Geometrical OpticsLecture:1 Introduction

Optics

Optics is the study of properties and nature of light and vision.

The external world is revealed to the eye by means of light.

What is light?

Light it is a form of energy, which produces sensation of sight.

We treat light as a ray while studying it interaction with moderate size objects, as electromagnetic wave while studying its interaction with small size objects and as photon while studying its interaction with subatomic particle. Accordingly, the subject is divided into three parts.

Geometrical Optics

In geometrical optics, light is treated as ray.

Geometrical optics primarily deals with phenomena like shadow formation and image formation due to reflection and refraction by various devices like mirrors and lenses. It also provides us with

All the laws of geometrical optics are derived from experiments.

Study of geometrical optics concerns with making use of these laws to explain various effects as shadow formation, image formation etc.

Ray and Beam Straight line path on which light travels is called a ray and a bunch or bundle of rays is called a beam.

Source Source is a body, which emits light. It may be a self-luminous body as the sun, a light bulb or a candle-flame or it may be a body which reflects light fallen upon as the moon.

Point source

It is a point object, which sends light either by its own or by reflection. It is an idealized concept because in practice we do not have point objects. But extremely distant sources as sun, stars etc can be treated as point sources on the earth.

Extended Source

All the practical sources, which we encounter in everyday life, have finite size. They are known as extended source.

They can be considered as infinitely large number of point sources placed adjacent to each other.

Parallel Beam Convergent beam Divergent beam

Has constant width

It spreads It shrinks

Types of Beam

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Isotropic and anisotropic Source

A source, which emits light, equally in all directions, is known as isotropic source and a source, which emits light in different amount in different directions, is known as anisotropic source.

Sun is an isotropic source and torch is an anisotropic source.

Transparent, opaque and translucent materials In general when light falls on a material body some amount is reflected back, some amount of light that enter the body is absorbed and remaining is transmitted through the body.

Transparent Materials

A substance such as air, clear water or glass, through which whole of the falling light is transmitted is known as perfectly transparent material. No amount of light is absorbed within a perfectly transparent material. We can see clearly through them.

No material is perfectly transparent but we generally use the term transparent for perfectly transparent material.

Opaque Materials

Through opaque materials, no light is transmitted. We cannot see through opaque material at all. Common examples of opaque materials are wood, iron etc.

Translucent Materials

Through translucent material some light is absorbed and transmitted light follows zigzag path within the material due scattering from particles of the medium. Through translucent materials, we cannot see clearly. In fact, we can feel presence of an object trough a translucent material but cannot clearly recognize it. Common examples are butter paper, ground glass, muddy water etc.

Optical medium An optical medium is an empty space or space filled with transparent material. In geometrical optics, it is generally assumed that the media are not only homogeneous and isotropic, but perfectly transparent as well.

A homogenous and isotropic medium possesses the same optical properties everywhere in all directions. Vacuum, air, glass, water and are the very common examples.

Boundary It is interface between two optical media.

Fundamental Laws of Geometrical Optics

Geometrical optics is based on the following five laws. These laws are derived from experience. In geometrical optics we do not account for these laws but assuming them true make use of them.

I. Rectilinear propagation of lightII. Mutual independence of raysIII. Law of optical reversibilityIV. Laws of reflection

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V. Laws of refraction

Rectilinear propagation of light

In a homogeneous isotropic medium light travels in straight-line path. Rectilinear propagation explains shadow formation, eclipses of the sun and the moon etc.

Example 1. Find the length of shadow of a vertical pole of height h due sunrays falling on the ground at angle with the horizontal.

Ans.

Mutual independence of rays

Light rays do not disturb each other upper intersection.

Law of optical reversibility

Rays retraces their path when their direction in reversed.

Lecture:2 Reflection and Its laws

When light falls on boundary between two media, some of its amount is turned back. This phenomenon is known as reflection and the boundary is known as reflecting boundary of surface. Falling light is known as incident light and the light that is turned back is known as reflected light.

For good reflecting surfaces almost all, the light is reflected back. When the entire incident light is reflected, the reflection is known as perfect reflection and the reflecting surface as perfectly reflecting surface. In geometrical optics, the term reflection is used for perfect reflection.

Regular and diffused reflections

Diffused reflection Reflection from rough surfaces like wall, wood etc is known regular or specular reflection.

In surfaces capable of exhibiting diffused reflection, we cannot see images but we can see the surface.

Regular reflection Reflection from smooth surfaces like mirrors is known regular or specular reflection.

In surfaces capable of exhibiting regular reflection, we can see images but we cannot see the surface.

Diffused reflection

Incident light

Reflected light

Reflecting surface

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Laws of Reflection

Ist Law: Angle of incidence is equal to angle of reflection.

i = r IInd Law: Incident ray, reflected and normal lies in same

plane

Note

● Laws are valid for curved surface as well.

● None of wavelength, frequency and speed changes due to reflection.

● Intensity or amplitude decreases.

Smaller is the angle of incidence, greater is the reduction in intensity. This effect we usually neglect in geometrical optics.

Example 2. A ray of light is incident on a horizontal reflecting floor. Find height of light spot made on the wall by the reflected light.

Ans.

Example 3. A ray of light is incident on a horizontal plane mirror M1. After first reflection at M1, the ray falls on the vertical mirror M2.

(a) What is the angle of incidence on M2?(b) In which direction ray would go after reflection from

M2.

Ans. (a) (b) Antiparallel to initial incident ray

Example 4. Initial incident ray is parallel to the mirror M2.

(a) Find angle of incidence on mirror M1.(b) Find angle for normal incidence on M2.(c) What will happen, if incidence on M2 is normal?

Ans. (a) (b) 45º (c) Ray will retrace itself after the second reflection

Example 5. Initial incident ray is on mirror M1.

Regular reflection

Incident light Reflected light

Reflecting surface

4

i r i r

i r

Incident Ray Reflected Ray

Normal

Incident Ray

d

d

M1

M2

M2

M1

M1

M2

i1

(a) Find expression for angle of incidence on mirror M2. Consider the cases , and

.(b) If the ray retraces itself after reflection from M2, find expression for angle of incidence i1 on

M1.

Ans. (a) , and (b)

Example 6. A plane mirror is placed as shown in the figure. If a ray incident parallel to the x-axis becomes vertical after reflection, find angle , which the mirror makes with the x-axis.

Ans. 45º

Example 7. A cylindrical reflecting surface is placed with its axis along the z-axis. The principle plane of the cylinder is a circle . A ray incident parallel to the x-axis becomes vertical after reflection.

(a) Find slope of the normal at the point of incidence.(b) Find slope of the tangent to the circle at the point of

incidence.(c) Find coordinates of the point of incidence.

Ans. (a) (b) (c)

Lecture:3

DeviationRotation of Plane Mirror

Deviation Change in direction of ray after reflection.

It is usually the smaller angle.

Normal incident (i = 0) max = 180º

Grazing incident (i → 90º) min = 0º

Total deviation after multiple reflections

Total deviation after several reflections equals to algebraic sum of deviations after every reflection.

Slope

i

5

x

y

x

y

i

Example 1. Initial incident ray is on mirror M1. Find conditions for clockwise and anticlockwise deviations after reflection at M2.

Ans. Clockwise 2

Anticlockwise 2

Note If ray turns towards the intersection point

If ray turns away from the intersection point

Example 2. Initial incident ray is on mirror M1 at angle .

(a) Decide directions of deviations at both the mirrors(b) Find deviations after every reflection.(c) Find total deviation.

Ans. (a) M1: Clockwise M2: Clockwise

(b) (c)

Example 3. Initial incident ray is on mirror M1 at angle .

(a) Decide directions of deviations at both the mirrors(b) Find deviations after every reflection.(c) Find total deviation.

Ans. (a) M1: Clockwise M2: Anticlockwise

(b) (c)

Rotation of Mirror If mirror is rotated by angle keeping incident ray fixed, the reflected ray rotates by double angle 2 in same sense as that of rotation of mirror.

Rotation of Incident Ray If incident ray is rotated by angle keeping the mirror fixed, the reflected ray rotates by angle in the opposite sense as that of rotation of the incident ray.

Example 4. If a plane mirror is rotated by angle in clockwise sense, by how much angle the incident ray must be rotated to keep reflected ray unchanged?

Ans. 2 clockwise

Example 5. (Exercise S-2, Q-1)

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M1

M2

i1

M1

M2

i1

M1

M2

i1

Original

New

2

OriginalNew

x

M2

The mirror is rotating at constant angular velocity keeping the incident ray fixed. When the mirror is perpendicular to the wall, find speed of light spot formed on the wall.

Ans. ↑

Lecture: 4

Vector form of laws of reflection

Formation of Images

Field of view

Vector form of Laws of Reflection

Example 1. A ray incident in the direction of vector is reflected by mirror placed in the x-y

plane. Find unit vector in the direction of the reflected ray.

Ans.

Example 2. A cylindrical reflecting surface is placed with its axis along the z-axis. The principle plane of the cylinder is a circle . A ray is incident parallel to the x-axis at the point (4, 3).

(d) Find unit vector .(e) Find unit vector in the direction of the reflected ray.(f) Find coordinates of the point of incidence.

Ans. (a) (b)

Example 3. A ray of light is incident on a horizontal plane mirror M1. After first reflection at M1, the ray falls on the vertical mirror M2.

(a) Express unit vector in the direction of the incidence ray.

(b) Express unit vector in the direction of reflected ray after reflection form M1.

(c) Express unit vector in the direction of reflected ray after reflection form M2.

(d) What regularity do you observe regarding components parallel and normal to the mirror in reflection?

Ans. (a) (b) (c)

(d) Component parallel to the mirror remains unchanged and the component normal to the mirror is reversed.

Image formation by Plane Mirror

Example 1. Consider the point objects A and B.

(a) Can the mirror make images of both the objects?(b) Locate their images.(c) Find regions from where one can see image of A only.(d) Find regions from where one can see image of B only.(e) Find regions from where one can see images of both the objects A and B.

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d

M1

M2y

x

y

xx

Incident Ray Reflected Ray

Normal

ie re

ne

x

y

A

B

(f) Find the region in front of the mirror from no one can see either of the images.

Ans. (a) Yes (b) Images of A and B

(c, d, e, f)

Lecture: 5

Formation of Images

Field of view

Multiple images in parallel mirrors and inclined mirrors

Formation of Images and Field of view

Example 1. Find coordinates of the image of point P for different orientation of the plane mirror shown in the figure.

(a) (b) (c)

(d) (e) (f)

A

B B’

A’

B’

A’

FOV of B’ only

FOV of A’ only

FOV of both A’ and B’

FOV of neither A’ nor B’

x

y

(a, b)

P

x

y

(a, b)

P

x

y

(a, a)

P

45º

x

y

(a, 0)

P

30º

x

y

(a, 0)

P

60º

x

y

P

r

8

Ans. (a) (a, b) (b) (a, b) (c) (a, b)

(d) (e) (f)

Example 2. Consider the extended objects in each of the following cases and draw their images.

(a) (b) (c)

(d) (e) (f)

Example 3. Find positions between which an insect moving on the x-axis can see its image in the mirror. The mirror has square shape of side 100 cm.

(a) (b)

Ans. (a) 50 cm to (50 + 100√2) cm (b) 50√3 cm to (50√3 + 200) cm

Example 4. A man is walking on the y-axis parallel to a plane mirror M of length l. A point object P is symmetrically placed in front of the mirror at a distance d on the x-axis. Find the positions between which the man can see the image of the object.

Ans. Between the points (0, 2l) and (0, 2l)

Example 5. Find the lowest and the highest eye levels of a man standing on A to see image of a point P on the ground in the plane mirror mounted on a wall. All dimensions are in centimeters.

Ans. 75 cm, 105 cm

Example 6. A point light source P is placed on the axis of a circular mirror of radius r. The mirror is mounted on a wall. Find radius of light spot obtained on the front wall.

Ans. 4r

Example 7. Minimum size of mirror

Find minimum size of mirror to see full image of him.

45º

45º

x

y

45º

50

x

y

30º

50

9

M

25P

10

50

A

50

M

3d

d

P

O

My

3d

dl

P

Ox

Ans. Vertical length of the mirror must be half of the height of the man.

Example 8. Position of the mirror to see full image

A man of height 1.8 m wants to use a mirror of minimum size to see his full image. If his has eyes are 1.7 m above the ground, what should he position the bottom of the mirror?

Ans. 0.85 m above the ground

Example 9. A man of height h is standing at the center of a room of height 2h and a mirror is installed on the front wall.

(a) To see full height of the back wall, what should be minimum size of the mirror?(b) To see full height of the back wall as well as full image of him, what should be minimum

size of the mirror?

Ans. (a) h/3 (b) 5h/6

Multiple images in parallel mirrors

A point object O is placed between two plane mirrors M1 and M2 as shown.

In mirror M1 first image is at distance 2a from O then separation between successive images

2a, 2b, 2a, 2b …………. ∞

In mirror M2 first image is at distance 2b from O then separation between successive images

2b, 2a, 2b, 2a …………. ∞

Example 1. A point object is placed between two parallel mirrors as shown in the figure.

(a) Find distance of 2nd image in M1 from O.(b) Find distance of 2nd image in M2 from O.(c) Find distance of 5th image in M1 from O.(d) Find distance between 3rd image in M1 and 4th image in M2.

Ans. (a) 60 cm (b) 60 cm (c) 140 cm (d) 200 cm

Lecture: 6

Multiple images in inclined mirrors

Motion of object and Mirror

Multiple images in inclined mirrors

A point object S is placed between two plane mirrors M1 and M2 as shown.

All the images are on a circle of radius OS and center at O

Angular position of the images from the object

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O

a b

M1 M2

M1

M2

O

S

O

10 20

M1 M2

Angular positions of images made by M1 (In clockwise direction)

, , , ….. till

Angular positions of images made by M2 (In anticlockwise direction)

, , , ……till

Example 1. A point object is placed between two mirrors inclined at angle 60º as shown in the figure.

(a) Find angular positions of all the images made by M1.(b) Find angular positions of all the images made by M2.(c) Do the last two images coincide?(d) Find total number of images.

Ans. (a) 60º, 120º, 180º

(b) 180º, 240º, 300º

(c) Yes(d) 5

Example 2. A point object is placed between two mirrors inclined at angle 60º as shown in the figure.

(a) Find angular positions of images made by M1.(b) Find angular positions of images made by M2.(c) Do the last two images coincide?(d) Find total number of images.

Ans. (a) 40º, 120º, 160º(b) 80º, 120º, 200º(c) Yes(d) 5

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M1

M2

40ºO

S20º

M1

M2

30ºO

S

30º

Q1

P2

P3

P1

Q2

Q3

S

30º

M2

M1

90º

30º

150º

210º 30

º

90º

150º

P1

Q2

P3

Q1

P2

Q3

S

40º

M1

M2

20º

80º

160º

200º

40º

80º

160º

Number of total images

Example 3. Find total number of images, if angle between the mirrors is 30º, 45º, 60º, 72º, 75º, 90º, 120º.

Ans.

Example 4. In which condition do the last two images

coincide?

Ans. is even

is odd and the object is symmetrically placed.

Example 5. A ray reflected successively from two mirrors inclined at certain angle undergoes a deviation of 240º. The after second reflection deviates away from intersection of the mirrors.

(a) Find the angle between the mirrors.(b) Find the total number of images of an object placed between the mirrors.

Ans. (a) 60º (b) 5

Motion of object and mirror

Example 1. A point object is moving in front of a fixed plane mirror as shown in the figure. Find velocity of its image with respect to the ground and with respect to the object.

Ans. ,

Example 2. Find velocity of the image relative to the ground and relative to the mirror.

Position of object No. of images

Even Any where Add Symmetric

Symmetric nFraction Anywhere Integral Part

Number of ImagesSymmetrical Object Asymmetrical

Object30º 12 11 1145º 8 7 750º 7.2 7 760º 6 5 572º 5 4 575º 4.8 4 490º 4 3 3110

º3.27 3 3

120º

3 2 3

12

3 m/s

2 m/s

vo

y

x

y

xx

Ans. m/s towards the left, m/s towards the left

Example 3. Find velocity of the image relative to the object.

Ans. 12 m/s towards the right

Example 4. Find velocity of the image relative to the ground.

Ans.

Example 5. A point object is moving with velocity in front of a

plane mirror that is moving with velocity as shown in the figure.

(a) Find velocity of the image.(b) Find speed of approach between the object and

the image.

Ans. (a) (b)

Example 6. A point object is moving with uniform velocity before an arrangement of two mirrors as shown in the figure.

(a) Find speed of image made by M1.(b) Find speed of image made by M2.(c) Find speed of its image in mirror M1 relative to its

image in mirror M2.

Ans. (a) vo towards right (b) along normal of M2

13

3 m/s5 m/s

37º

y

x

y

xx

vo

vm

y

x

y

xx

vo

M1

M2