Chapter 7: Geometrical

Optics

The branch of physics which studies the properties of

light using the ray model of light.

Overview Geometrical

Optics

Spherical

Mirror Thin Lens Refraction

Ray Diagram

vuf

111

2

rf and

Snell’s Law

r

nn

v

n

u

n 1221

Ray Diagram Thin lens

equation

&

Lens maker’s

equation

7.1 Reflection at a Spherical

Surface

State the law of reflection

Sketch and use ray diagrams to determine the

characteristics of image formed by spherical

mirrors

Use for real object only

Learning Objectives

vuf

111

Law of Reflection

The incident ray, the reflected ray and the normal all lie

in the same plane.

The angle of incidence, i equal the angle of reflection, r

as shown in figure below.

Normal

Reflection at a plane surface

Objecti

v

i

rr

u

Image

ihoh

heightobject :oh

height image :ih

distance image :v

distanceobject :u

A 'A

i

u vi

r

i

Object

Point object

Vertical (extended) object

Reflection at a plane surface

Reflection at a Spherical Surface A spherical mirror is a reflecting surface with spherical

geometry.

Two types:

Convex, if the reflection takes place on the outer

surface of the spherical shape.

Concave, if the reflecting surface is on the inner

surface of the sphere.

Ray Diagrams for Spherical

Mirrors

Ray diagram is defined as the simple graphical method

to indicate the positions of the object and image in a

system of mirrors or lenses.

For concave mirror – Focus point, F

is defined as a point where the

incident parallel rays converge

after reflection on the mirror.

For convex mirror – Focus point, F

is defined as a point where the

incident parallel rays seem to

diverge from a point behind the

mirror after reflection.

Mirror – Approximation

Width of the mirror is smaller to the curvature of the

mirror

Reflected rays make a small angle to the incident rays

All rays cross each other at nearly a single point

C

C

Front back

F P

θ

Mirror – How to draw

When draw the Ray Diagram, adjust

the size of the image so that it does

not go over the “border” line.

The position of the image

has shift significantly

When draw the mirror, set the

“border” line first, use straight

line for the mirror within the

“border” line

Ray Diagram for Concave Mirror

Ray 1

A ray parallel to the

principal axis passes

through or diverges

from the focal point F

after reflection.

Ray 2

A ray which passes

through or is directed

towards the focal

point F is reflected as

a ray parallel to the

principal axis.

Ray Diagram for Concave Mirror

Ray 3

A ray which passes

through or is directed

towards the centre of

curvature C will be

reflected back along

the same path.

At least any two

rays for drawing

the ray diagram.

Image Formed by Concave

Mirror

The characteristics of the images formed by concave

mirror depends on object’s location.

The relationship between the object distance and object

size and the image distance and image size are depicted in

the diagram below:

Object

Image

Ray Diagram for Convex Mirror

Ray 1

A ray parallel to the

principal axis passes

through or diverges

from the focal point F

after reflection.

Ray 2

A ray which passes

through or is directed

towards the focal

point F is reflected as

a ray parallel to the

principal axis.

Ray Diagram for Convex Mirror

Ray 3

A ray which passes

through or is directed

towards the centre of

curvature C will be

reflected back along

the same path.

At least any two

rays for drawing

the ray diagram.

Image Formed by Convex Mirror

Unlike concave mirrors, convex mirrors always produce

images that share same characteristics:

virtual

upright

diminished (smaller than the object)

formed at the back of the mirror (behind the mirror)

As the object distance is decreased, the image distance is

decreased and the image size is increased.

Convex mirror always being used as a driving mirror

because it has a wide field of view and providing an

upright image.

How to Describe an Image?

L represents the relative location

O represents the orientation (either upright or inverted)

S represents the relative size (either magnified, diminished or the same size as the object)

T represents the type of image (either real or virtual).

The Mirror Equation

Spherical mirror’s equation:

for spherical mirror (only):

Therefore the equation can also be written as:

2

rf

vur

1

12

vuf

1

11 Real object only

Linear Magnification, m

Linear magnification of the spherical mirror, m is defined

as the ratio between image height, hi and object height,

ho.

m is a positive value if the image formed is upright and it

is negative if the image formed is inverted.

Height, h is a positive value if the image formed is

upright and it is negative if the image formed is inverted.

u

v

h

hm

o

i

Sign Convention for Spherical

Mirror’s Equation:

Physical Quantity Positive sign (+) Negative sign (-)

Object Distance, u Real object

(in front of the mirror)

Virtual object

(at the back of the mirror)

Image Distance, v Real image

(same side of the object)

Virtual image

(Opposite side of the object)

Focal length, f Concave mirror Convex mirror

Linear magnification, m Upright image Inverted image

Object/ Image Height, h Upright image Inverted image

VERY Important!!

(and r)

Example 1

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. Determine

a. the focal length of the mirror and state the type of the

mirror used

b. the magnification of the image

Example 1 – Solution

Example 2

An upright image is formed 20.5 cm from the real object

by using the spherical mirror. The image’s height is one

fourth of object’s height.

a. Where should the mirror be placed relative to the

object?

b. Calculate the radius of curvature of the mirror and

describe the type of mirror required.

c. Sketch and label a ray diagram to show the formation

of the image

Example 2 – Solution

Example 2 – Solution

Example 2 – Solution

Example 3

A mirror on the passenger side of your car is convex and

has a radius of curvature 20.0 cm. Another car is seen in

this side mirror and is 11.0 m in front of the mirror (behind

your car). If this car is 1.5 m tall, calculate the height of

the car image.

Example 3 – Solution

Example 4

A person of 1.60 m height stands 0.60 m from a surface of

a hanging shiny globe in a garden.

a. If the diameter of the globe is 18 cm, where is the

image of the person relative to the surface of the

globe?

b. How tall is the person’s image?

c. State the characteristics of the person’s image.

Example 4 – Solution

Example 4 – Solution

7.2 Refraction at a Plane and

Spherical Surface

State and use the laws of refraction (Snell’s

Law) for layers of materials with different

densities.

Use for spherical surface.

Learning Objectives

r

nn

v

n

u

n 1221

Refraction Refraction is defined as the changing of direction of a

light ray and its speed of propagation as it passes from

one medium into another.

Laws of refraction state:

The incident ray, the refracted ray and the normal all lie

in the same plane.

For two given media,

where n1 : refractive index of medium containing incident ray

n2 : refractive index of medium containing refracted ray

constant sin

sin

1

2 n

n

r

irnin sin sin 21 or

Refraction at a Plane n1 < n2

(Medium 1 is less dense medium 2)

n1 > n2

(Medium 1 is denser than medium 2)

The light ray is bent toward the

normal (i > r)

The light ray is bent away from the

normal (i < r)

Refraction at a Plane

(Special case)

When i = 0°, no refraction take place

Refraction at a Plane

(Special case)

i = critical angle

r = 90°

i > critical angle

i = r

(Must be from denser to less dense medium)

Refractive Index (Index of

Refraction) Refractive index is defined as the constant ratio for

the two given media.

The value of refractive index depends on the type of

medium and the colour of the light.

It is dimensionless and its value greater than 1.

Consider the light ray travels from medium 1 into

medium 2, the refractive index can be denoted by

2

121

2 mediumin light ofvelocity

1 mediumin light ofvelocity

v

vn

(Medium containing

the incident ray) (Medium containing

the refracted ray)

Refractive Index (Index of

Refraction)

If medium 1 is vacuum, then the refractive index is called

absolute refractive index, written as

Note!!

If the density of medium is greater hence the refractive

index is also greater

v

cn

mediumin light ofvelocity

in vacuumlight ofvelocity

Relationship between n and λ:

As light travels from one medium to another, its

wavelength, changes but its frequency, f remains

constant.

The wavelength changes because of different material.

The frequency remains constant because the number of

wave cycles arriving per unit time must equal the

number leaving per unit time so that the boundary

surface cannot create or destroy waves.

By considering a light travels from medium 1 (n1) into

medium 2 (n2), the velocity of light in each medium is

given by

11 fv or 22 fv

Relationship between n and λ:

If medium 1 is vacuum or air:

2

1

2

1

f

f

v

v where

1

1n

cv

2

2n

cv and

2

1

2

1

n

c

n

c

2211 nn Refractive index is

inversely proportional

to the wavelength

0

mediumin light of wavelength

in vacuumlight of wavelengthn

Relationship between n and d:

Other equation for absolute refractive index in term of

depth is given by

depthapparent

depth realn

Example 5

A fifty cent coin is at the bottom of a swimming pool of

depth 3.00 m. The refractive index of air and water are

1.00 and 1.33 respectively. Determine the apparent depth

of the coin.

Example 5 – Solution

Refraction at a Spherical Surface

O I C

r i

n1 n2

O C

Convex Surface towards the object

P

P

Normal line

Refraction at a Spherical Surface Concave Surface towards the object

O I C

r i

n1 n2

O C

P

P

Normal line

Refraction at a Spherical Surface

Equation of spherical refracting surface:

where

n1 : refractive index of medium containing incident ray

n2 : refractive index of medium containing refracted ray

u : object distance from pole P

v : image distance from pole P

r

nn

v

n

u

n 1221

O I C

r i

n1 n2

P

u v

Refraction at a Spherical Surface

• If the refraction surface is flat (plane):

then

• The equation (formula) of linear magnification for

refraction by the spherical surface is given by

r 0v

n

u

n 21

un

vn

h

hm

o

i

2

1

Sign Convention for Refraction

or Thin Lenses:

Physical Quantity Positive sign (+) Negative sign (-)

Object Distance, u Real object

(in front of the refracting

surface)

Virtual object

(at the back of the

refracting surface)

Image Distance, v Real image

(opposite side of the

object)

Virtual image

(same side of the object)

Radius of Curvature, r Convex surface Concave surface

Focal length, f Converging lens Diverging lens

Linear magnification, m Upright image Inverted image

Object/ Image Height, h Upright image Inverted image

VERY Important!!

Example 6 A cylindrical glass rod in air has a refractive index of 1.52.

One end is ground to a hemispherical surface with radius,

r = 3.00 cm as shown in figure below.

Calculate,

a. the position of the image for a small object on the axis

of the rod, 10.0 cm to the left of the pole as shown in

figure.

b. the linear magnification.

(Given the refractive index of air , na = 1.00)

Example 6 – Solution

Example 6 – Solution

Example 7 Figure below shows an object O placed at a distance 20.0

cm from the surface P of a glass sphere of radius 5.0 cm

and refractive index of 1.63.

Determine

a. the position of the image formed by the surface P of the

glass sphere,

b. the position of the final image formed by the glass

sphere.

(Given the refractive index of air , na= 1.00)

Example 7 – Solution

a) u1 = 20.0 cm, r = +5.0 cm, nglass 1.63, nair = 1.00

O C 1I

cm 0.201 u cm 5.211 v

P

2n1n

r

cm 5.21

5

00.163.163.1

0.20

00.1

1221

v

v

r

nn

v

n

u

n

Light

Example 7 – Solution

b) u2 = ‒11.5 cm, r = ‒ 5.0 cm, nglass 1.63, nair = 1.00

O C

2Icm 1.52

P

2n

First surface

1n

Q

cm 1.512 u

Second surface

cm 74.3

5

63.100.100.1

5.11

63.1

1221

v

v

r

nn

v

n

u

n

The image is 3.74 cm at the

back of the second surface Q.

Light

= O2 I1

Try it now!

A point source of light is placed at a distance of

25.0 cm from the centre of a glass sphere of radius

10 cm. Find the image position of the source.

(Given refractive index of glass = 1.50 and

refractive index of air = 1.00)

Answer : 27.5 cm at the back of the concave

surface (second refracting surface)

7.3 Thin Lenses

Sketch and use ray diagrams to determine the

characteristic of image formed by concave and convex

lenses

Use thin lens equation for real object only

Use lens maker’s equation

Use the thin lens formula for a combination of two

convex lenses

Learning Objectives

Thin Lens Thin lens is defined as a transparent material with two

spherical refracting surfaces whose thickness is thin compared

to the radii of curvature of the two refracting surfaces.

Convex (Converging) lens which are thicker at the centre

than the edges

Concave (Diverging) lens which are thinner at the centre then

at the edges

Thin Lenses

For converging (convex) lens

Focal point is defined as the point on

the principal axis where rays which

are parallel and close to the principal

axis converges after passing through

the lens. Its focus is real (principal).

For diverging (concave) lens

Focal point is defined as the point on

the principal axis where rays which are

parallel to the principal axis seem to

diverge from after passing through the

lens. Its focus is virtual.

Ray Diagram for Convex Lens

Ray 1

Ray which is parallel

to the principal axis

will be deflected by

the lens towards/ away

from the focal point F.

Ray 2

Ray passing through

the optical centre is

un-deflected.

Ray Diagram for Convex Lens Ray 3

Ray which passes

through the focal

point becomes

parallel to the

principal axis after

emerging from lens.

At least any two

rays for drawing

the ray diagram.

Image Formed by Convex Lens

The characteristics of the images formed by convex lens

depends on object’s location.

The relationship between the object distance and object

size and the image distance and image size are depicted in

the diagram below:

Object

Image

Ray Diagram for Concave Lens

Ray 1

Ray which is parallel

to the principal axis

will be deflected by

the lens towards/ away

from the focal point F.

Ray 2

Ray passing through

the optical centre is

un-deflected.

Ray Diagram for Concave Lens

Ray 3

Ray which appear to

converge to the focal

point becomes

parallel to the

principal axis after

emerging from lens.

At least any two

rays for drawing

the ray diagram.

Image Formed by Concave Lens

Unlike convex lens, concave lens always produce images

that share same characteristics:

virtual

upright

diminished (smaller than the object)

formed in front of the lens (between focal point and lens)

As the object distance is decreased, the image distance is

decreased and the image size is increased.

Thin Lens Equation & Lens

Maker’s Equation

Thin lens equation:

Lens Maker’s Equation:

where f : focal length

r1 : radius of curvature of first refracting surface

r2 : radius of curvature of second refracting surface

nmaterial : refractive index of lens material nmedium : refractive index of medium

fvu

111 For real object only

21medium

material 111

1

rrn

n

f

Thin Lens Equation & Lens

Maker’s Equation

If the medium is air, then the lens maker’s equation can

be written as

The radius of curvature of flat refracting surface is

infinity, r = ∞.

For thin lens formula and lens maker’s equation, Use

the sign convention for refraction.

21

111

1

rrn

f

Sign Convention for Refraction

or Thin Lenses:

Physical Quantity Positive sign (+) Negative sign (-)

Object Distance, u Real object

(in front of the refracting

surface)

Virtual object

(at the back of the

refracting surface)

Image Distance, v Real image

(opposite side of the

object)

Virtual image

(same side of the object)

Radius of Curvature, r Convex surface Concave surface

Focal length, f Converging lens Diverging lens

Linear magnification, m Upright image Inverted image

Object/ Image Height, h Upright image Inverted image

VERY Important!!

Lens Maker’s Equation (Example)

A convex meniscus lens is made from a glass with a

refractive index n = 1.50 . The radius of curvature of the

convex surface is 22.4 cm and the concave surface is

46.2 cm. What is the focal length of the lens?

22.4 cm

lens inside the circle

46.2 cm

lens outside the circle

Lens Maker’s Equation (Example)

Method 1

r1 = +22.4 cm ; r2 = +46.2 cm

Light

21

111

1

rrn

f

cm 96.86

2.46

1

4.22

115.1

1

f

f

Lens Maker’s Equation (Example)

Method 2

r1 = ‒46.2 cm ; r2 = ‒ 22.4 cm

Light

21

111

1

rrn

f

cm 96.86

4.22

1

2.46

115.1

1

f

f

Linear Magnification, m

Linear magnification of the spherical mirror, m is defined

as the ratio between image height, hi and object height,

ho.

Since , the linear magnification equation

can be written as

u

v

h

hm

o

i

fvu

111

vfvu

1111

f

vm

Example 8

A biconvex lens is made of glass with refractive index 1.52

having the radii of curvature of 20 cm respectively.

Calculate the focal length of the lens in

a. water,

b. carbon disulfide.

(Given nw = 1.33 and nc = 1.63)

Example 8 – Solution

Example 9

A person of height 1.75 m is standing 2.50 m in from of a

camera. The camera uses a thin biconvex lens of radii of

curvature 7.69 mm. The lens made from the crown glass of

refractive index 1.52.

a. Calculate the focal length of the lens.

b. Sketch a labeled ray diagram to show the formation of

the image.

c. Determine the position of the image and its height.

d. State the characteristics of the image.

Example 9 – Solution

Example 9 – Solution

Example 9 – Solution

Example 9 – Solution

Example 10

An object is placed 90.0 cm from a glass lens (n = 1.56)

with one concave surface of radius 22.0 cm and one

convex surface of radius 18.5 cm. Determine

a. the image position.

b. the linear magnification.

Example 10 – Solution

Example 10 – Solution

Combination of Two Convex

Lenses

Many optical instruments, such as microscopes and

telescopes, use two converging lenses together to

produce an image.

In both instruments, the 1st lens (closest to the object)is

called the objective and the 2nd lens (closest to the eye)

is referred to as the eyepiece or ocular.

The image formed by the 1st lens is treated as the

object for the 2nd lens and the final image is the image

formed by the 2nd lens.

The position of the final image in a two lenses system

can be determined by applying the thin lens formula to

each lens separately.

Combination of Two Convex

Lenses

The overall magnification of a two lenses system is the

product of the magnifications of the separate lenses.

21mmm where

m1 : magnification due to the first lens

m2 : magnification due to the second lens

Example 11

An object is 15.0 cm from a convex lens of focal length

10.0 cm. Another convex lens of focal length 7.5 cm is

40.0 cm behind the first. Find the position and

magnification of the image formed by

a. the first convex lens

b. both lenses

Example 11 – Solution

a) f1 = 10.0 cm, u1 = 15.0 cm

111

111

vuf

cm 0.30

15

1

10

11

1

1

v

v

1

11

u

vm

21 m

d

1u

1f1f 2f2f

F1 F1 F2 F2 O

(at the back of the 1st lens)

Example 11 – Solution d

1u

1f1f 2f2f

F1 F1 F2 F2 O1I

1v 2u

cm 10

3040

12

vdu

Note!!

I1 = O2 BUT v1 ≠ u2

Example 11 – Solution

b) f2 = 7.5 cm, u2 = 10.0 cm

Total linear magnification:

cm 0.30

10

1

5.7

11

2

2

v

v

222

111

vuf

2

22

u

vm

32 m

0.621 mmm

(at the back of the 2nd lens)

Summary Geometrical

Optics

Spherical

Mirror Thin Lens Refraction

Ray Diagram

vuf

111

2

rf and

Snell’s Law

r

nn

v

n

u

n 1221

Ray Diagram

Thin lens

equation

&

Lens maker’s

equation

21medium

material 111

1

rrn

n

f

IMPORTANT !

Front Back

Front Back

Real object Virtual object

Virtual image Real image

Real object Virtual object

Virtual image Real image

IMPORTANT!

Object

Convex towards the object

→ r +ve

Object

Concave towards the object

→ r -ve

IMPORTANT!

r1 → +ve

r2 → ‒ve

Light

r1 → -ve

Light r2 → ∞