CHAPTER 6 GEOMETRICAL OPTICS

Chapter 6
Geometrical Optics

[ 10 Hours ]

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Subtopics

6.1 Reflection at a spherical surface
6.2 Thin Lenses

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Geometric Optics

The study of light propagation and is
concerned with the laws controlling the
reflection and refraction
of rays of light

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-- Assumptions made :
1. Light travels in a fixed direction
in a straight line as it passes through a
uniform medium.
2. It changes its direction when it
meets the surface of a different
medium.

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6.1 Reflection at a spherical surface (mirror)

Reflection of light
• is defined as the return of all or part of a beam of

light when it encounters the boundary between two
media.

Law of Reflection

i=r The incident ray, the reflected
ray & the normal all lie in the
same plane.

Angle of reflection, r equals
the angle of incidence, i

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Rays reflection from a mirror or other shiny surfaces
obeying the law of reflection.
Reflecting surfaces do not have to be flat.
Most common curved mirrors are spherical
Spherical mirror is a reflecting surface with spherical
geometry.

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There are two types of spherical mirror. There are convex
mirror and concave mirror.

Figures (a) and (b) show the shape of concave and convex mirrors.

(a) Concave mirror (b) Convex mirror
(Converging type) (Diverging type)

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1. Center of curvature, C 8
-- the center of the sphere of which a curved
mirror forms a part.

2. vertex, V
-- the point at the center of the mirror

3. Principal axis of the mirror
-- a straight line drawn through C & V

4. Focal point, F
-- a point at midway between C & V

5. Radius of curvature , r
-- distance from the vertex, V to the center of
curvature, C.

6. Focal length, f

-- distance from the vertex of the mirror to the focal
point.

Focal length for a spherical mirror of radius r

r = 2 f OR f =r
2

7. Object distance, u

-- distance from the vertex of the mirror to object.

8. Image distance, v

-- distance from the vertex of the mirror to image. 9

Concave mirror – After reflection from the
mirror, parallel incident rays converge (come
together) at the focal point , F.

Also known as converging mirror

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Convex mirror -- Parallel incident rays diverge after
reflection as if they had originated from a focal
point, F behind the mirror as indicated by the
dashed lines.

Also known as diverging mirror

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Check Your Conceptual

Knowledge

Suppose you would like to use the Sun to start a
fire in the wilderness. Which type of mirror,
concave or convex would work best ? Give your
reason.

Answer : Concave mirror

Reason :

Rays from Sun are parallel.

Concave mirror will converge the rays together

at focal point where it stands a better chance of

starting a fire. 12

Ray Diagrams for mirror

- To find the orientation, size
& location of an image

To draw ray diagram, we need to know :
1. the position of the object
2. the location of the mirror’s focal point, F & the

center of curvature, C.

3 rays are drawn to locate the image. All of the rays
start from the same object point.

The intersection of any two of these rays locates the
image. 3rd ray serves as a check of the construction.

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1st ray : drawn from the top of the object
parallel to the principal axis & is
reflected through the focal point F.

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14

2nd ray : drawn from the top of the object
through the focal point F & is
reflected parallel to the principal
axis.

2
2

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3rd ray : drawn from the top of the object
through the center of curvature C &
is reflected back along its incoming
path.

3

3

16

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Images formed in a concave mirror for different u

Object Ray diagram Image
characteristic
distance, u CI V
OF back RID
u>r  Real
Front  Inverted
 Diminished
 Formed
between point
C and F.

u=r O V RIS
back
F  Real
C  Inverted
 Same size
I  Formed at point

Front C.

18 17

Object Ray diagram Image
characteristic
distance, u IC V
OF RIM
f<u<r  Real
Front back  Inverted
u=f  Magnified
O  Formed at a

distance greater
than CV.

IN

 Formed at
infinity.

CF V

Front back 19 18

Object Ray diagram Image
characteristic
distance, u F
VUM
u<f C OV  Virtual
 Upright
 Magnified
 Formed at the

back of the
mirror

I

Front back

Mirrors used when applying makeup and mirror

used by dentist to observe the teeth of a patient

are concave mirror; in use, the distance from the

face or teeth to the mirror is less than focal

length and an enlarged, erect (upright) image is

seen. 20

Application of Concave Mirror : Enlargement mirror & reflector

Dental Mirror Make-up Mirror Shaving Mirror

Car Headlight

Concave reflectors are used in car headlights. The bulb
of the head light is placed at the focal point of the
reflector. The reflected light emerges in a parallel beam
and gives more concentrated visibility to the driver at 21
night.

Parabolic Dish

Concave mirrors are used in solar powered gadgets. The parallel rays of the sun
are reflected to focus at the focal point F. The solar energy concentrated at F is
then used or converted (for example into electrical energy by a solar cell) by the
gadget

Look at the mirror! A
shinny and smooth
surfaced mirror act as a
good reflector

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Images formed in a convex mirror

V Use broken lines For any position in front of
for any rays or
O image behind the convex mirror, the
mirror
u image formed are
IF o virtual
front o upright
v
C odiminished (smaller
back
than the object)
oformed at the back of

the mirror

VUM

This type of mirror is often used 24
in stores to foil shoplifters. A
single mirror can be used to
survey a large field of view
because it forms a smaller image
of the interior of the store.

Applications

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The easy way to memorize the characteristics
of image formed by spherical mirror

Concave

u> 2f RIDuan (Real, Inverted, Diminish)

u=2f RISau (Real, Inverted, Same size)

u<2f RIMau ( Real, Inverted, Magnified)

u=f Intai ( Infinity)

u<f VUMuttu (Virtual, Upright,Magnified)

Convex
u any position VUD (Virtual,Upright,Diminish)

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Real Image Virtual Image

formed when light rays formed when the light rays

pass through & diverge do not pass through the

from the image point. image point but appear to

diverge from that point.

(Light reflected and form image in (Light ‘somehow appear’ to pass

the same region with the object as it through a mirror to form an image

is supposed to) inside the mirror)

can be displayed on a cannot be displayed on a
screen. screen.

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Check Your Conceptual Knowledge

(1) How are the virtual images produced by a

concave mirror and convex mirror

differentiated? Answer :

Virtual image

Convex Concave formed by convex
mirror are

diminished

(reduced in size)

but the virtual

image formed by

concave mirror at

u<f is magnified

(enlarged). 28

EXAMPLE

F
CO

Figure 1 shows an object O placed in front of a concave

mirror where C and F are center of curvature and focal

point.

(i) Copy Figure 1 and construct the ray diagram for

formation of the image.

(ii) State the characteristics of the image. 29

2 rays with arrow (incident rays and reflected rays) ..J1
Interception of 2 rays produce virtual image …J1

SOLUTIONLabel & correct position of C, F, O, I …J1

C FO I

The image characteristics: 30
(1) Virtual
(2) Upright
(3) Magnified

Spherical Mirror Equation

Used to calculate the image distance v
from a knowledge of the object distance,
u and radius of curvature r or focal
length f.



1+1= 1 where f =r
uv f 2

u : Object distance 31
v : Image distance
f : Focal length
r : radius of curvature for the mirror

Linear Magnification

m = − image distance, v = height of image, hi
object distance, u height of object, ho

If m > 1.0, image is larger than object or magnified
If m < 1.0, image is smaller than object or diminished
If m = 1.0, image is same size as the object
If m = +ve, image is upright
If m = -ve, image is inverted

Example: if m= -2.5 it means that the
image is inverted (negative value),
magnified (bigger than 1) and real (v is
positive)

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Quantities + –

u In front of mirror Back of the mirror
(real object ) (virtual object )
v Back of the mirror
In front of mirror (virtual image)
( real image )
Convex mirror
r , f Concave mirror
inverted
m upright

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Example

Assume that a certain spherical mirror has a focal
length of + 10.0 cm. Locate & describe the image for
object distances of

(a) 25.0 cm
(b) 5.0 cm

Solution

Focal length, f is + ve → concave mirror

(a) Given : u = + 25.0 cm = + 0.25 m
f = + 10 cm = + 0.1 m

from : 1 + 1 = 1 34
uv f

1 +1= 1 35
0.25 v 0.1

1 =6
v
v = 0.167 m

Magnification,

m = − v = − 0.167 = − 0 .6 6 8
u 0.25

m < 1.0 → image is smaller than the object
(Diminished)

+ v → Image is Real, Inverted

Answer for (b) :
v = – 0.1 m ; m = 2.0
Image is virtual, located behind the
mirror, upright, enlarged

Example

A woman who is 1.5 m tall is located 3.0 m
from an anti shoplifting mirror. The focal
length of the mirror is – 0.25 m. Find :
(a) the position of her image
(b) the magnification
(c) the height of the image.

Given : ho = 1.5 m , u = 3.0 m ; f = –0.25 m 36

(a) from : 1 + 1 = 1
uv f
1 +1= 1

+ 3.0 v − 0.25

1 = − 13  v = − 0 .2 3 1 m
v3

(b) Magnification, m = − v = − (- 0.231) = 0 .0 7 7

u 3.0

(c) from : m = hi  hi = m h0 = 0.077 (1.5)
ho h i = 0 .1 2 m

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Example

A pill bottle 3.0 cm tall is placed 12.0 cm in front of a
mirror. A 9.0 cm tall upright image is formed.
(a) The mirror is convex, concave or flat ? Why ?
(b) What is its radius of curvature ?

Solution

Given : ho = 3.0 cm , hi = 9.0 cm
u = 12.0 cm, Upright → virtual image

from : m = hi = 9 = 3
ho 3

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Image is enlarged & upright → concave mirror
Reason:
convex mirror → image always reduced in
size.
Flat mirror → image is same size.
Only concave mirror formed enlarged,

upright image when u < f

(b) Magnification :

m=−v 39
u

3=− v
12

v = −36 cm

from : 1 + 1 = 1
uv f

1+ 1 =1
12 − 36 f

1= 2
f 36
f = 18 cm

from : f = r  r = 2 f
2
= 2 (1 8 )
r = 36 cm

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Follow Up exercise

1. A candle with flame 1.5 cm tall is placed 5 cm from the front of a concave
mirror. A virtual image is produced that is 10 cm from the vertex of the
mirror.
(a) Find the focal length & radius of curvature of the mirror.
(b) How tall is the image of the flame ?

Ans: (a) 10 cm ; (b) 3 cm

2. The erect image of an object 18 cm in front of a mirror is half the size of
the object. (a) mirror is convex, concave or plane ? Why ? (b) What is
the focal length of the mirror ?

Ans: (b) –18 cm

3. A dentist uses a small mirror attached to a thin rod to examine one of
your teeth. When the tooth is 1.20 cm in front of the mirror, the image
it forms is 9.25 cm behind the mirror. Find (a) focal length of the mirror,
(b) the magnification of the image.

Ans: (a) 1.38 cm ; (b) 7.71

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4. A boy 1.20 m tall sees his image in a shining ball hanging from a wall.
The ball is 8.50 cm in diameter and 1.10 m away from the boy as
shown in Figure.

u

ho

(a) Where is the image of the boy relative to the surface of the 42
globe?

(b) How tall is the boy’s image?
(c) State the characteristics of the boy’s image.

Ans: (a) –2.085 cm ; (b) 2.275 cm

6.2 Thin Lenses

Lens – an optical system with 2 refracting surfaces.

Thin Lens – simplest lens having 2 spherical surfaces

close enough together that we can neglect the

thickness of the lens. 43

2 types of thin lenses :
(1) Converging or convex lenses
-- a lens which converges rays of light.

-- thicker across their middle.

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(2) Diverging or concave lenses
-- a lens which diverges rays of light.

-- thinner across their middle.

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A lens has 2 spherical surfaces with radii of
curvature r1 & r2 .

OO

Principal axis – a line through the center of curvature
C1 & C2 .

Center of the lens , O – geometric center of a lens

where all rays pass through this point will continue

in a straight line. 46

Focal point , F – a point where the beam of parallel
rays incident on the lens either
(a)converge after refraction for a convex lens or
(b)diverge after refraction for a concave lens.

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Focal length, f – distance between focal point , F &

center of the lens, O. 48

Ray Diagrams for Thin Lenses
To locate image of a converging lens, 2 of 3
following rays are drawn from the top of the object :

-- through the center of the
lens & continue in straight
line.

-- Parallel to principal axis,
after refracted, passes
through F

-- through F on the front

side & emerges from the

lens parallel to principal

axis. 49

Images formed by a convex (converging) lens
• Table shows the ray diagrams of locating an image

formed by a convex lens for various object distance, u.

Object Ray diagram Image
characteristic
distance, u F1 I
Front  Real
u > 2f O2F1 F2 2F2  Inverted
 Diminished
back  Formed between

point F2 and 2F2.
(at the back of the
lens)

RID

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