes through centre of curvature will follow the same path after reflection.

 SIGN CONVENTION FOR SPHERICAL MIRRORS
Sign convention rules are same for both the mirrors as shown in below figure

A sign convention for measuring distances. (3marks)
1) All distances are measured from the pole of the mirror or the optical centre of the
lens.
2)The distances measured in the same direction as the incident light are taken as
positive .
3)The measured in the direction opposite to the direction of incident light are taken as
negative.
4) The heights measured upwards are taken as positive
5)The heights measured downwards are taken as negative.
u-Object distance, v= image distance , R= radius of curvature, f= focal length
Note: Sign convention is useful in calculation of image distance(v), object distance(u),
focal length(f) etc while solving numericals.

 Image formation in concave mirror at different places
(1)
(2)
Nature of image: The object AB is placed
at far distance from mirror, hence parallel
beam of light incident and meet at focal
point to form an image.
 Image formed is real
 Inverted
 Point size
 At focal point infront of mirror
Nature of image: The object AB is
placed beyond center of curvature.
 Inverted
 Diminished(Smaller size)
 Image is formed between C and F

(3)
(4)
(5)
placed at center of curvature.
 Inverted
 Same size that of object
 Image is formed at C only
placed between center of curvature
and focal point.
 Inverted
 Enlarged(Magnified)
 Image formed beyond C
Concave mirror-
Object placed at F
placed at focal point(F)
 Inverted
 Highly Enlarged(Magnified)
 Image formed at infinity

(6)
Summary of the image formation by concave mirror
Position of the
Object
Position of the
Image
Size of the Image
Nature of the
image
(1)At infinity At the focus, F
Highly diminished
Point size
Real, Inverted
(2)Beyond C Between F and C
Diminished(Smaller
than object)
Real, Inverted
(3)At C At C Same size Real, Inverted
(4)Between C and F Beyond C Enlarged Real, Inverted
(5)At F At Infinity Highly Enlarged Real, Inverted
(6)Between P and F Behind the mirror Enlarged Virtual, Erect
Image formation in convex mirror at different places
In Convex mirror no matter where the object placed infront of the mirror the image
formed by it always has same nature.
 Virtual and erect
 Diminished
 Formed behind mirror
placed between focal point(F) and
pole(P).
 Image formed is Virtual
 Erect
 Enlarged(Magnified)
 Image formed behind mirror

RELATION BETWEEN FOCAL LENGTH(F) AND RADIUS OF
CURVATURE (R)(3marks)
The distance between the focus F and the pole P of the mirror is called the focal length of
the mirror, denoted by f.
The geometry of reflection of an incident ray is shown. Let C be the centre of
curvature of the mirror. Consider a ray parallel to the principal axis striking the mirror at
M. Then CM will be perpendicular to the mirror at M(Normal). Let θ be the angle of
incidence, and MD be the perpendicular from M on the principal axis.
From ray diagram we can write
∠𝑀𝐶𝑃 = 𝜃 𝑎𝑛𝑑 ∠𝑀𝐹𝑃 = 2𝜃
Now consider two right angle triangle, ⊿𝑀𝐷𝐶 𝑎𝑛𝑑 ⊿𝑀𝐷𝐹
tan 𝜽 =
𝑴𝑫
𝑪𝑫
and tan 𝟐𝜽 =
𝑴𝑫
𝑭𝑫
----(1)
For small angle , then tan 𝜽 ≈ 𝜽 and tan𝟐𝜽 ≈ 𝟐𝜽
MD
FD
= 2
𝑀𝐷
𝐶𝐷
𝐹𝐷 =
𝐶𝐷
2
Now, for small θ, the point D is very close to the point P. Therefore, FD = f and CD = R.
𝒇 =
𝑹
𝟐
M
D
𝟐𝜽

THE MIRROR EQUATION(5marks)
The formula which establish the relation between object distance(u), image distance(v)
and focal length(f).
Object distance(u): The distance of object from the pole of the mirror.
Image distance(v): The distance of image from the pole of the mirror.
Consider an Object(AB) is placed infront of a concave mirror at distance ‘u’ from
the pole. By the ray diagram the image(𝐴𝚤𝐵𝚤) formed at distance ‘v’ from the pole.
Hence from the sign convention we have,
𝐵𝚤
𝑃=Image distance=-v FP=focal length=-f BP=Object distance=-u---(1)
From ray diagram consider two similar triangles ⊿𝐴𝚤
𝐵𝚤
𝐹 and ⊿MPF, hence
ratio of their sides must be equal.
𝐵𝚤
𝐴𝚤
PM
=
𝐵𝚤
𝐹
𝐹𝑃
𝑜𝑟
𝑩𝚤
𝑨𝚤
𝐁𝐀
=
𝐵𝚤
𝐹
𝐹𝑃
(∵ 𝑃𝑀 = 𝐴𝐵) − − − (2)
Similar ⊿𝐴𝑃𝐵 𝑎𝑛𝑑 ⊿𝐴𝚤𝑃𝐵𝚤 are right angle triangles are also similar , hence
𝑩𝚤𝑨𝚤
𝐁𝐀
=
𝐵𝚤𝑃
BP
− − − − − −(3)
Comparing equation (2) and (3) we get

𝐵𝚤𝐹
𝐹𝑃
=
𝐵𝚤𝑃
BP
but(∵ 𝐵𝚤𝐹 = 𝐵𝚤𝑃 − 𝐹𝑃)
𝐵𝚤𝑃 − 𝐹𝑃
𝐹𝑃
=
𝐵𝚤𝑃
BP
𝑓𝑟𝑜𝑚 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛(1)𝑟𝑒𝑝𝑙𝑎𝑐𝑖𝑛𝑔
−𝑣 − (−𝑓)
−𝑓
=
−𝑣
−𝑢
−𝑣 + 𝑓
−𝑓
=
𝑣
𝑢
𝑂𝑅
𝑣 − 𝑓
𝑓
=
𝑣
𝑢
On solving we get
𝟏
𝒇
=
𝟏
𝒗
+
𝟏
𝒖
This relation is known as the mirror equation.
Magnification(m): It is defined as the ratio of height of the image(ℎ𝑖) to the height of
the object(ℎ𝑜).
𝑚 =
𝐻𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑖𝑚𝑎𝑔𝑒
𝐻𝑖𝑒𝑔ℎ𝑡 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡
=
ℎ𝑖
ℎ𝑜
Magnification is given by following equations,
𝒎 = −
𝒗
𝒖
𝑶𝒕𝒉𝒆𝒓 𝒇𝒐𝒓𝒎𝒔 𝒐𝒇 𝒎𝒂𝒈𝒏𝒊𝒇𝒊𝒄𝒂𝒕𝒊𝒐𝒏 , 𝑚 =
𝑓
𝑓 + 𝑢
𝑜𝑟 𝑚 =
𝑓 − 𝑣
𝑓
Note : Magnification is positive for virtual image, m=+ve
Magnification is negative for real image, m=-ve
The above mirror equation and magnification
equation is valid(applicable) for both concave and convex mirror. There
are no separate set of equations for convex mirror.

NUMERICALS
1)An object is placed at (i) 10 cm, (ii) 5 cm in front of a concave mirror of radius of
curvature 15 cm. Find the position, nature, and magnification of the image in each case.
2)An object of 4cm in size, is placed at 25cm from concave mirror of focal length 15cm. At
what distance from the mirror should a screen be placed to get sharp image ? Find the
nature and size of the image.(Ans- v=-37.5cm hi=-6cm)
3)Find the focal length of the convex mirror whose radius of curvature is 32cm.
4) A small candle, 2.5 cm in size is placed at 27 cm in front of a concave mirror of radius of
curvature 36 cm. At what distance from the mirror should a screen be placed in order to
obtain a sharp image? Describe the nature and size of the image. If the candle is moved
closer to the mirror, how would the screen have to be moved?
5) A 4.5 cm needle is placed 12 cm away from a convex mirror of focal length 15 cm. Give
the location of the image and the magnification. Describe what happens as the needle is
moved farther from the mirror.
6) A concave mirror produce three times magnified real image of an object placed at 10cm
from the mirror. Find the location of the image.
7)A concave mirror of focal length 20cm is placed 50 cm from a wall. How far from the
wall an object be placed to form its real image on the wall?
8)An object is placed at distance of 25 cm from a spherical mirror and its image is formed
behind the mirror at distance of 5 cm. Find focal length? Is it concave or convex mirror?
9)An object is placed in front of a convex mirror of radius of curvature 40 cm at a distance
of 10 cm. Find the position, nature and magnification of mirror.
10)An object is kept in front of a concave mirror of focal length of 15 cm. the image formed
is 3 times the size of the object. Calculate the two possible distances of the object from the
mirror.

Refraction of light: The phenomena in which change in the direction of
light when it passes from one medium to another medium obliquely.
Refraction of light in everyday life are:
 When a thick glass slab is placed over some printed matter the letters appear
raised when viewed through the glass slab
 A lemon kept in water in a glass tumbler appears to be bigger than its actual size
when viewed from the sides.
 A pencil which is partially immersed in a water in a glass tumbler appears to be
displaced at the interface of air and water.
 An object placed under water or a tank or a pond appears to be raised.
 The stars appear to Twinkle on a clear night.
 The pool of water appears to be less deep than actually it is.
LAW’S OF REFRACTION(Snell’s Law)
1)The incident ray, the refracted ray and the normal all lie in the same plane.
(2) The ratio of the sine of the angle of incidence to the sine of angle of refraction is
constant. The angles of incidence (i ) and refraction (r ) are the angles that the incident
and its refracted ray make with the normal, respectively.
sin 𝑖
sin 𝑟
= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝐬𝐢𝐧 𝒊
𝐬𝐢𝐧 𝒓
= 𝒏𝟐𝟏 =
𝒏𝟐
𝒏𝟏
𝑛21 is called as refractive index of the second medium with respect to first medium.
Refractive index can also defined as the opposition offered by the medium to
propagation of light. The inherent property of medium which tends to decrease the speed of
light.
𝑛1
𝑛2

 Refractive index is just a number and unit less quantity.
 More is the value of refractive index, less is the speed of light.
Refractive index of some materials are given below
Refraction through a rectangular glass slab
When light travels from rarer medium to denser medium( air to glass) it bends
toward normal.
When light travels from denser medium to rarer medium ( glass to air) it bends
away from normal.

Note: Refractive index(n) also defined as the ratio of speed of light in vacuum(c) to
the speed of light in medium(v).
𝒏 =
𝒄
𝒗
 LATERAL SHIFT: When a ray of light is incident obliquely on a parallel sided glass
slab the emergent ray shifts laterally . The perpendicular distance between the
direction of the incident ray and emergent ray is called “lateral shift’’.
 NORMAL SHIFT: The apparent shift in the position of an object placed in one medium
and viewed along the normal, from the other medium
AO-Incident ray
BC-Emergent ray
OB- Refracted ray
i=angle of incident
r=angle of refraction
e=angle of emergence
XY=d= Lateral shift
T=Thickness of glass slab
𝒅 =
𝒕𝒔𝒊𝒏(𝒊 − 𝒓)
𝐜𝐨𝐬 𝒓
= 𝒕𝒔𝒆𝒄(𝒓)𝐬𝐢𝐧)(𝒊 − 𝒓)

Advance sunrise and delayed sunset due to atmospheric refraction. Time difference
between actual sunset and apparent sunset is about 2 minutes. Similarly for sunrise also.
Hence apparently sunset 2min later and sunrise 2 min early .
TOTAL INTERNAL REFLECTION
When light travels from an optically denser medium to a rarer medium at the interface,
it is completely reflected back into the same medium without any refraction. This is called
as Total internal reflection.
Explanation
 When a ray of light AO1 enters from a denser medium to a rarer medium, it bends
away from the normal.
 The incident ray AO1 is partially reflected as (O1C) and partially transmitted (O1B)
or refracted, the angle of refraction (r) being larger than the angle of incidence (i).

𝒊𝒄
 As the angle of incidence increases, so, the angle of refraction goes on increases
and finally when it becomes, 𝑟 = 900
. The refracted ray is bent such that it traces
along the interface between the two media. This is shown by the ray AO3 D in Fig.
 If the angle of incidence is increased still further (e.g., the ray AO4), refraction is
not possible, and the incident ray is totally reflected back in the same medium.
Condition for total internal reflection(2marks):
1) The incident ray must travel from denser medium to rarer medium.
2) The incident angle must be greater than the critical angle ,i.e 𝑖 > 𝑖𝑐
Critical angle(𝒊𝒄): The angle at which the refracted ray is perpendicular the normal.
Or
The angle above which the total internal reflection take place.
 Relation between refractive index (n) and the critical angle(𝒊𝒄) for the pair of
media(3marks)
The angle at which the refracted ray is perpendicular the normal. i.e 𝑟 = 900
From snell’s law,
sin𝑖
sin𝑟
= 𝑛12 =
𝑛1
𝑛2
For the above figure the pair of media are water and air, hence refractive index for water
is 𝑛2 = 𝑛 and for air 𝑛1 = 1 also 𝑖 = 𝑖𝑐 then above equation becomes
sin 𝑖𝑐
sin 90
=
1
𝑛
On solving we get
𝒏 =
𝟏
𝒔𝒊𝒏 𝒊𝒄
Air(𝑛1)
water(𝑛2)
𝑟
Refracted ray
Incident ray

Application of total internal reflection(2marks)
1) Diamond sparkling(shine) due to TIR.
2) Mirage phenomena
3) Fiber optics works on the principle of TIR.
4) Endoscopy in medical
5) Total reflecting prism
CRITICAL ANGLE OF SOME TRANSPARENT MEDIA WITH RESPECT TO AIR
Material medium Refractive index(n) Critical angle(𝑖𝑐)
Water 1.33 48.75
Crown glass 1.52 41.14
Flint glass 1.62 37.31
Diamond 2.42 24.41
From above table it conclude that as refractive index is more, critical angle is small
LENSES
It is transparent glass bounded by two spherical surfaces.
 Concave lens: A concave lens is flat in the middle and thicker at the edges.
 Convex lens: Convex lenses are thick in the middle and thinner at the edges.
Lenses are also used to form a real and virtual image depending on the type of
lens and the position of the object from the lens. The size of the image also vary as the
position of object changes.

Characteristics of the spherical lenses
 Optical centre(O): It is the centre point of a lens through which incident light passes
undeviated.
 Centre of curvature(C1 and C2):The centre of the sphere, of which spherical lens
form a part is called as centre of curvature. Since lenses has two surface hence they
have two centre of curvature.
 Radius of curvature(R1 and R2): The Radius of the sphere, of which spherical lens
form a part is called as radius of curvature. Since lenses has two surface hence they
have two radius of curvature.
 Principle axis: The line which joins the centre of curvature and the optical centre of
the lens is known as principle axis.
 Principle focus(F1 and F2): The point on the principle axis where the parallel beam
of light after passing through the lenses actually meet in case of convex lens and
appears to meet in case of concave lens that point is known as principle focus. All
lenses have two principle focus on either side of the lenses.
Refraction through spherical lenses
(i)In case of convex lens a parallel beam of light incident on spherical surface and
undergoes refraction and meet(converge) at principle focal point .Hence this lens is also
called as converging lens.
(i)In case of concave lens a parallel beam of light incident on spherical surface and
undergoes refraction and move away from (converge)principle axis but appears to meet
at focal point .Hence this lens is also called as diverging lens.
Convex lens Concave lens

Rules for Image formation in spherical lenses

Sign convention in spherical lenses
Rule 3-
Ray passes through optical centre will emerge without
deviation

Image formation in convex lens at different places

Image formation in concave lense at different places
For concave lens there are two cases as shown below. Does not matter what is
position of object from the lense , the nature of the image always same.
Refraction at a spherical surface
A relation between object and image distance in terms of refractive index of the
medium and the radius of curvature of the curved spherical surface. It holds for any
curved spherical surface.

Let a spherical surface AB , having centre of curvature C , radius of curvature R and
refractive index 𝑛2. The above figure shows the geometrical formation of image ‘I’ of an
object ‘O’. The rays are incident from a medium of refractive index 𝑛1 to another medium
of refractive index 𝑛2.
We assumed that the aperture of the curved surface is very small hence the
length NM and NP are equal. Consider right angled triangle ⊿𝑁𝑃𝑂, ⊿𝑁𝑃𝐶 𝑎𝑛𝑑 ⊿𝑁𝑃𝐼
tan 𝛼 =
𝑁𝑃
𝑀𝑂
tan 𝛽 =
𝑁𝑃
𝑀𝐼
tan 𝛾 =
𝑁𝑃
𝑀𝐶
For small angle ,
𝛼 =
𝑁𝑃
𝑀𝑂
𝛽 =
𝑁𝑃
𝑀𝐼
𝛾 =
𝑁𝑃
𝑀𝐶
Now for triangle 𝑁𝐶𝑂, ∠𝑖 is an exterior angle hence we can write
𝑖 = ∠𝑁𝑂𝑀 + ∠𝑁𝐶𝑀 = 𝛼 + 𝛾
𝒊 =
𝑵𝑷
𝑴𝑶
+
𝑵𝑷
𝑴𝑪
− − − − − (𝟏)
Similarly, triangle 𝑁𝐶𝐼, ∠𝛾 is an exterior angle hence we can write
∠𝛾 = ∠𝐶𝑁𝐼 + ∠𝐶𝐼𝑁 = 𝑟 + 𝛽
𝑟 = 𝛾 − 𝛽
𝒓 =
𝑵𝑷
𝑴𝑪
−
𝑵𝑷
𝑴𝑰
− − − −(𝟐)
From snell’s law,
sin𝑖
sin𝑟
=
𝑛2
𝑛1
=
𝑖
𝑟
For small angle, sin 𝑖 ≈ 𝑖 𝑎𝑛𝑑 sin𝑟 ≈ 𝑟
𝑖 𝑛1 = 𝑟 𝑛2--------(3)
Substituting the value of 𝑖 and 𝑟 from eqn (1) and (2) in (3)
𝑛1 (
𝑁𝑃
𝑀𝑂
+
𝑁𝑃
𝑀𝐶
) = 𝑛2 (
𝑁𝑃
𝑀𝐶
−
𝑁𝑃
𝑀𝐼
)

(
𝑛1
𝑀𝑂
+
𝑛1
𝑀𝐶
) = (
𝑛2
𝑀𝐶
−
𝑛2
𝑀𝐼
)
(
𝑛1
𝑀𝑂
+
𝑛2
𝑀𝐼
) = (
𝑛2
𝑀𝐶
−
𝑛1
𝑀𝐶
) =
𝑛2 − 𝑛1
𝑀𝐶
From above ray diagram and using sign convention
𝑀𝑂 = −𝑢 𝑀𝐼 = +𝑣 𝑀𝐶 = +𝑅
(
𝑛1
−𝑢
+
𝑛2
𝑣
) =
𝑛2 − 𝑛1
𝑅
Note: If the object is in rare medium and image formed in denser medium then
If the object is in denser medium and image formed in rare medium then
Numerical-
Light from a point source in air falls on a spherical glass surface (n = 1.5 and radius
of curvature = 20 cm). The distance of the light source from the glass surface is 100 cm. At
what position the image is formed? (Ans: v = +100 cm)
(
𝒏𝟐
𝒗
−
𝒏𝟏
𝒖
) =
𝒏𝟐 − 𝒏𝟏
𝑹
(
𝑛2
𝑣
−
𝑛1
𝑢
) =
𝑛2 − 𝑛1
𝑅
(
𝑛1
𝑣
−
𝑛2
𝑢
) =
𝑛1 − 𝑛2
𝑅

Refraction by a lens(lens maker’s formula)and lens formula
Lenses of different focal lengths are used for various optical instruments.
The focal length of a lens depends upon the refractive index of the material of the lens
and the radius of curvatures of the two surfaces. The lens maker formula is commonly
used by lens manufacturers for manufacturing lenses of desired focal length.
Derivation: Consider a thin lens with two refracting surfaces having radius of
curvature 𝑅1 and 𝑅2 as shown in figure below. Let the refractive index of surrounding
medium is 𝑛1 and for lens 𝑛2.
The geometrical ray diagram of formation of image I of an point object ‘O’ is traced
below.
 The image I’ is formed at distance v’ by the refraction of light from object O due to
curved surface of radius 𝑅1.
 But the final image I of an object O is formed at distance v due to another
refraction of light at second curved surface of radius 𝑅2
For the first surface,
(
𝑛2
𝑣′
−
𝑛1
𝑢
) =
𝑛2 − 𝑛1
𝑅1
− − − (1)
For the second surface ,
(
𝑛1
𝑣
−
𝑛2
𝑣′
) =
𝑛1 − 𝑛2
𝑅2
− − − (2)
Adding eqn (1) and (2)
Object
Image

(
𝑛1
𝑣
−
𝑛1
𝑢
) =
𝑛2 − 𝑛1
𝑅1
+
𝑛1 − 𝑛2
𝑅2
𝑛1 (
1
𝑣
−
1
𝑢
) =
𝑛2 − 𝑛1
𝑅1
−
𝑛2 − 𝑛1
𝑅2
𝑛1 (
1
𝑣
−
1
𝑢
) = (𝑛2 − 𝑛1) (
1
𝑅1
−
1
𝑅2
)
On solving we get
(
1
𝑣
−
1
𝑢
) =
(𝑛2 − 𝑛1)
𝑛1
(
1
𝑅1
−
1
𝑅2
) − − − (3)
When 𝑢 = ∞ then 𝑓 = 𝑣 hence above equation reduced to
𝟏
𝒇
=
(𝒏𝟐 − 𝒏𝟏)
𝒏𝟏
(
𝟏
𝑹𝟏
−
𝟏
𝑹𝟐
) − − − (4)
The above equation is known as lens makers formula
Comparing equation (3) and (4) we get lens formula as
𝟏
𝒇
=
𝟏
𝒗
−
𝟏
𝒖
Limitations of the lens maker’s formula
 It is applicable to only thin lenses .The lens should not be thick.
 The medium used on both sides of the lens should always be same
Magnification(m)
Magnification of a lens is defined as the ratio of the height of image to the height
of object.
𝒎 =
𝑯𝒆𝒊𝒈𝒉𝒕 𝒐𝒇 𝒊𝒎𝒂𝒈𝒆
𝑯𝒆𝒊𝒈𝒉𝒕 𝒐𝒇 𝒐𝒃𝒋𝒆𝒄𝒕
=
𝒉𝒊
𝒉𝒐
=
𝒗
𝒖
 For erect (and virtual) image , m is positive.
 For an inverted (and real) image, m is negative.

Power of a lens(P): The power of lens is defined as the how strongly rays converge of
diverge from the lens.
Or
It is the reciprocal of the focal length.
𝑃 =
1
𝑓
1 Dioptre: The power of a lens of focal length of 1 metre is one dioptre.
Combination of thin lenses:
Combination of the lenses include two or more lenses placed co-axially. It is required
to increase the sharpness of the image, decrease the chromatic aberration in image and
also increase the field of view.
Expression for Equivalent focal length of the two lenses of different focal length
kept in contact co-axially(5marks)
Consider two lenses 𝐿1and 𝐿2 of focal length f1 and f2 placed in contact with each
other. Let the object be placed at a point O beyond the focus of the first lens 𝐿1.
The first lens produces an image at I1. Since image I1 is real, it serves as a virtual
object for the second lens 𝐿2, producing the final image at I .
For the image formed by the first lens 𝐿1, we get
1
𝑓1
=
1
𝑣1
−
1
𝑢
− − − −(1)
For the image formed by the first lens 𝐿2, we get
Unit of power is Dioptre(D)
 For convex lens power
is positive.
 For concave lens
power is negative.

1
𝑓2
=
1
𝑣
−
1
𝑣1
− − − −(2)
Adding Equation (1) and (2)
1
𝑓1
+
1
𝑓2
=
1
𝑣
−
1
𝑢
− − − (3)
If the two lens-system is regarded as equivalent to a single lens of focal length f, we have
1
𝑓
=
1
𝑣
−
1
𝑢
− − − −(4)
Comparing (3) and (4) we get
𝟏
𝒇
=
𝟏
𝒇𝟏
+
𝟏
𝒇𝟐
For ‘n’ number of lenses the equivalent focal length is given by,
𝟏
𝒇
=
𝟏
𝒇𝟏
+
𝟏
𝒇𝟐
+
𝟏
𝒇𝟑
… … . . +
𝟏
𝒇𝒏
𝒊𝒏 𝒕𝒆𝒓𝒎𝒔 𝒐𝒇 𝒑𝒐𝒘𝒆𝒓 𝑷 = 𝑷𝟏 + 𝑷𝟐 + 𝑷𝟑 + … … . . . +𝑷𝒏
Magnification of combination: The total magnification of two lenses kept in contact
with each other is the product of magnification produced by each lens.
𝒎 = 𝒎𝟏 × 𝒎𝟐 × 𝒎𝟑 × … … . . .× 𝒎𝒏
Note: Combination of lenses is commonly used in designing lenses for cameras,
microscopes, telescopes and other optical instruments.
PRISM
Prism is a transparent medium bounded by two inclined refracting surfaces and base.
𝑷 = 𝑷𝟏 + 𝑷𝟐
The angle between the two inclined plane is
known as Angle of prism(A).

REFRACTION THROUGH A PRISM
Expression for angle of minimum deviation 𝜹 in terms of refractive index(n)
Angle of deviation(δ): The angle between the emergent ray QR and the direction of the
incident ray OP is called the angle of deviation, δ.
Consider a triangular prism ABC , and angle of prism is ∠𝐴. Let a incident ray OP
incident on surface AB with an angle ∠𝑖 . The ray undergoes another refraction at surface
AC and emerges out with an angle ∠𝑒 called as emergent angle.
From figure, in quadrilateral APNQ
∠𝑃𝐴𝑄 + ∠𝐴𝑃𝑁 + ∠𝑃𝑁𝑄 + ∠𝐴𝑄𝑁 = 360𝑜
(∵ ∠𝐴𝑃𝑁 = ∠𝐴𝑄𝑁 = 90𝑜
)
𝐴 + ∠𝑃𝑁𝑄 = 180𝑜
− − − (1)
From triangle PNQ ,
𝑟1 + 𝑟2 + ∠𝑃𝑁𝑄 = 180𝑜
− − − (2)
Comparing (1) and (2) we get, 𝑟1 + 𝑟2 = A
The total deviation δ is the sum of deviations at the two faces, 𝜹 = 𝛿1 + 𝛿2
𝜹 = (𝑖 − 𝑟1) + (𝑒 − 𝑟2) [∵ 𝜹𝟏 = (𝒊 − 𝒓𝟏) & 𝜹𝟐 = (𝒆 − 𝒓𝟐)]
𝜹 = 𝒊 + 𝑒 − (𝑟2 + 𝑟2)
𝜹 = 𝒊 + 𝒆 − 𝑨
At minimum angle of deviation the refracted ray PQ is parallel to the base of the prism.

Then , 𝜹 = 𝜹𝒎 and 𝒊 = 𝒆 also r1=r2=r hence above equation becomes
𝜹𝒎 = 𝟐𝒊 − 𝑨 or 𝑖 =
𝜹𝒎+𝑨
𝟐
and 𝑟 =
𝐴
2
From snell’s law,
sin 𝑖
sin 𝑟
=
𝑛2
𝑛1
𝑏𝑢𝑡 𝑓𝑜𝑟 𝑎𝑖𝑟 𝑛1 = 1
𝒏 =
𝐬𝐢𝐧 (
𝜹𝒎 + 𝑨
𝟐
)
𝐬𝐢𝐧 (
𝑨
𝟐
)
The graph between angle of deviation (δ) and angle of incidence (i) for a triangular
prism is represented by.
For the prism as the angle of incidence (i) increases, the angle of deviation (δ) first
decreases goes to minimum value (δm )and then increases. For a prism there is only one
angle of incident at which minimum deviation occurs.
Note: For thin prism , 𝜹𝒎 = (𝒏 − 𝟏)𝑨 , It implies that, thin prisms do not deviate light
much.

SOME NATURAL PHENOMENA DUE TO SUNLIGHT:
 The spectacles of colours that we see around us all the time is possible due to
sunlight.
 In the visible spectrum, red light is at the long wavelength end (~700 nm) while
the violet light is at the short wavelength end (~ 400 nm).
 Dispersion(Splitting of white light into its component) takes place because the
refractive index of medium for different frequencies (colours) is different. For
example, the bending of red component of white light is least while it is more for
the violet.
 Dispersive media: The medium in which the white light splits into its component
is called as dispersive media. Example-Glass, water droplets
 Non-Dispersive media: The medium in which the white light do not splits into its
component is called as non-dispersive media. Example-Vacuum
THE RAINBOW
The rainbow is an example of the dispersion of sunlight by the water drops in the
atmosphere. This is a phenomenon due to combined effect of dispersion, refraction and
reflection of sunlight by spherical water droplets of rain.
Dispersion of sunlight through water droplets:
The water in the atmosphere act as dispersive
medium for the splitting of sunlight into its component.
When sunlight enters water drop first refracted,which
causes the different wavelengths (colours) of white
light to separate. Longer wangelength of light (red) are
bent the least while the shorter wavelength (violet) are
bent the most. Next, these component rays strike the
inner surface of the water drop and get totally
internally reflected .The reflected light is refracted again when it comes out of the drop.
It is found that the violet light emerges at an angle of 40° and red light emerges at an
angle of 42° with reference to incident light. For other colours, angles lie in between
these two values.

Types of rainbow
1) Primary rainbow:
 The primary rainbow is a result of three-step process, that is, two refraction
and one total internal reflection.
 The observer sees a rainbow with red colour on the top and violet on the
bottom.
 Primary rainbow is sharp and intense.
 It is always formed at lower altitude.
 Visible in the sky frequently
2) Secondary rainbow:
 The secondary rainbow is a result of four-step process, that is, two refraction
and two total internal reflection.
 The observer sees a rainbow with violet colour on the top and red on the
bottom.
 Primary rainbow is fainter and low intense.
 It is always formed at higher altitude.
 Visible in the sky rarely.
Primary rainbow Secondary rainbow

Scattering of light
As sunlight travels through the earth’s atmosphere, it gets scattered
(changes its direction) by the atmospheric particles. Light of shorter wavelengths is
scattered much more than light of longer wavelengths.
Rayleigh scattering law :The intensity of scattered light is inversely proportional to the
fourth power of the wavelength.
𝑰 ∝
𝟏
𝝀𝟒
Phenomena associated with scattering of white light
1)Sky appaers blue in colour: The bluish colour predominates in a clear sky, since blue
has a shorter wavelength than red and is scattered much more strongly
2) Reddish appearance of the sun and full moon near the horizon: At sunset or sunrise, the
sun’s rays have to pass through a larger distance in the atmosphere. Most of the blue and
other shorter wavelengths are removed by scattering. The least scattered red light
reaching our eyes, therefore, the sun looks reddish.
OPTICAL INSTRUMENTS
Optical instruments are the devices which utilize reflection , refraction, TIR etc
phenomena of light to enhance an image for more clear view. Use of an optical
instruments, such as a magnifying lens or any complicated device like microscope or
telescope usually makes things bigger and helps us to see in a more detailed manner.
The microscope: A microscope is an instrument that makes an magnify image of a
small object.
Types of microscope
Simple microscope: A simple magnifier or microscope is a converging lens of small focal
length carrying handle.
Through simple microscope an erect, magnified and virtual image of the object
can be seen.
The image formed here at two different position, one is at least distance of
distinct vision i.e. D=25cm, and at infinity. In both the case magnification and comfort to
see the image is different.
Least Distance of Distinct vision(D): The minimum distance of an object from eye to
have its clear image is called "Least Distance of Distinct Vision". This distance for normal
human eye is about 25 cm from eye.

1)Magnification of simple microscope when the image is formed at LDDV i.e D=v
2)Magnification of simple microscope when the image is formed at infinity.
Application of simple microscope:
1)It is used in watch makers and jewelers for fine work. It is also used in forensic lab to
examine fingerprints and palm lines.
2)It is also used to read the reading on vernier scale by student.
The linear magnification m, for the
image formed at the near point D, by
a simple microscope can be
obtained by using the relation
𝑚 = (1 +
𝐷
𝑓
)
Where f is focal length of lens
The magnification is given by
𝑚 =
𝐷
𝑓
Any simple microscope has a limit
of magnification ≤ 𝟗 𝒕𝒊𝒎𝒆𝒔. To get
the larger magnification we need
compound microscope made of two
or more lense.

COMPOUND MICROSCOPE
A microscope that consists of two lenses in series, the first serving as the ocular
lens ,and the second act as the objective lens .
Construction-A compound microscope consists of two convex lenses. The lens near to
the object is called an objective having small focal length(𝑓
𝑜). The lens near to the eye of
an observer is called as eyepiece, having large focal length(𝑓
𝑒) compare to objective lens
.This combination of lens forms a magnified lenses forms a magnified image of object.
Working-The object AB is placed at a distance slightly more than focal length of the
objective so that it’s real, inverted & magnified image A’B’ is obtained beyond the centre
of Curvature (C) of the objective lens. The image A’B’ becomes an virtual object for the
eyepiece. The position of image A’B’ is adjusted such that it lies within the focal length of
eyepiece. The eyepiece piece forms a virtual, erect & magnified image A’’B’’ of the object.
Thus, The final image formed by a compound microscope is virtual, inverted
& magnified behind the object
1)Magnification of compound microscope when the image is formed at LDDV i.e D=v
𝑚 =
𝐿
𝑓
𝑜
(1 +
𝐷
𝑓
𝑒
)
1)Magnification of compound microscope when the image is formed at LDDV i.e D=v
𝑚 =
𝐿
𝑓
𝑜
(
𝐷
𝑓
𝑒
)

TELESCOPE
Telescope is an optical instrument that makes distant objects appear magnified by
using an arrangement of lenses or curved mirrors.
It also has an objective and an eyepiece. But here, the objective has a large focal
length and a much larger aperture than the eyepiece.
Light from a distant object enters the objective and a real image is formed in the
tube at its second focal point. The eyepiece magnifies this image producing a final
inverted image.
Types of Telescope
1)Refractor telescope :Refracting telescopes can be used both for terrestrial and
astronomical observations. The refractor telescope uses a lens to gather and focus light.
Advantages
1. Refractor telescopes are rugged.
2. The glass surface inside the tube is sealed from the atmosphere so it rarely needs
cleaning.
3. Since the tube is closed off from the outside, air currents and effects due to changing
temperatures are eliminated.
4. This means that the images are steadier and sharper than those from a reflector
telescope of the same size.
Final image Magnification of refracting telescope is given by,
𝒎 =
𝒇𝒐
𝒇𝒆

2)Reflector telescope: A reflector telescope uses a mirror as its objective. The mirror is
close to the rear of the telescope and light is bounced off (reflected) as it strikes the
mirror. This is also known as a Cassegrain telescope.
It has the advantages of a large focal length in a short telescope. The largest
telescope in India is in Kavalur, Tamil Nadu. It is a 2.34 m diameter reflecting telescope
(Cassegrain). It was ground, polished, set up, and is being used by the Indian Institute of
Astrophysics, Bangalore. The largest reflecting telescopes in the world are the pair of
Keck telescopes in Hawaii, USA, with a reflector of 10 metre in diameter.
Advantages: Reflector telescopes do not suffer from chromatic aberration because all
wavelengths will reflect off the mirror in the same way. It can be build in bigger size.
Reflector telescopes are cheaper to make than refractors of the same size.
How telescope meakes an image of large distance object

09.Ray optics.pdf

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