Ep notes

OPTICAL FIBER

Introduction Fiber optics deals with the light propagation through thin glass fibers.

Fiber optics plays an important role in the field of communication to transmit voice,

television and digital data signals fro one place to another. The transmission of light

along the thin cylindrical glass fiber by total internal reflection was first demonstrated

by John Tyndall in 1870 and the application of this phenomenon in the field of

communication is tried only from 1927. Today the applications of fiber optics

are also extended to medical field in the form of endoscopes and to instrumentation

engineering in the form of optical sensors. The Basic principle of optical fiber

Principle: The basic principle of optical fiber in the transmission of optical signal is total

internal reflection.

Total internal reflection:-

When the light ray travels from denser medium to rarer medium the refracted

ray bends away from the normal. When the angle of incidence is greater than the

critical angle, the refracted ray again reflects into the same medium. This

phenomenon is called total internal reflection.

The refracted ray bends towards the normal as the ray travels from rarer medium

to denser medium. The refracted ray bends away from the normal as it travels

from denser medium to rarer medium.

Conditions for Total Internal Reflection

(a) the refractive index n1 of the core must always be greater than the refractive index n2

of the cladding.

(b) The angle of incidence i must be greater than critical angle C

it can be define as when light travels from a more optically dense material [larger index of

refraction] to a less dense material the angle of refraction is larger than the incident angle.

Because the refracted angle is always larger than the incident angle, it is possible for the

refracted angle to reach 90° before the incident angle reaches 90°. If the light were to refract out

of the denser medium, it would then run along the surface. Larger angles would then yield

situations which would force the sine function to be larger than 1.00, which is mathematically

impossible.

When the incident angle reaches the condition whereby the refracted ray would bend to an angle

of 90°, it is called the CRITICAL ANGLE. The critical angle obeys the following equation:

This reflected ray changes in intensity as we vary the angle of incidence. At small incident

angles (almost perpendicular to the surface) the reflected ray is weak and the refracted ray is

strong.

Construction of optical fiber:-

The optical fiber mainly consists the following six parts as shown in figure Core:

A typical glass fiber consists of a central core material. Generally core

diameter is 50 . The core is surrounded by cladding. The core medium

refractive is always greater than the cladding refractive index.

Cladding Cladding refractive index is lesser than the cores refractive index. The

over all diameter of cladding is 125 to 200 .

Silicon Coating Silicon coating is provided between buffer jacket and cladding. It

improves the quality of transmission of light.

Buffer Jacket Silicon coating is surrounded by buffer jacket. Buffer jacket is made of

plastic and protects the fiber cable from moisture.

Strength Member Buffer jacket is surrounded by strength member. It provides strength to the

fiber cable.

Outer Jacket Finally the fiber cable is covered by polyurethane outer jacket. Because

of this arrangement fiber cable will not be damaged during pulling,

bending, stretching and rolling through the fiber cable is made up of glasses.

NA & ACCEPTANCE ANGLE DERIVATION

“In optics, the numerical aperture (NA) of an optical system is a dimensionless number

that characterizes the range of angles over which the system can accept or emit light.”

optical fiber will only propagate light that enters the fiber within a certain cone, known as

the acceptance cone of the fiber. The half-angle of this cone is called the acceptance angle,

θmax. on

where n1 is the refractive index of the fiber core, and n2 is the refractive index of the cladding.

When a light ray is incident from a medium of refractive index n to the core of index n1, Snell's

law at medium-core interface gives

From the above figure and using trigonometry, we get :

Where is the critical angle for total internal reflection, since

Substituting for sin θr in Snell's law we get:

By squaring both sides

Thus,

from where the formula given above follows.

θmax =

This has the same form as the numerical aperture in other optical systems, so it has become

common to define the NA of any type of fiber.

Definition:-

Acceptance angle:- Acceptance angle is defined as the maximum angle of incidence at the interface

of air medium and core medium for which the light ray enters into the core and

travels along the interface of core and cladding.

Acceptance Cone:-

There is an imaginary cone of acceptance with an angle .The light that enters the fiber at

angles within the acceptance cone are guided down the fiber core Numerical aperture:- Numerical aperture is defined as the light gathering capacity of an optical fiber and it is

directly proportional to the acceptance angle.

Numerically it is equal to the sin of the acceptance angle

Classification of fibers:-

Based on the refractive index of core medium, optical fibers are classified into

two categories.

i. Step index fiber ii. Graded index fiber

Based on the number of modes of transmission, optical fibers are classified into

two categories

i. Single mode fiber

ii. Multimode fiber

Based on the material used, optical fibers are may broadly classified into four

categories

i. All glass fibers

ii. All plastic fibers iii. Glass core with plastic cladding fibers iv.

Polymer clad silica fibers.

Step index fiber:- In step index fibers the refractive index of the core medium is uniform and

undergoes an abrupt change at the interface of core and cladding as shown in figure.

The diameter of core is about 10micrometers in case of single mode fiber and 50 to 200

micrometers in multi mode fiber. Attenuation is more for step index multi mode fibers but less in step index single

mode fibers

Numerical aperture is more for step index multi mode fibers but it is less in step

index single mode fibers

Graded index fiber:- In graded index fibers, the refractive index of the core medium is varying in the

parabolic manner such that the maximum refractive index is present at the center

of the core.

The diameter of the core is about 50 micro meters.

Attenuation is very less in graded index fibers Numerical aperture is less in graded index

fibers Graded index Figure Two types of fiber: (Top) step index fiber; (Bottom) Graded index fiber

Single mode optical fiber In single mode optical fibers only one mode of propagation is possible.In case of single mode

fiber the diameter of core is about 10micrometers.The difference between the refractive

indices of core and cladding is very small. In single mode fibers there is no dispersion,

so these are more suitable for

Communication. The single mode optical fibers are costly, because the fabrication is

difficult.The process of launching of light into single mode fibers is very difficult.

Multi mode optical fiber

In multi mode optical fibers many mummer of modes of propagation are possible. In case of

in multi mode fiber the diameter of core is 50 to 200 micrometers. The difference between

the refractive indices of core and cladding is also large compared to the single mode

fibers. Due to multi mode transmission, the dispersion is large, so these fibers are not used

for communication purposes. The multi mode optical fibers are cheap than single mode

fibers, because the fabrication is easy. The process of launching of light into single mode

fibers is very easy.

Based on the material:- Three common type of fiber in terms of the material used:

Glass core with glass cladding –all glass or silica fiber Glass core with plastic cladding –plastic cladded/coated silica (PCS) Plastic core with plastic cladding – all plastic or polymer fib

Attenuation:-

Definition: a loss of signal strength in a lightwave, electrical or radio signal usually related to

the distance the signal must travel.

Attenuation is caused by:

Absorption

Scattering

Radiative loss

Losses:-

Losses in optical fiber result from attenuation in the material itself and from scattering,

which causes some light to strike the cladding at less than the critical angle

Bending the optical fiber too sharply can also cause losses by causing some of the light to

meet the cladding at less than the critical angle

Losses vary greatly depending upon the type of fiber

Plastic fiber may have losses of several hundred dB per kilometer

Graded-index multimode glass fiber has a loss of about 2–4 dB

per kilometer

Single-mode fiber has a loss of 0.4 dB/km or less

Macrobending Loss:

The curvature of the bend is much larger than fiber diameter. Lightwave suffers sever loss due

to radiation of the evanescent field in the cladding region. As the radius of the curvature

decreases, the loss increases exponentially until it reaches at a certain critical radius. For any

radius a bit smaller than this point, the losses suddenly becomes extremely large. Higher order

modes radiate away faster than lower order modes.

Microbending Loss:

microscopic bends of the fiber axis that can arise when the fibers are incorporated into cables.

The power is dissipated through the microbended fiber, because of the repetitive coupling of

energy between guided modes & the leaky or radiation modes in the fiber.

Dispersion:-

The phenomenon in an optical fibre whereby light photons arrive at a distant point in different

phase than they entered the fibre. Dispersion causes receive signal distortion that ultimately

limits the bandwidth and usable length of the fiBer cable

The two main causes of dispersion are:

Material (Chromatic) dispersion

Waveguide dispersion

Intermodal delay (in multimode fibres)

Dispersion in fiber optics results from the fact that in multimode propagation, the signal travels

faster in some modes than it would in others.Single-mode fibers are relatively free from

dispersion except for intramodal dispersion .Graded-index fibers reduce dispersion by taking

advantage of higher-order modes.One form of intramodal dispersion is called material

dispersion because it depends upon the material of the core.Another form of dispersion is called

waveguide dispersion .Dispersion increases with the bandwidth of the light source

The advantage of fiber optic cable over metallic cable:-

1. Extremely wide (large) bandwidth.

The bandwidth available with a single glass fibre is more than 100GHZ. With such a large

bandwidth, it is possible to transmit thousands of voice conversations or dozens of video signals

over the same fibre simultaneously. Irrespective of whether the information is voice, data or

video or a combination of these, it can be transmitted easily over the optical fibre. Less no of

independent signals alone can be sent through metallic cables.

2. Immunity to electrostatic interference.

As optical fibres are being made of either glass or plastic external electric noise and lightning

do not affect the energy in a cable. The result is noise free transmission. While this is not true

for metallic cables made up of metals, as they are good conductors of electricity.

3. Elimination of cross Talk.

Fibre systems are immune to cross talk between cables caused by magnetic induction. Whereas

in a metallic cable cross talk results from the electromagnetic coupling between two adjacent

wires.

4. Lighter weight and smaller size.

Fibres are very smaller in size. This size reduction makes fibre the ideal transmission medium

for ships, aircraft and high rise buildings where bulky copper cables occupy to much space.

Reduction in size so reduction in weight also.

5. Lower cost.

The material used in fibres is silica glass or silicon dioxide which is one of the most abundant

materials on earth. So available in lower cost.

6. Security.

Fibre cables are more secure than metallic cables. Due to its immunity to electromagnetic

coupling and radiation, optical fibre can be used in most secure environment. Although it can be

intercepted or tapped, it is very difficult to do so because, at the receiving users end an alarm

would be sounded.

7. Greater safety.

In many wired system the potential hazard of short circuits requires precautionary designs.

Whereas, the dielectric nature of optical fibres eliminates the spark hazard.

8. Corrosion

Fibre cables are more resistive to environmental extremes. They operate over large temperature

variation than their metallic counter parts, and are less affected by corrosive liquids and gases.

9. Longer life span and ease of maintenance.

A longer life span of 20 to 30 years is predicted for the fibre optic cables as compare to 12to 15

years of metallic cables.

Differences between step index fibers and graded index fibers:-

Step index fiber

Graded index fiber

1. In step index fibers the refractive index of the

core medium is uniform through and

undergoes an abrupt change at the interface of

core and cladding.

1. In graded index fibers, the refractive index of

the core medium is varying in the parabolic

manner such that the maximum refractive index

is present at the center of the core.

2. The diameter of core is about

10micrometers in case of single mode fiber and

50 to 200 micrometers in multi mode fiber.

2. The diameter of the core is about 50 micro meters.

3. The transmitted optical signal will cross the

fiber axis during every reflection at the core

cladding boundary.

3. The transmitted optical signal will never cross

the fiber axis at any time.

4. The shape of propagation of the optical

signal is in zigzag manner.

4. The shape of propagation of the optical signal

appears in the helical or spiral manner

5. Attenuation is more for multi mode step

index fibers but Attenuation is less in single

mode step index fibers

5. Attenuation is very less in graded index fibers

6. Numerical aperture is more for multi

mode step index fibers but it is less in single

mode step index fibers

6. Numerical aperture is less in graded index fibers

Differences between single mode fibers and Multy mode fibers:-

Single mode fiber Multimode fiber

Single Mode cable is a single strand (most

applications use 2 fibers) of glass fiber with a

diameter of 8.3 to 10 microns that has one mode of

transmission.

Multi-Mode cable has a little bit bigger diameter,

with a common diameters in the 50-to-100 micron

range for the light carry component

Single Modem fiber is used in many applications

where data is sent at multi-frequency (WDM

Wave-Division-Multiplexing) so only one cable is

needed

Most applications in which Multi-mode fiber is used,

2 fibers are used (WDM is not normally used on

multi-mode fiber).

Example:- step index fiber Example:- multimode step index fiber

The small core and single light-wave virtually

eliminate any distortion that could result from

overlapping light pulses, providing the least signal

attenuation and the highest transmission speeds of

any fiber cable type.

multiple paths of light can cause signal distortion at

the receiving end, resulting in an unclear and

incomplete data transmission

Applications of optical fibers

1. Optical fibers are extensively used in communication system. 2. Optical fibers are in exchange of information between different computers 3. Optical fibers are used for exchange of information in cable televisions,

space vehicles, submarines etc.

4. Optical fibers are used in industry in security alarm systems, process control

and industrial auto machine.

5. Optical fibers are used in pressure sensors in biomedical and engine control. 6. Optical fibers are used in medicine, in the fabrication in endoscopy for

the visualization of internal parts of the human body.

7. Sensing applications of optical fibers are

Displacement sensor

Fluid level detector Liquid

Temperature and pressure sensor

Chemical sensors

8. Medical applications of optical fibers are

Gastroscope

Orthoscope Couldo

EXAMPLE:-

1.

A silica optical fiber has a core of refractive index 1.55 and a cladding of refractive index

1.47. Determine (i) the critical angle at the core-cladding interface (ii) the numerical

aperture for the fiber and (iii) the acceptance angle in the air for the fiber.

Given,

n1=1.55,

n2=1.47

Øin(max)=?

NA=?

Øc=?

Acceptance angle Øin(max)= sin-1

(n12 – n2

2)

1/2

Øin(max)= sin-1

(1.552 –1.47

2)

1/2

= sin-1

(2.41-2.16)1/2

= sin-1

(0.25)1/2

=

sin

-1 (0.316)

Øin(max) =30°00’

Numerical aperture NA= (n12 – n2

2)

1/2

= 1.552 –1.47

2)

1/2

= (2.41-2.16)1/2

= (0.25)1/2

= 0.316

critical angle Øc = sin-1

(n2 / n1)

= sin-1

(1.47 / 1.55)

= sin-1

(0.9483)

= 71°.55’

2.

An optical fiber has refractive index of core and cladding is 1.514 and 1.48 Respectively.

Calculate the acceptance angle and the fractional index Change

Given

,n1=1.514,

n2=1.48

Øin(max)=? ∆=?

Acceptance angle Øin(max ) = sin-1

(n12 – n2

2)

1/2

Øin(max) = sin-1

(1.5142 –1.48

2)

1/2

= sin-1

(2.29-2.19)1/2

= sin

-1 (0.1)

1/2

=

sin

-1 (0.316)

Øin(max)=18°42’

Numerical aperture NA= (n12 – n2

2)

1/2

=(1.5142 –1.48

2)

1/2

=(2.29-2.19)1/2

=(0.1)1/2

=0.316

NA=n1√2∆

0.316/1.514=√2∆

(0.2087)2=2∆

∆=0.0435/2

∆=0.0217

Dielectrics

Introduction

Dielectrics are the materials having electric dipole moment permanently.

Dipole: A dipole is an entity in which equal positive and negative charges are separated by a small distance..

DIPOLE moment (µEle ):The product of magnitude of either of the charges and separation distance b/w them

is called Dipole moment.

µe = q . x coulmb.m

All dielectrics are electrical insulators and they are mainly used to store electrical energy.

Ex: Mica, glass, plastic, water & polar molecules…

Dielectric const. of medium

The relative permittivity(εr) is often known as dielectric const. of medium it can given by,

εr=ε/ε0

Dielectric constant is ratio of permittivity of medium to permittivity of free space.

The value of capacitance of capacitor is given by,

C0=εrε0A/d

By this eqn we can say that high εr increases capacity of capacitor.

Polar and Nonpolarized Molecules

Non-polar Molecules : The Dielectric material in which there is no permanent dipole existence in absence

of an external field is …..

2 – Compounds made of molecules which are symmetrically shaped

Polar Molecules :The Dielectric material in which there is permanent dipole existence even in absence of

an external field is …..

Polarization of Dielectrics

As shown in fig. when an electric field is applied to dielectric material their negative & positive charges tend

to align in equilibrium position.

Gauss’s Law In Dielectrics

In absence of dielectric In presence of dielectric

0 0

d

0

0

0 0

0 0 0

0

0

E Vk

E V

E qE

k kA

q q 'E

A A

q q q 'So,

kA A A

1then , q ' q (1 )

k

So, E.ds

V Ed

S

q q '

1q q (1 )

k

q

k

k E.ds q

o,

N ow

This relation true is for parallel plate capacitor Which is Gauss’s law for dielectrics.

0

0

0

0

0

E.ds q

qE A

qE

A

0

0 0

0 0

E.ds q q '

q q 'E A

q q 'E

A A

Three Electric vectors

The resultant dielectric field is given by,

Where,

E=Electric field

D=Flux Density or

Displacement vector

P=Polarization

Electric susceptibility:

The polarization vector P is proportional to the total electric flux density and direction of

electric field.

Therefore the polarization vector can be written as:

Relation between εr &

Displacement vector,

Types of polarization

1. Electron polarization

2. Ionic polarization

3. Orientation polarization

4. Space charge polarization

0 0

0 0

0

0

'

',

,

, D

p

q qE

A A

qnow P

A

q PE

A

qE P

A

qnow D

A

So E P

0

0

0

0

( 1)

1

e

e

r

e r

P E

P

E

E

E

0

0

0

r 0 0

0

D E P

N ow ,P=

( - ) E P

(or) ( . - ) E P

( 1) . P

W here,( 1)

r

r

E

E

1. Electronic polarization

When no external field is applied nucleus of atom is like in fig. (a)

When external field is applied, displacement in opposite direction is observed between nucleus &

electrons due to this dipole moment is induced.

This type of polarization is called Electronic polarization.

Ex. Germanium, Silicon, Diamond etc…

2. Ionic polarization

Some materials like ionic crystal does not possess permanent dipole moment.

Fig. (a) shows natural arrangement of ionic crystal. When Ele. Field is applied on this type of

material displacement of ions is observed.

Due to an external electric field a positive & negative ion displaces in the direction opposite to

each other due to which distance between them is reduced & ionic polarization is generated.

Ionic polarization is observed in materials like NaCl, KBr, KCl etc…

Let us consider simple example of NaCl crystal.

As shown in fig. when crystal is placed in an external electric field Na+ ion displaces in one

direction & Cl- ion goes in opposite direction.

3. Orientation polarization

Some molecules like H2O, HCl having permanent dipole moment p0.

In the absence of a field, individual dipoles are arranged in random way, so net average dipole

moment in a unit volume is zero as shown in fig. (b).

A dipole such as HCl placed in a field experiences a torque that tries to rotate it to align p0 with

the field E.

In the presence of an applied field, the dipoles try to rotate to align parallel to each other in

direction of electric field fig (d).

This type of polarization is Orientation polarization.

This type of polarization occurs only in polar substances like H2O, CH3Cl when they are placed

in external field.

4. Space charge polarization (Interfacial polarization)

A crystal with equal number of mobile positive ions and fixed negative ions.

In the absence of a field, there is no net separation between all the positive charges and all the

negative charges.

In the presence of an applied field, the mobile positive ions migrate toward the negative charges

and positive charges in the dielectric.

The dielectric therefore exhibits Space charge or interfacial polarization.

Energy stored in dielectric field

Work done is,

.

?

.

.

dW F dr

F

dW qE dr

dW E dp

p pP

lA V

0

0

0

2

0

2

0

( 1) .

. .( 1) .

. .( 1) .

1( 1) E

2

1( 1) E

2

?

r

r

r

r

r

p PV

dW EVdP

P E

dW E V dE

dW E V dE

W V

W

V

U

Band Theory of Solid

Objectives

• Effective Mass of electron

• Concept of Holes

• Energy Band Structure of Solids:

Conductors, Insulators and Semiconductors

• Semiconductors

Intrinsic and Extrinsic Semiconductors

• Type of diodes

Simple Diode

Zener Diode

Effective Mass of electron

An electron moving in the solid under the influence of the crystal potential is subjected to

an electric field.

We expect an external field to accelerate the electron, increasing E and k and change the

electron’s state.

------ (1)

But, dx/dt = vg ------ (2)

------ (3)

dk

dgv

1

gvdx

dVe

dt

dk

dk

d

dt

dx

dx

dVe

dt

dVe

dt

d

eV

and

----- (4)

------ (5)

------ (6)

------ (7)

----- (8)

Concept of Holes

Consider a semiconductor with a small number of electrons excited from the valence

band into the conduction band.

If an electric field is applied,

• The conduction band electrons will participate in the electrical current

• The valence band electrons can “move into” the empty states, and thus can also

contribute to the current.

If we describe such changes via “movement” of the “empty” states – the picture will be

significantly simplified. This “empty space” is called a Hole.

“Deficiency” of negative charge can be treated as a positive charge.

dx

dVek

dt

d

gvdx

dVe

dt

dkgv

eEkdt

d

dt

dk

dk

d

dk

d

dk

d

dt

d

dt

dva

g

11

k

dt

d

dk

d

dt

dk

dk

d

2

2

22

211

Holes act as charge carriers in the sense that electrons from nearby sites can “move” into

the hole.

Holes are usually heavier than electrons since they depict collective behavior of many

electrons.

To understand hole motion, one requires another view of the holes, which represent them

as electrons with negative effective mass m*.

For example the movement of the hole think of a row of chairs occupied by people with

one chair empty, and to move all people rise all together and move in one direction, so the

empty spot moves in the same direction.

Energy Band Structure of Solids Conductor, Semiconductor and Insulator

In isolated atoms the electrons are arranged in energy levels.

Energy Band in Solid

The following are the important energy band in solids:

Valence band

Conduction band

Forbidden energy gap or Forbidden band

Valance band

The band of energy occupied by the valance electrons is called valence band. The

electrons in the outermost orbit of an atom are known as valance electrons. This band may be

completely or partial filled.

Electron can be move from one valance band to the conduction band by the

application of external energy.

Conduction band

The band of energy occupied by the conduction electrons is called conduction

band. This is the uppermost band and all electrons in the conduction band are free electrons.

The conduction band is empty for insulator and partially filled for conductors.

Forbidden Energy Gap or Forbidden band

The gap between the valance band and conduction band on energy level diagram

known as forbidden band or energy gap.

Electrons are never found in the gap. Electrons may jump from back and forth

from the bottom of valance band to the top of the conduction band. But they never come to rest

in the forbidden band.

According to the classical free electron theory, materials are classified in to three types:

Conductors

Semiconductors

Insulators

Conductors

There is no forbidden gap and the conduction band and valence band are

overlapping each other between and hence electrons are free to move about. Examples are Ag,

Cu, Fe, Al, Pb ….

Conductor are highly electrical conductivity

So, in general electrical resistivity of conductor is very low and it is of the order of 10-6

Ω

cm.

Due to the absence of the forbidden gap, there is no structure for holes.

The total current in conductor is simply a flow of electrons.

For conductors, the energy gap is of the order of 0.01 eV.

Semiconductors:

Semiconductors are materials whose electrical resistivity lies between insulator

and conductor. Examples are silicon (Si), germanium (Ge) ….

The resistivity of semiconductors lies between 10-4

Ω cm to 103 Ω cm at room

temperature.

At low temperature, the valence band is all most full and conduction band is almost

empty. The forbidden gap is very small equal to 1 eV.

Semiconductor behaves like an insulator at low temperature. The most commonly used

semiconductor is silicon and its band gap is 1.21 eV and germanium band gap is 0.785

eV.

When a conductor is heated its resistance increases; the atoms vibrate more and

the electrons find it more difficult to move through the conductor but, in a semiconductor

the resistance decreases with an increase in temperature. Electrons can be excited up to the

conduction band and Conductivity increases.

Insulators

Here the valence band is full but the conduction band is totally empty. So, a free electron

from conduction band is not available.

In insulator the energy gap between the valence and conduction band is very large and

it’s approximately equal to 5 eV or more.

Hence electrons cannot jump from valence band to the conduction band. So, a very high

energy is required to push the electrons to the conduction band.

Therefore the electrical conductivity is extremely small.

The resistivity of insulator lie between 103 to 10

17 Ωm, at the room temperature

Examples are plastics, paper …..

Types of semiconductors

Intrinsic Semiconductor

The intrinsic semiconductors are pure semiconductor materials. These

semiconductors possess poor conductivity. The elemental and compound semiconductor can be

intrinsic type. The energy gap in semiconductor is very small. So, even at the room temperature,

some of electrons from valance band can jump to the conduction band by thermal energy.

The jump of electron in conduction band adds one conduction electron in

conduction band and creates a hole in the valence band. The process is called as “generation of

an electron–hole pair”.

In pure semiconductor the no. of electrons in conduction band and holes in holes

in valence bands are equal.

Extrinsic Semiconductor

Extrinsic semiconductor is an impure semiconductor formed from an intrinsic

semiconductor by adding a small quantity of impurity atoms called dopants.

The process of adding impurities to the semiconductor crystal is known as doping.

This added impurity is very small of the order of one atom per million atoms of

pure semiconductor.

Depending upon the type of impurity added the extrinsic semiconductors are

classified as:

• p – type semiconductor

• n – type semiconductor

p – type semiconductor

The addition of trivalent impurities such as boron, aluminum or gallium to

an intrinsic semiconductor creates deficiencies of valence electrons, called "holes". It is typical to

use B2H6 di-borane gas to diffuse boron into the silicon material.

n – type semiconductor

The addition of pentavalent impurities such as antimony, arsenic or phosphorous

contributes free electrons, greatly increasing the conductivity of the intrinsic semiconductor.

Phosphorous may be added by diffusion of phosphine gas (PH3).

Simple Diode

The two terminals are called Anode and Cathode. At the instant the two materials

are “joined”, electrons and holes near the junction cross over and combine with each other.

Holes cross from P-side to N-side and free electrons cross from N-side to P-side.

At P-side of junction, negative ions are formed.

At N-side of junction, positive ions are formed.

Depletion region is the region having no free carriers. Further movement of

electrons and holes across the junction stops due to formation of depletion region. Depletion

region acts as barrier opposing further diffusion of charge carriers. So diffusion stops within

no time. Current through the diode under no-bias condition is zero.

Reverse bias

Positive of battery connected to n-type material (cathode).

Negative of battery connected to p-type material (anode).

Free electrons in n-region are drawn towards positive of battery; Holes in p-region

are drawn towards negative of battery.

Depletion region widens, barrier increases for the flow of majority carriers.

Majority charge carrier flow reduces to zero.

Minority charge carriers generated thermally can cross the junction – results in a

current called “reverse saturation current” Is , Is is in micro or nano amperes or less. Is does

not increase “significantly” with increase in the reverse bias voltage

Zener Diode

A diode which is heavily doped and which operates in the reverse breakdown

region with a sharp breakdown voltage is called a Zener diode.

This is similar to the normal diode except that the line (bar) representing the cathode is bent

at both side ends like the letter Z for Zener diode. In simple diode the doping is light; as a

result, the breakdown voltage is high and not sharp. But if doping is made heavy, then the

depletion layers becomes very narrow and even the breakdown voltage gets reduced to a

sharp value.

Working Principle

The reverse breakdown of a Zener diode may occur either due to Zener effect or

avalanche effect. But the Zener diode is primarily depends on Zener effect for its working.

When the electrical field across the junction is high due to the applied voltage, the

Zener breakdown occurs because of breaking of covalent bonds. This produces a large

number of electrons and holes which constitute a steep rise in the reverse saturation current

(Zener current IZ). This effect is called as Zener effect.

Zener current IZ is independent of the applied voltage and depends only on the

external resistance.

The I-V characteristic of a Zener diode is shown in this figure. The forward

characteristic is simply that of an ordinary forward biased junction diode.

Under the reverse bias condition, the breakdown of a junction occurs. Its depends

upon amount of doping. It can be seen from above figure as the reverse voltage is increased

the reverse current remains negligibly small up to the knee point (K) of the curve.

At point K, the effect of breakdown process beings. The voltage corresponding to

the point K in figure is called the Zener breakdown voltage or simply Zener voltage (VZ),

which is very sharp compared to a simple p-n junction diode. Beyond this voltage the

reverse current (IZ) increases sharply to a high value.

The Zener diode is not immediately burnt just because it has entered the

breakdown region.

The Zener voltage VZ remains constant even when Zener current IZ increases

greatly. This ability of a diode is called regulating ability and it enables us to use Zener

diode for voltage regulation.

The maximum value of current is denoted by IZ max and the minimum current to

sustain breakdown is denoted by IZ min. By two points A and B on the reverse VI

characteristic, the Zener resistance is given by the relation

rz = ( Δ VZ / Δ IZ). ------- (1)

Zener diode Applications:

1) Zener diodes are used as a voltage regulator.

2) They are used in shaping circuits as peak limiters or clippers.

3) They are used as a fixed reference voltage in transistor biasing and for

comparison purpose.

4) They are used for meter protection against damage from accidental application

of excessive voltage.

LASER

Light Amplification by Stimulated Emission of Radiation

Introduction

The full form of LASER is Light Amplification by Stimulated Emission of

Radiation.

Laser light is highly powerful and it is capable of propagating over long distances

and it is not easily absorbed by water.

Light having following Properties:

• Wavelength

• Frequency

• Amplitude

• Phase

• Coherence/Incoherence

• Velocity

• Direction

The characteristics or properties of Laser Light are:

• Coherence

• High Intensity

• High directionality

• High monochromaticity

Absorption

According to Bohr’s law atomic system is characterized by discrete energy level. When

atoms absorb or release energy it transit upward or downward.

Here lower level is E1 and excited level is E2, the photon energy hƒ = E2 – E1.

The atom absorbed an incident photon. As the result of absorption atom absorbed energy

and the atom jumped to excited state E2. This transition is called absorption. It is also referred to

as induced absorption.

We may express the process as,

A + hν = A*

Where A is an atom in lower state and A* is an excited atom.

The rate of absorption depends on no. of atoms N1 present in E1 and spectral energy density u(ƒ)

of radiation.

P12 α N1 u(ƒ) -----(1)

So, P12 = B12N1 u(ƒ) -----(2)

In each absorption transition event, an atom in the medium is excited and one photon is

subtracted from the incident beam, which result in attenuation of light in the medium.

Spontaneous Emission

An atom cannot stay in the excited state for a longer time. Ina time of 10-8

sec, the atom

come back to the ground state by releasing a photon of energy hν, and hν = E = E2 – E1. Where

E1 = Ground State and E2 = Excited State.

The emission of photon by an atom without any external impetus is called spontaneous

emission.

We may write the process as,

A* → hν + A

Here system having atoms in excited state. Atom goes to downward transition with emitting

photons, hƒ = E1 – E2.

Emission is random, so if not in same phase becomes incoherent.

The transition depends on atoms in excited state N2.

P12 (spont) α N2 = A21 N2 ------- (1)

Where, A21 = Einstein coefficient for spontaneous Emission. We get Incoherent radiation forms

heat by light amplification of radiation by spontaneous emission.

Stimulated Emission

An atom in the excited state need not wait for spontaneous emission of photon. Well before

the atom can make a spontaneous transition, it may interacts with a photon with energy hν = E2 –

E1, and make a downward transition. The photon is said to stimulated of induced the excited

atom to emit a photon of energy hν = E2 – E1. The passing photon does not disappear and in

addition to it there is a second photon which is emitted by the excited atom.

The phenomenon of forced photon emission by an excited atom due to the action of an

external energy is called stimulated emission or induced emission.

The process may be expressed as,

A* + hν → A + 2hν

Here system having atoms in excited state. The atom goes to downward transition with

emitting photons.

2hƒ = E1 – E2. After applying photon energy hƒ.

Emission is depends on energy density u(ƒ) & No. of atoms in excited state N2

P12 (stimul) α u(ƒ) N2 - -------- (1)

= B21 N2 u(ƒ) -------- (2)

Where, B21 = Einstein coefficient for Stimulated Emission.

Thus one photon of energy hƒ stimulates two photons of energy hƒ in same phase & directions.

So, we get coherent light amplification of radiation by stimulated emission.

Population Inversion

It is the process of increasing exited electrons in higher energy levels. Due to this

process the production of laser is possible. The energy level between the ground state E1 (1st

level) and exited state E3 (3rd

level) is known as metastable state E2 (2nd

level).

By the optical pumping electrons from ground state jumps to excited state by

absorbing photons. The electrons remain only for 10-8

sec in exited state E3, so most of them

jump back to the ground state E1 by emitting photons. But some of them jump to the

metastable state E2.

They (electron) stay in metastable state for more then 10-3

sec. So electron density

increases in metastable state. Thus the transitions are possible it takes more no. of electrons

together and ν – (knew)

12 photon beam is produced which constitute laser beam.

Optical Pumping

There are no of techniques for pumping a collection of atoms to an inverted state.

• Optical pumping

• Electrical discharge

• Direct conversion

When photon of blue green light incident on Ruby crystal, electrons from ground

state absorbs and exited and jumps on higher energy state levels and comes back to metastable

state. They increase population of electrons in metastable state.

This process is called “optical pumping” which is done by flash tube.

Relation between Einstein’s ‘A’ and ‘B’ coefficients

Einstein obtained a mathematical expression for the existence of two different kinds of

processes,

(1) Spontaneous emission

(2) Stimulated emission

Consider all atoms r in thermal equilibrium at T and radiation of frequency (ƒ) and energy

density u(ƒ). Here N1 and N2 r atoms in E1 and E2 respectively.

In equilibrium absorption rates and emission rates must be same. i.e.

B12 N1 u(ƒ) = A21 N2+ B21 N2 u(ƒ)

→ A21 N2= u(ƒ) [B12N1 – B21N2]

→ u(f) = [A21 N2 / (B12 N1 – B21 N2)] --------- (1)

---------- (2)

So Boltzmann distribution law is,

---------- (3)

21

21

12 1

21 2

( )

[ ]

ƒ

1

A

Bu

B N

B N

1

2

/

1 0

/

2 0

E kT

E kT

N N e

N N e

And

----------- (4)

But, E2 – E1 = hf ----------- (5)

So, ----------- (6)

------------ (7)

According to plank’s radiation formula,

------------ (8)

Where, B12 = B21 & A21 / B21 = 8∏hf3/c

3 ------------ (9)

So, Ratio of spontaneous to stimulated emission:

----------- (10)

So,

------------ (11)

------------ (12)

R = e hf/KT

- 1 -------------- (13)

So,

• If hƒ << kT, in thermal equilibrium,

Then R = ehf/KT

- 1 << 1

• hƒ<<kT – Stimulated emission

– Valid in microwave region (MASER)

• hƒ>>kT – Spontaneous emission

– Valid in visible region, incoherent Valid

2 1( ) /1

2

E E kTNe

N

h /1

2

ƒ kTNe

N

21

21

ƒ12

21

h /

ƒ

1

( )

[ ]kT

e

A

Bu

B

B

3

3 ƒh /

8 1( ) ( )

[ ]

ƒƒ

1kT

uc

h

e

2 21 21

2 21 21

3

3

8

( ) ( ) ( )

ƒ

ƒ ƒ ƒ

N A A hR

B u B u ucN

3

3 /

3

3

ƒh

8( )

8

ƒƒ

&

ƒƒ

1

1( ) ( )

[ ]kT

h

uc

uR

h

e

c

Types of LASER

There are three types of lasers

1. Solid Laser (Ruby Laser)

2. Liquid Laser

3. Gas Laser ( He – Ne Laser, CO2 Laser)

Ruby Laser

To produce laser from solid, Ruby crystal is used. Ruby is an aluminum oxide crystal

(Al2O

3) in which some of the aluminum atoms have been replaced with Cr

+3

chromium atoms

(0.05% by weight).

It was the first type of laser invented, and was first operated by Maiman in research

laboratories on 1960.

Chromium gives ruby its characteristic pink or red color by absorbing green and blue

light.

For a ruby laser, a crystal of ruby is formed into a cylinder. The ruby laser is used as a

pulsed laser, producing red light at 6943 Å.

Ruby crystal is surrounded by xenon tube. Ruby crystal is fully silvered at one side and

partially silvered at the other end.

A strong beam of blue green light is made to fall up on crystal from xenon tube and this

light is absorbed by the crystal.

Because of this, many electrons from ground state or normal state are raised to the

excited state or higher state and electron falls to metastable state.

During this transition photon is not emitted but excess energy of the electrons absorbed in

crystal lattice.

As electron drops to metastable state they remain there for certain time ~ 10-6

sec.

Thus, the incident blue green light from tube increases the number of electron in

metastable state and then the population inversion can be achieved.

If a light of different frequency is allowed to fall on this material, the electrons move

back and forth between silvered ends of the crystal.

While moving through they get stimulated and exited electrons radiate energy.

Thus readia photon has the same frequency as that of incident photon and is also in

exactly same phase.

When the intensity of light beam is increased the same process is repeated.

Finally extremely intensified beam of light energies from the semi silvered side of the

crystal.

This way it is possible to get extremely intensified and coherent beam of light from the

crystal. This beam is nothing but higher energetic beam – ie. LASER beam.

Applications of Ruby Laser

Ruby lasers have declined in use with the discovery of better lasing media. They are still

used in a number of applications where short pulses of red light are required. Holography's

around the world produce holographic portraits with ruby lasers, in sizes up to a meter squared.

Many non-destructive testing labs use ruby lasers to create holograms of large objects

such as aircraft tires to look for weaknesses in the lining.

Ruby lasers were used extensively in tattoo and hair removal.

Drawbacks of Ruby Laser

The laser requires high pumping power because the laser transition terminates at the

ground state and more than half of ground state atoms must be pumped to higher state to achieve

population inversion.

The efficiency of ruby laser is very low because only green component of the pumping

light is used while the rest of components are left unused.

The laser output is not continues but occurs in the form of pulses of microseconds

duration.

The defects due to crystalline imperfections are also present in this laser.

Gaseous Laser (He – Ne Laser)

Helium - neon laser, usually called a He-Ne laser, is a type of small gas laser. He-Ne

lasers have many industrial and scientific uses, and are often used in laboratory demonstrations

of optics.

He-Ne laser is an atomic laser which employs a four-level pumping scheme.

The active medium is a mixture of 10 parts of helium to 1 part of neon.

Neon atoms are centers and have energy levels suitable for laser transitions while helium

atoms help efficient excitation of neon atoms.

The most common wavelength is 6328 Å. These lasers produced powers in the range 0.5

to 50 mW in the red portion of the visible spectrum.

They have long operating life of the order of 50,000 hrs.

Construction

It consists of a glass discharge tube of about typically 30 cm long and 1.5 cm diameter.

The tube is filled with a mixture of helium and neon gases in the 10:1.

Electrodes are provided in the tube to produce a discharge in the gas.

They are connected to a high voltage power supply. The tube is hermetically sealed with

glass windows oriented at Brewster angle to the tube. The cavity mirrors are arranged externally.

Working

When the power is switched on, a high voltage of about 10 kV is applied across the gas.

It is sufficient to ionize the gas.

The electrons and ions are produced in the process of discharge are accelerated toward

the anode and cathode respectively.

The electron have a smaller mass, they acquire a higher velocity. They transfer their

kinetic energy to helium atoms through inelastic collisions.

The initial excitation effects only the helium atoms. They are in metastable state and

cannot return in ground state by the spontaneous emission.

The excited helium atoms can return to the ground state by transforming their energy to

neon atoms through collision. These transformations take place when two colliding atoms have

initial energy state. It is called resonant transfer of energy.

So, the pumping mechanism of He-Ne Laser is when the helium atom in the metastable

state collides with neon atom in the ground state the neon atom is excited and the helium atom

drops back to the ground state.

The role of helium atom is thus to excite neon atom and cause, population inversion. The

probability of energy transfer from helium atoms to neon atoms is more as there are 10 atoms of

helium per 1 neon atom in gas mixture.

Without the Brewster windows, the light output is unpolarized; because of it laser output

to be linearly polarized.

When the excited Ne atom passes from metastable state (3s) to lower level (2p), it emits

photon of wavelength 632 nm.

This photon travels through the gas mixture parallel to the axis of tube; it is reflected

back and forth by the mirror ends until it stimulates an excited Ne atom and causes it to emit a

photon of 632 nm with the stimulating photon.

The stimulated transition from (3s) level to (2p) level is laser transition.

Although 6328 Å is standard wavelength of He-Ne Laser, other visible wavelengths 5430

Å (Green) 5940 Å (yellow-orange), 6120 Å (red-orange) can also produce.

Overall gain is very low and is typically about 0.010 % to 0.1 %.

The laser is simple practical and less expensive.

The Laser beam is highly collimated, coherent and monochromatic.

Applications of He-Ne Laser

The Narrow red beam of He-Ne laser is used in supermarkets to read bar codes.

The He-Ne Laser is used in Holography in producing the 3D images of objects.

He-Ne lasers have many industrial and scientific uses, and are often used in laboratory

demonstrations of optics.

Semiconductor Laser (Diode Laser)

A semiconductor laser is a laser in which a semiconductor serves as a photon source.

The most common semiconductor material that has been used in lasers is gallium

arsenide.

Einstein’s Photoelectric theory states that light should be understood as discrete lumps of

energy (photons) and it takes only a single photon with high enough energy to knock an electron

loose from the atom it's bound to.

Stimulated, organized photon emission occurs when two electrons with the same energy

and phase meet. The two photons leave with the same frequency and direction.

P-type Semiconductors

In the compound GaAs, each Ga atom has three electrons in its outermost shell of

electrons and each As atom has five.

When a trace of an impurity element with two outer electrons, such as Zn (zinc), is added

to the crystal.

The result is the shortage of one electron from one of the pairs, causing an imbalance in

which there is a “hole” for an electron but there is no electron available. This forms a p-type

semiconductor.

N-type Semiconductors

When a trace of an impurity element with six outer electrons, such as Se (selenium), is

added to a crystal of GaAs, it provides on additional electron which is not needed for the

bonding.

This electron can be free to move through the crystal. Thus, it provides a mechanism for

electrical conductivity. This type is called an n-type semiconductor.

Under forward bias (the p-type side is made positive) the majority carriers, electrons in

the n-side, holes in the p-side, are injected across the depletion region in both directions to create

a population inversion in a narrow active region. The light produced by radioactive

recombination across the band gap is confined in this active region.

Application of Lasers

1. Laser beam is used to measure distances of sun, moon, stars and satellites very

accurately.

2. It can be used for measuring velocity of light, to study spectrum of matters, to study

Raman effect.

3. It can be is used for increasing speed and efficiency of computer.

4. It is used for welding.

5. It is used in biomedical science.

6. It is used in 3D photography.

7. It is used for communication, T. V. transmission, to search the objects under sea.

8. It can be used to predict earthquake.

9. Laser tools are used in surgery.

10. It is used for detection and treatment of cancer.

11. It is used to aline straight line for construction of dam, tunnels etc.

12. It is used in holography.

13. It is used in fiber optic communication.

14. It is also used in military, like LIDAR.

15. It is used to accelerate some chemical reactions.

Special Theory of Relativity

Introduction to Relativity

o The dependence of various physical phenomena on relative motion of the observer

and the observed objects, especially regarding the nature and behaviour of light,

space, time, and gravity is called relativity.

o When we have two things and if we want to find out the relation between their

physical property i.e.velocity,accleration then we need relation between them that

which is higher and which is lower.In general way we reffered it to as a relativity.

o The famous scientist Einstein has firstly found out the theory of relativity and he has

given very useful theories in relativity.

o In 1905, Albert Einstein determined that the laws of physics are the same for all non-

accelerating observers, and that the speed of light in a vacuum was independent of the

motion of all observers. This was the theory of special relativity.

FRAMES OF REFERENCE

o A Reference Frame is the point of View, from which we Observe an Object.

o A Reference Frame is the Observer it self, as the Velocity and acceleration are

common in Both.

o Co-ordinate system is known as FRAMES OF REFERENCE

o Two types:

1. Inertial Frames Of Reference.

2. non-inertial frame of reference.

o We have already come across idea of frames of reference that move with constant

velocity. In such frames, Newton’s law’s (esp. N1) hold. These are called

inertial frames of reference.

o Suppose you are in an accelerating car looking at a freely moving object (I.e., one

with no forces acting on it). You will see its velocity changing because you are

accelerating! In accelerating frames of reference, N1 doesn’t hold – this is a non-

inertial frame of reference.

Galilean Transforms

o Parallel axes (for convenience)

o K’ has a constant relative velocity in the x-direction with respect to K

o Time (t) for all observers is a

Fundamental invariant,

i.e., the same for all inertial observers

o Galilean Transformation Inverse Relations

o Step 1. Replace with .

o Step 2. Replace “primed” quantities with

“unprimed” and “unprimed” with “primed.”

o General Galilean Transformations

o Newton’s Eqn of Motion is same at face-value in both reference frames

Einstein’s postulates of special theory of relativity

o The First Postulate of Special Relativity

The first postulate of special relativity states that all the laws of nature are the

same in all uniformly moving frames of reference.

o The Second Postulate of Special Relativity

The second postulate of special relativity states that the speed of light in empty

space will always have the same value regardless of the motion of the source

or the motion of the observer.

The speed of a light flash emitted by either the spaceship or the space station is

measured as c by observers on the ship or the space station. Everyone who measures

the speed of light will get the same value, c.

The Ether

o Light is a wave.

o Waves require a medium through which to propagate.

o Medium as called the “ether.” (from the Greek aither, meaning upper air)

o Maxwell’s equations assume that light obeys the Newtonian-Galilean transformation.

The Ether: Since mechanical waves require a medium to propagate, it was generally

accepted that light also require a medium. This medium, called the ether, was

assumed to pervade all mater and space in the universe.

The Michelson-Morley Experiment

o Experiment designed to measure small changes in the speed of light was performed

by Albert A. Michelson (1852 – 1931, Nobel ) and Edward W. Morley (1838 – 1923).

o Used an optical instrument called an interferometer that Michelson invented.

o Device was to detect the presence of the ether.

o Outcome of the experiment was negative, thus contradicting the ether hypothesis.

o Michelson developed a device called an inferometer.

o Device sensitive enough to detect the ether.

o Apparatus at rest wrt the ether.

o Light from a source is split by a half silvered mirror (M)

o The two rays move in mutually perpendicular directions

o The rays are reflected by two mirrors (M1 and M2) back to M where they recombine.

o The combined rays are observed at T.

o The path distance for each ray is the same (l1=l2).

o Therefore no interference will be observed

o Apparatus at moving through the ether.

o First consider the time required for the parallel ray

o Distance moved during the first part of the path is

o Similarly the time for the return trip is

o The total time

o For the perpendicular ray ,we can write, from fig.

||

Lt

(c u )

ct L ut

Lt

(c u )

||

2 2

2 2

( ) ( )

2

( )

2 /

1

L Lt

c u c u

Lc

c u

L c

u c

o The return path is the same as the initial leg therefore the total time is

o The time difference between the two rays is,

o The expected time difference is too small to be measured directly!

o Instead of measuring time, Michelson and Morley looked for a fringe change.

o as the mirror (M) was rotated there should be a shift in the interference fringes.

Results of the Experiment

A NULL RESULT

o No time difference was found!

o Hence no shift in the interference patterns

Conclusion from Michelson-Morley Experiment

o the ether didn’t exist.

The Lorentz Transformation

We are now ready to derive the correct transformation equations between two inertial

frames in Special Relativity, which modify the Galilean Transformation. We consider

two inertial frames S and S’, which have a relative velocity v between them along the

x-axis.

2 2 2

2 2 2 2 2

2 2 2

2 2

)

( )

( initial leg of the patct L ut

L c t u t

c u t

Lt

c u

h

2 2

2 2

2

2 /

1

Lt

c u

L ct

u c

12

12 2

|| 2 2

2 2

2 3

21 1

2

2

L u ut t t

c c c

After a binom ial expansi

L u Lut

c c c

on

Now suppose that there is a single flash at the origin of S and S’ at time , when the

two inertial frames happen to coincide. The outgoing light wave will be spherical in

shape moving outward with a velocity c in both S and S’ by Einstein’s Second

Postulate.

We expect that the orthogonal coordinates will not be affected by the horizontal

velocity:

But the x coordinates will be affected. We assume it will be a linear transformation:

But in Relativity the transformation equations should have the same form (the laws of

physics must be the same). Only the relative velocity matters. So,

Consider the outgoing light wave along the x-axis (y = z = 0).

Now plug these into the transformation equations:

Plug these two equations into the light wave equation:

x

y

z

S

x'

y'

z '

S' v

2 2 2 2 2

2 2 2 2 2

x y z c t

x y z c t

y y

z z

x k x vt

x k x vt

k k

x ct in fram e S '

x ct in fram e S

x k x vt k ct vt kct 1 v / c and

x k x vt k ct vt kct 1 v / c

ct x kct 1 v / c

ct x kct 1 v / c

t kt 1 v / c

t kt 1 v / c

o Plug t’ into the equation for t:

o So the modified transformation equations for the spatial coordinates are:

o Now what about time?

o Solve for t’:

o So the correct transformation (and inverse transformation) equations are:

2

2 2 2

2 2

t k t 1 v / c 1 v / c

1 k 1 v / c

1k

1 v / c

x x vt

y y

z z

x x vt

x x vt inverse transform ation

Plug one into the other:

x x vt vt

2 2

2 2

2 2

2

2 2

2 2 2 2

2 2 2 2

2

x x vt vt

x 1 vt vt

1 v / c 1x vt vt

1 v / c

xv / c vt vt

1t xv / c vt

v

t t vx / c

2 2

x x vt x x vt

y y y y

z z z z

t t vx / c t t vx / c

The Lorentz

Transformation

Application of Lorentz Transformation

Time Dilation

We explore the rate of time in different inertial frames by considering a special kind

of clock – a light clock – which is just one arm of an interferometer. Consider a light

pulse bouncing vertically between two mirrors. We analyze the time it takes for the

light pulse to complete a round trip both in the rest frame of the clock (labeled S’),

and in an inertial frame where the clock is observed to move horizontally at a velocity

v (labeled S).

In the rest frame S’

Now put the light clock on a spaceship, but measure the roundtrip time of the light

pulse from the Earth frame S:

So the time it takes the light pulse to make a roundtrip in the clock when it is moving

by us is appears longer than when it is at rest. We say that time is dilated. It also

doesn’t matter which frame is the Earth and which is the clock. Any object that

moves by with a significant velocity appears to have a clock running slow. We

summarize this effect in the following relation:

m irror

m irror

L

L

c t / 2

v t / 2

1

2

1 2

Lt = tim e up

c

Lt = tim e dow n

c

2L=t t

c

1

2

2 2 2 2 2

2 2 2 2

2

2

2 2

2 2 2 2

tt tim e up

2

tt tim e dow n

2

T he speed of light is still c in this fram e, so

L v t / 4 c t / 4

L c v t / 4

4Lt

c v

2L 1t

c 1 v / c 1 v / c

2 2

1t , 1

1 v / c

Length Contraction

o Now consider using a light clock to measure the length of an interferometer arm. In

particular, let’s measure the length along the direction of motion.

o In the rest frame S’:

o Now put the light clock on a spaceship, but measure the roundtrip time of the light

pulse from the Earth frame S:

o In other words, the length of the interferometer arm appears contracted when it moves by us.

This is known as the Lorentz-Fitzgerald contraction. It is closely related to time dilation. In

fact, one implies the other, since we used time dilation to derive length contraction.

A A’ C C’

vt1 L 1

2

1 2

1 1 1

2 2 2

1 2 2 2 2 2

2 2

2 2

1 2 2 2 2 2

2 2

t tim e out

t tim e back

t t t

LL vt ct t

c v

LL vt ct t

c v

2Lc 2L 1t t t

c v c 1 v / c

ctL 1 v / c

2

But, t from tim e dilation

1 v / c

2Lc 2L 1t t t

c v c 1 v / c

ctL 1 v / c

2

But, t

1 v2 2

0

2 2

from tim e dilation

/ c

L 1L 1

1 v / c

Superconductivity

Introduction of superconductivity

Superconductivity is a phenomenon of exactly zero electrical resistance and expulsion of

magnetic fields occurring in certain materials when cooled below a characteristic critical

temperature. It was discovered by Dutch physicist Heike Kamerlingh Onnes on April 8,

1911 in Hg, which has critical temperature of 4.2 K.

Properties of Superconductors

(1) Electrical Resistance

Zero Electrical Resistance

Defining Property

Critical Temperature

Quickest test

10-5

Ωcm

(2) Effect of Magnetic Field

Critical magnetic field (HC) – Minimum magnetic field required to destroy

the superconducting property at any temperature.

2

01

C

C

TH H

T

H0 - Critical field at 0K

T - Temperature below TC

TC - Transition Temperature

Element HC at 0K

(mT)

Nb 198

Pb 80.3

Sn 30.9

(3) Effect of Electric Current

Large electric current – induces magnetic field – destroys superconductivity

Induced Critical Current iC = 2πrHC

Persistent Current

Steady current which flows through a superconducting ring without any

decrease in strength even after the removal of the field.

Diamagnetic property.

Meissner effect

When Superconducting material cooled bellow its Tc it becomes resistenceless

& perfect diamagnetic.

When superconductor placed inside a magnetic field in Tc all magnetic flux is

expelled out of it the effect is called Meissner effect.

Perfect diamagnetism arises from some special magnetic property of

Superconductor.

If there is no magnetic field inside the superconductor relative permeability

or diamagnetic constant μr =0.

Total magnetic induction B is,

If magnetic induction B=0 then,

Magnetic Flux Quantization

Magnetic flux enclosed in a superconducting ring = integral multiples of fluxon

Φ = nh/2e = n Φ0 ; (Φ0 = 2x10-15

Wb)

Effect of Pressure

Pressure ↑, TC ↑

High TC superconductors – High pressure

0( )B H M

00 ( )H M

M H

1m

M

H

Thermal Properties

Entropy & Specific heat ↓ at TC

Disappearance of thermo electric effect at TC

Thermal conductivity ↓ at TC – Type I superconductors

Stress

Stress ↑, dimension ↑, TC ↑, HC affected

Frequency

Frequency ↑, Zero resistance – modified, TC not affected

Impurities

Magnetic properties affected

Size

Size < 10-4

cm – superconducting state modified

General Properties

No change in crystal structure

No change in elastic & photo-electric properties

No change in volume at TC in the absence of magnetic field

Isotope Effect

Maxwell

TC = Constant / Mα

TC Mα = Constant (α – Isotope Effect coefficient)

α = 0.15 – 0.5

α = 0 (No isotope effect)

TC√M = constant

Classification & characterization of superconductor

Type - I or soft superconductor

o Exhibit complete Meissner effect.

o Bellow Hc super conductor above Hc Normal

o Value of Hc is order of 0.1 T.

o Aluminum, lead & Indium are type - I super conductor

o Not used as strong electromagnets

Type - II or Hard superconductor

o Exhibit complete Meissner effect bellow a certain critical field Hc1 at

this point diamagnetism & superconductivity ↓. This state is mix state

called vortex state.

o At certain critical field Hc2 superconductivity disappears.

o Niobium, Aluminum, Silicon, Ceramic are type - II superconductors.

o Pb is type I superconductor ac Hc = 600 gauss at 4º K when a small

impurity of In is added it becomes type II superconductor with Hc1 =

400 gauss & Hc2 = 1000 gauss.

London equation

According to London’s theory there are two type of

electrons in SC.

o Super electrons

o Normal electrons

o At 0º K there are only Super electrons.

o With increasing temp. Super electrons ↓ Normal electrons ↑ .

o Let nn, un & ns, us are no. density & drift velocity of normal electrons

& super electrons respectively.

Equation of motion of Super electrons under electric field is,

Now current & drift velocity are related as,

sdu

m eEdt

s s s

s s s

s

s

s

I n eAu

J n eu

Ju

n e

2

( )

s

s

s s

Jd

n ee E

dt

n e Ed J

dt m

This is London's first equation.

- London's first equation gives absence of resistance. If E = 0 then,

- Now from Maxwell's eqns.

0s

dJ

dt

( )

d BE

dt

B A

d AE

dt

d AE

dt

d AE

dt

2

2

2

2

2

2

( )

( )

s s

s

s

s

s

s

s

s

s

ss

n e Ed J

dt m

d J mE

dt n e

d J m d A

dt n e dt

d m d AJ

dt n e dt

mJ A

n e

n eJ A

m

This is London's Second equation

- Again from ampere Law,

- Take curl on both sides

λ is called London penetration depth.

BCS Theory of Superconductivity

The properties of Type I superconductors were modeled successfully by the

efforts of John Bardeen, Leon Cooper, and Robert Schrieffer in what is

commonly called the BCS theory.

A key conceptual element in this theory is the pairing of electrons close to

the Fermi level into Cooper pairs through interaction with the crystal lattice.

0

2

0( )

s

s

B J

n eB A

m

2

0

2

2

2

0

( )

&

( )

s

s

n eB A

m

Now

B B B A B

n eB B B

m

A B

This pairing results form a slight attraction between the electrons related to lattice

vibrations; the coupling to the lattice is called a phonon interaction.

Pairs of electrons can behave very differently from single electrons which are

fermions and must obey the Pauli exclusion principle.

Cooper Pairs:

The transition of a metal from the normal to the superconducting state has the

nature of a condensation of the electrons into a state which leaves a band gap

above them.

This kind of condensation is seen with super fluid helium, but helium is made up

of bosons -- multiple electrons can't collect into a single state because of the Pauli

exclusion principle.

Froehlich was first to suggest that the electrons act as pairs coupled by lattice

vibrations in the material.

This coupling is viewed as an exchange of phonons, phonons being the quanta of

lattice vibration energy.

Experimental corroboration of an interaction with the lattice was provided by the

isotope effect on the superconducting transition temperature.

The boson-like behavior of such electron pairs was further investigated by

Cooper and they are called "Cooper pairs".

The condensation of Cooper pairs is the foundation of the BCS theory of

superconductivity.

s

In the normal state of a metal, electrons move independently, whereas in the BCS

state, they are bound into "Cooper pairs" by the attractive interaction. The BCS

formalism is based on the "reduced" potential for the electrons attraction.

You have to provide energy equal to the 'energy gap' to break a pair, to break one pair

you have to change energies of all other pairs.

This is unlike the normal metal, in which the state of an electron can be changed by

adding a arbitrary small amount of energy.

The energy gap is highest at low temperatures but does not exist at temperatures

higher than the transition temperature.

The BCS theory gives an expression of how the gap grows with the strength of

attractive interaction and density of states.

The BCS theory gives the expression of the energy gap that depends on the

Temperature T and the Critical Temperature Tc and is independent of the material:

Applications of Superconductors

Engineering:

Transmission of power

Switching devices

Sensitive electrical instruments

Memory (or) storage element in computers.

Manufacture of electrical generators and transformers

Medical:

Nuclear Magnetic Resonance (NMR)

Diagnosis of brain tumor

Magneto – hydrodynamic power generation

Josephson effect or Devices

Principle: persistent current in d.c. voltage.

Josephson junctions

A type of electronic circuit capable of switching at very high speeds when

operated at temperatures approaching absolute zero.

Named for the British physicist who designed it,

A Josephson junction exploits the phenomenon of superconductivity.

Construction

A Josephson junction is made up of two superconductors, separated by a non-

superconducting layer so thin that electrons can cross through the insulating barrier.

The flow of current between the superconductors in the absence of an applied voltage is

called a Josephson current,

The movement of electrons across the barrier is known as Josephson tunneling.

Two or more junctions joined by superconducting paths form what is called a Josephson

interferometer.

Consists of superconducting ring having magnetic fields of quantum values (1,2,3..)

Placed in between the two Josephson junctions.

Explanation:

Consists of thin layer of insulating material placed between two

superconducting materials.

Insulator acts as a barrier to the flow of electrons.

When voltage applied current flowing between super conductors by tunneling

effect.

Quantum tunneling occurs when a particle moves through a space in a manner

forbidden by classical physics, due to the potential barrier involved

Components of current

In relation to the BCS theory (Bardeen Cooper Schrieffer) mentioned earlier,

pairs of electrons move through this barrier continuing the superconducting

current. This is known as the dc current.

Current component persists only till the external voltage application. This is ac

current.

Uses of Josephson devices

Magnetic Sensors

Gradiometers

Oscilloscopes

Decoders

Analogue to Digital converters

Oscillators

Microwave amplifiers

Sensors for biomedical, scientific and defence purposes

Digital circuit development for Integrated circuits

Microprocessors

Random Access Memories (RAMs)

Super conducting Quantum Interference Devices

Discovery:

The DC SQUID was invented in 1964 by Robert Jaklevic, John Lambe, Arnold Silver,

and James Mercereau of Ford Research Labs

Principle:

Small change in magnetic field, produces variation in the flux quantum.

Construction:

The superconducting quantum interference device (SQUID) consists of two

superconductors separated by thin insulating layers to form two parallel Josephson

junctions.

Type:

Two main types of SQUID:

1) RF SQUIDs have only one Josephson junction

2) DC SQUIDs have two or more junctions.

Thereby,

More difficult and expensive to produce.

Much more sensitive.

Fabrication:

Lead or pure niobium, the lead is usually in the form of an alloy with 10% gold or

indium, as pure lead is unstable when its temperature is repeatedly changed.

The base electrode of the SQUID is made of a very thin niobium layer.

The tunnel barrier is oxidized onto this niobium surface.

The top electrode is a layer of lead alloy deposited on top of the other two, forming a

sandwich arrangement.

To achieve the necessary superconducting characteristics, the entire device is then

cooled to within a few degrees of absolute zero with liquid helium.

Uses:

Storage device for magnetic flux.

Study of earthquakes.

Removing paramagnetic impurities.

Detection of magnetic signals from brain, heart etc.

Cryotron:

The cryotron is a switch that operates using superconductivity.

The cryotron works on the principle that magnetic fields destroy superconductivity.

The cryotron is a piece of tantalum wrapped with a coil of niobium placed in a liquid

helium bath.

When the current flows through the tantalum wire it is superconducting, but when a

current flows through the niobium a magnetic field is produced.

This destroys the superconductivity which makes the current slow down or stop.

Magnetic Levitated Train:

Principle: Electro-magnetic induction

Introduction:

Magnetic levitation transport, or maglev, is a form of transportation that suspends

guides and propels vehicles via electromagnetic force.

This method can be faster than wheeled mass transit systems, potentially reaching

velocities comparable to turboprop and jet aircraft (500 to 580 km/h).

Why superconductor?

Superconductors may be considered perfect diamagnets (μr = 0), completely expelling

magnetic fields due to the Meissner effect. The levitation of the magnet is stabilized

due to flux pinning within the superconductor. This principle is exploited by EDS

(Electrodynamics suspension) magnetic levitation trains.

In trains where the weight of the large electromagnet is a major design issue (a very

strong magnetic field is required to levitate a massive train) superconductors are used

for the electromagnet, since they can produce a stronger magnetic field for the same

weight.

How to use a Super conductor?

Electrodynamics suspension

In Electrodynamic suspension (EDS), both the rail and the train exert a magnetic

field, and the train is levitated by the repulsive force between these magnetic fields.

The magnetic field in the train is produced by either electromagnets or by an array of

permanent magnets.

The repulsive force in the track is created by an induced magnetic field in wires or

other conducting strips in the track.

At slow speeds, the current induced in these coils and the resultant magnetic flux is

not large enough to support the weight of the train.

For this reason the train must have wheels or some other form of landing gear to

support the train until it reaches a speed that can sustain levitation.

Propulsion coils on the guide way are used to exert a force on the magnets in the train

and make the train move forwards.

The propulsion coils that exert a force on the train are effectively a linear motor: An

alternating current flowing through the coils generates a continuously varying

magnetic field that moves forward along the track.

The frequency of the alternating current is synchronized to match the speed of the

train.

The offset between the field exerted by magnets on the train and the applied field

create a force moving the train forward.

Advantages:

No need of initial energy in case of magnets for low speeds

One liter of Liquid nitrogen costs less than one liter of mineral water

Onboard magnets and large margin between rail and train enable highest recorded

train speeds (581 km/h) and heavy load capacity. Successful operations using high

temperature superconductors in its onboard magnets, cooled with inexpensive liquid

nitrogen

Magnetic fields inside and outside the vehicle are insignificant; proven, commercially

available technology that can attain very high speeds (500 km/h); no wheels or

secondary propulsion system needed

Free of friction as it is “Levitating”

Atomic Physics

“Classical Physics”:

developed in 15th

to 20th

century,provides very successful description of “every day, ordinary

objects”

motion of trains, cars, bullets,….

orbit of moon, planets

how an engine works,..

subfields: mechanics, thermodynamics, electrodynamics,

Quantum Physics:

developed early 20th

century, in response to shortcomings of classical physics in describing

certain phenomena (blackbody radiation, photoelectric effect, emission and absorption

spectra…)describes “small” objects (e.g. atoms )

QP is “weird and counterintuitive”

“Those who are not shocked when they first come across quantum theory cannot possibly

have understood it” (Niles Bohr)

“Nobody feels perfectly comfortable with it “ (Murray Gell-Mann)

“I can safely say that nobody understands quantum mechanics” (Richard Feynman)

BUT…

QM is the most successful theory ever developed by humanity underlies our

understanding of atoms, molecules, condensed matter, nuclei, elementary particles

Crucial ingredient in understanding of stars, …

Quantum physics is basically the recognition that there is less difference between waves

and particles than was thought before

key insights:

light can behave like a particle

particles (e.g. electrons) are indistinguishable

particles can behave like waves (or wave packets)

waves gain or lose energy only in "quantized amounts“

detection (measurement) of a particle wave will change suddenly into a new wave

quantum mechanical interference – amplitudes add

QP is intrinsically probabilistic

what you can measure is what you can know

WAVE-PICTURE OF RADIATION—ENERGY FLOW I S CONTI N UOUS

• Radio waves, microwaves, heat waves, light waves, UV-rays, x-rays and y-rays belong to

the family of electromagnetic waves. All of them are known as radiation.

• Electromagnetic waves consist of varying electric and magnetic fields traveling at the

velocity of 'c'. The proMaxwell's theory treated the emission of radiation by a source as a

continuous process.

• A heated body may be assumed to be capable of giving out energy that travels in the form

of waves of all possible wavelengths.

• In the same way, the radiation incident on a body was thought to be absorbed at all

possible wavelengths.

• The intensity of radiation is given by,

I = 1E12

• where E is the amplitude of the electromagnetic wave.

• pagation of electromagnetic waves and their interaction with matter can be explained

with the help of Maxwell's electromagnetic theory.

• The phenomena of interference, diffraction and polarization of electromagnetic radiation

proved the wave nature of radiation.

• Therefore, it is expected that it would explain the experimental observations made on

thermal (heat) radiation emitted by a blackbody.

Blackbody radiation and Planck hypothesis

• Two patches of clouds in physics sky at the beginning of 20th

century.

• The speed of light Relativity

• The blackbody radiation foundation of Quantum theory

• Convection is transfer of heat by actual motion of. The hot-air furnace, the hot-water

heating system, and the flow of blood in the body are examples.

• Radiation The heat reaching the earth from the sun cannot be transferred either by

conduction or convection since the space between the earth and the sun has no material

medium. The energy is carried by electromagnetic waves that do not require a material

medium for propagation. The kind of heat transfer is called thermal radiation.

• Blackbody is defined as the body which can absorb all energies that fall on it. It is

something like a black hole. No lights or material can get away from it as long as it is

trapped. A large cavity with a small hole on its wall can be taken as a blackbody.

LAWS OF BLACK BODY RADIATION

1. Stefan and Boltzmann’s law: it is found that the radiation energy is proportional to the fourth

power of the associated temperature. 4M (T ) T

2. Wien’s displacement law: the peak of the curve shifts towards longer

wavelength as the temperature falls and it satisfies

where b is called the Wien's constant. b=2.89X10-3

4M (T ) T

peakT b

This law is quite useful for measuring the temperature of a blackbody with a very high

temperature. You can see the example for how to measure the temperature on the surface of the

sun.

• The above laws describes the blackbody radiation very well.

• The problem exists in the relation between the radiation power Mλ(T) and the

wavelength λ.

• Blackbody radiation has nothing to do with both the material used in the blackbody

concave wall and the shape of the concave wall.

• Two typical theoretical formulas for blackbody radiation : One is given by Rayleigh and

Jeans and the other by Wein.

3.Rayleigh and Jeans

• In 1890, Rayleigh and Jeans obtained a formula using the classical electromagnetic

(Maxwell) theory and the classical equipartition theorem of energy in thermotics. The

formula is given by

2

3

8 kTE ( )

c

Rayleigh-Jeans formula was correct for very long wavelength in the far infrared but hopelessly

wrong in the visible light and ultraviolet region. Maxwell‟s electromagnetic theory and

thermodynamics are known as correct theory. The failure in explaining blackbody radiation

puzzled physicists! It was regarded as ultraviolet Catastrophe (disaster).

4. Planck Radiation Law:

Where,

E=Quantum energy

h= Planck constant

v= frequency

PLANCK'S QUANTUM HYPOTHESIS — Energy is quantized

• Max Planck empirical formula explained the experimental observations.

• In the process of formulation of the formula, he assumed that the atoms of the walls of

the blackbody behave like small harmonic oscillators, each having a characteristic

frequency of vibration, lie further made two radical assumptions about the atomic

oscillators.

• An oscillating atom can absorb or mends energy in discrete units. The indivisible discrete

unit of energy hs, is the smallest amount of energy which can be absorbed or emitted by

the atom and is called an energy quantum. A quantum of energy has the magnitude given

by

E = hv

4M (T ) T

hcE h

where v is the frequency of radiation and „h' is a constant now known as the Planck's

constant.

• The energy of the oscillator is quantized. It can have only certain discrete amounts of

energy En.

En= nhv n=1,2,3……

• The hypothesis that radiant energy is emitted or absorbed basically in a discontinuous

summer and in the form of quanta is known as the Planck's quantum hypothesis.

• Planck's hypothesis states that radiant energy Is quantized and implies that an atom exists

in certain discrete energy states. Such states arc called quantum stales and n is called the

quantum number.

• The atom emits or absorbs energy by jumping from one quantum state to another

quantum state. The assumption of discrete energy states for an atomic oscillator (Fig.a)

was a departure from the classical physics and our everyday exper

• If we take a mass-spring harmonic oscillator, it can receive any amount of energy form

zero to some maximum value (Fig.b). Thus, in the realm of classical physics energy

always appears to occur with continuous values and energy exchange between bodies

involves any arbitrary amounts of energy.

PARTICLE PICTURE OF RADIATION —Radiation is a stream of photons

Max Planck introduced the concept of discontinuous emission and absorption of radiation

by bodies but he treated the propagation through space as occurring in the form of

continuous waves as demanded by electromagnetic theory.

• Einstein refined the Planck's hypothesis and invested the quantum with a clear and

distinct identity.

• He successfully explained the experimental results of the photoelectric effect in 1905 and

the temperature dependence of specific heats of solids in 1907 basing on Planck's

hypothesis.

• The photoelectric effect conclusively established that light behaves as a swam of

particles. Einstein extended Planck's hypothesis as follows:

1 Einstein assumed that the light energy is not distributed evenly over the whole

expanding wave front but rather remains concentrated in discrete quanta. He

named the energy quanta as photons. Accordingly, a light beam is regarded as a

stream of photons travelling with a velocity ' c' .

2 An electromagnetic wave having a frequency f contains identical photons, each

having an energy hƒ. The higher the frequency of the electromagnetic wave, the

higher is the energy content of each photon.

3. An electromagnetic wave would have energy hƒ if it contains only one photon.

2hv if it contains 2 photons and so on. Therefore, the intensity of a

monochromatic light beam I. is related to the concentration of photons. N. present

in the beam. Thus,

I = N hƒ

Note that according to electromagnetic theory, the intensity of a light beam is given by

I = 1E12

4. When photons encounter matter, they impart all their energy to the panicles of matter and

vanish. That is why absorption of radiation is discontinuous. The number of photons

emitted by even a weak light source is enormously large and the human eye cannot

register the photons separately and therefore light appears as a continuous stream. Thus,

the discreteness of light is not readily apparent.

The Photon

• As the radiant energy is viewed as made up of spatially localized photons. we may

attribute particle properties to photons.

1. Energy: The energy of a photon is determined by its frequency v and is given by E = hƒ.

Using the relation ω= 2π and writing h/2π = ħ. we may express E= ħω

2. Velocity: Photons always travel with the velocity of light „c'.

3. Rest Mass: The rest mass of photon is zero since a photon can never be at rest. Thus, m0=

0

4. Relativistic mass: As photon travels with the velocity of light, it has relativistic mass.

given by m= E/c2 = hv/c

2

5. Linear Momentum: The linear momentum associated with a photon may be expressed as

p=E/c=hv/c= h/λ

As the wave vector k= 2π/λ , p = hk/ 2π = ħk.

6. Angular Momentum: Angular momentum is also known as spin which is the intrinsic

property of all microparticles. Photon has a spin of one unit. Thus. s = lħ.

7. Electrical Charge: Photons are electrically neutral and cannot be influenced by electric or

magnetic fields. They cannot ionize matter.

Example: 1

Calculate the photon energies for the following types of electromagnetic radiation:

(a) a 600kHz radio wave; (b) the 500nm (wavelength of) green light; (c) a 0.1 nm

(wavelength of) X-rays.

Solution:

(a) for the radio wave, we can use the Planck-Einstein law directly

15 3

9

E h 4.136 10 eV s 600 10 Hz

2.48 10 eV

(b) The light wave is specified by wavelength, we can use the law explained in wavelength:

6

9

hc 1.241 10 eV mE 2.26eV

550 10 m

(c). For X-rays, we have

6

4

9

hc 1.241 10 eV mE 1.24 10 eV 12.4keV

0.1 10 m

Photoelectric Effect:-

The quantum nature of light had its origin in the theory of thermal radiation and was

strongly reinforced by the discovery of the photoelectric effect.

Fig. Apparatus to investigate the photoelectric effect that was first found in 1887 by Hertz.

In figure , a glass tube contains two electrodes of the same material, one of which is irradiated by

light. The electrodes are connected to a battery and a sensitive current detector measures the

current flow between them.

The current flow is a direct measure of the rate of emission of electrons from the irradiated

electrode.

The electrons in the electrodes can be ejected by light and have a certain amount of kinetic

energy. Now we change:

(1) the frequency and intensity of light,

(2) the electromotive force (e.m.f. or voltage),

(3) the nature of electrode surface.

It is found that:

(1). For a given electrode material, no photoemission exists at all below a certain frequency of

the incident light. When the frequency increases, the emission begins at a certain frequency. The

frequency is called threshold frequency of the material. The threshold frequency has to be

measured in the existence of e.m.f. (electromotive force) as at such a case the photoelectrons

have no kinetic energy to move from the cathode to anode . Different electrode material has

different threshold frequency.

(2). The rate of electron emission is directly proportional to the intensity of the incident light.

Photoelectric current ∝ The intensity of light

(3). Increasing the intensity of the incident light does not increase the kinetic energy of the

photoelectrons.

Intensity of light ∝ kinetic energy of photoelectron

However increasing the frequency of light does increase the kinetic energy of photoelectrons

even for very low intensity levels.

Frequency of light ∝ kinetic energy of photoelectron

(4). There is no measurable time delay between irradiating the electrode and the emission of

photoelectrons, even when the light is of very low intensity. As soon as the electrode is

irradiated, photoelectrons are ejected.

(5) The photoelectric current is deeply affected by the nature of the electrodes and chemical

contamination of their surface.

In 1905, Einstein solved the photoelectric effect problem by applying the Planck‟s hypothesis.

He pointed out that Planck‟s quantization hypothesis applied not only to the emission of

radiation by a material object but also to its transmission and its absorption by another material

object. The light is not only electromagnetic waves but also a quantum. All the effects of

photoelectric emission can be readily explained from the following assumptions:

Therefore we have the equation of photoelectric effect:

21

2h A mv

Using this equation and Einstein‟s assumption, you could readily explain all the results in the

photoelectric effect: why does threshold frequency exist (problem)? why is the number of

photoelectrons proportional to the light intensity? why does high intensity not mean high

photoelectron energy (problem)? why is there no time delay (problem)?

Example: Ultraviolet light of wavelength 150nm falls on a chromium electrode. Calculate

the maximum kinetic energy and the corresponding velocity of the photoelectrons (the

work function of chromium is 4.37eV).

Solution: using the equation of the photoelectric effect, it is convenient to express the energy in

electron volts. The photon energy is

6

9

1.241 108.27

150 10

hc eV mE h eV

m

2

2

1

2

1(8.27 4.37) 3.90

2

h A mv

mv eV eV

19 19 19 2 21 1.602 10 1.602 10 1.602 10eV J N m kg m s

2 19 2 213.90 3.90 1.602 10

2mv eV kg m s

19

6

31

2 3.90 12.496 101.17 10 /

9.11 10

eVv m s

m

EXERCISE:-

1. The wavelength of yellow light is 5890 A. What is the energy of the photons in the

beam? Empress in electron volts.

2. 77w light sensitive compound on most photographic films is silver bromide, Aglin A film

is exposed when the light energy absorbed dissociates this molecule into its atoms. The

energy of dissociation of Agllr is 23.9 k.catitnot Find the energy in electron volts, the

wavelength and the frequency of the photon that is just able to dissociate a molecule of

silver bromide.

3. Calculate the energy of a photon of blue light with a frequency of 6.67 x 1014

Hz. (State

in eV) [2.76eV]

4. Calculate the energy of a photon of red light with a wavelength of 630 nm. [1.97eV]

5. Barium has a work function of 2.48 eV. What is the maximum kinetic energy of the

ejected electron if the metal is illuminated by light of wavelength 450 nm? [0.28 eV]

6. When a 350nm light ray falls on a metal, the maximum kinetic energy of the

photoelectron is 1.20eV. What is the work function of the metal? [2.3 eV]

7. A photon has 3.3 x 10-19

J of energy. What is the wavelength of this photon?

8. What is the energy of one quantum of 5.0 x 1014

Hz light?

4M (T ) T

X-Rays

Objectives:

Introduction and production of X-Rays

Properties of X-Rays

Diffraction of X-Rays

The Bragg’s X-Ray spectrometer

Continuous spectra

Characteristics Radiation

Moseley’s law

Absorption of X-Ray

Compton effect

Applications of X-Rays

Introduction and production of X-Rays

Introduction of X- Rays

Wilhelm Rontgen discovered X-rays in 1985 during the course of some

experiments with a discharge tube. He noticed that a screen coated with barium

platinocyanide present at a distance from the discharge tube. Rontgen called these

invisible radiations X-rays. Finally he concluded that X-rays are produced due to the

bombardment of cathode rays on the walls of the discharge tube.

It is well known that X-rays are produced when the fast moving electrons, and

that metals or high atomic weight are most effective for this purpose.

X-rays are electromagnetic waves with very short wavelengths. X-rays are highly

penetrating and it can pass through many solids. They occur beyond the UV region in the

electromagnetic spectrum. Their wavelengths range from 0.01 to 10 Å.

Production or Generation of X-rays

X-rays are produced by an X-ray tube. The schematic of the modern type of X-ray

tube designed by Coolidge is shown in above figure.

It is an evacuated glass bulb enclosing two electrodes, a cathode and an anode.

The cathode consists of a tungsten filament which emits electrons when it

heated. The electrons are focused into a narrow beam with the help of a metal

cup S.

The anode consists of a target material, made of tungsten or molybdenum,

which is embedded in a copper bar.

Water circulating through a jacket surrounding the anode and cools the anode. Further

large cooling fins conduct the heat away to the atmosphere.

The face of the target is kept at an angle relative to the oncoming electron beam.

A very high potential difference of the order of 50 kV is applied across the electrodes.

The electrons emitted by the cathode are accelerated by the anode and acquire high

energies of order of 105

eV. When the target suddenly stops these electrons, X-rays are

emitted.

The magnetic field associated with the electron beam undergoes a change when

the electrons are stopped and electromagnetic waves in the form of X-rays are generated.

The grater of the speed of the electron beam, the shorter will be the wavelength of

the radiated X-rays. Only about 0.2 % of the electron beam energy is converted in to X-

rays and the rest of the energy transforms into heat. It is for the reason that the anode is

intensively cooled during the operation of X-ray tube.

The intensity of the electron beam depends on the number of electron leaving the

cathode. The hardness of the X-rays emitted depends on the energy of the electron beam

striking the target. It can be adjusted by varying the potential difference applied between

the cathode and anode. Therefore, the larger potential difference, the more penetrating or

harder X-rays.

Properties of X-Ray

They have relatively high penetrating power.

They are classified into Hard X-rays & Soft X-rays.

The X-rays which have high energy and short wavelength is known as Hard X-

rays.

The X-rays which have low energy and longer wavelength is known as Soft X-

rays.

X-rays causes the phenomenon of flouroscence.

On passing through a gas X-rays ionize the gas.

They are absorbed by the materials through which they traverse.

X-rays travel in straight line. Their speed in vacuum is equal to speed of light.

X-rays can affect a photographic film.

X-rays are undeflected by electric field or magnetic field.

Diffraction of X-Rays – Bragg’s law

Consider a crystal as made out of parallel planes of ions, spaced a distance d

apart. The conditions for a sharp peak in the intensity of the scattered radiation are:

1. That the X-rays should be secularly reflected by the ions in any one plane.

2. That the reflected rays from successive planes should interfere constructively.

Path difference between two rays reflected from adjoining planes:

2dsinθ, for the rays to interfere constructively, this path difference must be an integral

number of wavelength λ,

Suppose that a single monochromatic wave (of any type) is incident on aligned

planes of lattice points, with separation , at angle . Points A and C are on one plane,

and B is on the plane below. Points ABCC' form a quadrilateral.

There will be a path difference between the ray that gets reflected along AC' and

the ray that gets transmitted, and then reflected, along AB and BC respectively. This path

difference is:

The two separate waves will arrive at a point with the same phase, and hence

undergo constructive interference, if and only if this path difference is equal to any

integer value of the wavelength, i.e.

Where, the same definition of and apply as above.

Therefore,

from which it follows that,

Putting everything together,

Which simplifies to

Which is Bragg's law.

Bragg angle is just the half of the total angle by which the incident beam is deflected.

The Bragg’s X-Ray spectrometer

An X-ray diffraction experiment requires,

(1) X-ray source

(2) The sample

(3) The detector

Depending on method there can be variations in these requirements. The X-ray

radiation may either monochromatic or may have variable wave length.

Structures of polycrystalline sample and single crystals can be studied. The

detectors used in these experiments are photographic film.

The schematic diagram of Bragg’s X-ray spectrometer is given in above.

X-ray from an X-ray tube is collimated by passing team through slits S1 and S2. This

beam is then allowed to fall on a single crystal mounted on a table which can be rotated

about an axis perpendicular to the plane of incident of X-rays. The crystal behaves as a

reflected grating and reflects X-rays. By rotating the table, the glancing angle θ at which

the X-ray is incident on the crystal can be changed. The angle for which the intensity of

the reflected beam is maximum gives the value of θ. The experiment is repeated for each

plane of the crystal. For first order reflection n = 1 so that, λ = 2d sinθ; for n = 2, 2λ = 2d

sinθ; ……., and so on.

A photographic plate or an ionization chamber is used to detect the rays reflected by

the crystal.

Continuous X-rays or Bremsstrahlung X-rays

"Bremsstrahlung" means "braking radiation" and is retained from the original

German to describe the radiation which is emitted when electrons are decelerated or

"braked" when they are fired at a metal target. Accelerated charges give off

electromagnetic radiation, and when the energy of the bombarding electrons is high

enough, that radiation is in the x-ray region of the electromagnetic spectrum. It is

characterized by a continuous distribution of radiation which becomes more intense and

shifts toward higher frequencies when the energy of the bombarding electrons is

increased. The curves above are who bombarded tungsten targets with electrons of four

different energies.

The continuous distribution of x-rays which forms the base for the two sharp

peaks at left is called "Bremsstrahlung" radiation.

The bombarding electrons can also eject electrons from the inner shells of the

atoms of the metal target, and the quick filling of those vacancies by electrons dropping

down from higher levels gives rise to sharply defined characteristic x-rays.

Characteristic X-rays

Characteristic x-rays are emitted from heavy elements when their electrons make

transitions between the lower atomic energy levels. The characteristic x-rays emission

which shown as two sharp peaks in the illustration at left occur when vacancies are

produced in the n=1 or K-shell of the atom and electrons drop down from above to fill the

gap. The X-rays produced by transitions from the n=2 to n=1 levels are called Kα X-rays,

and those for the n=3->1 transition are called Kβ X-rays.

Transitions to the n=2 or L-shell are designated as L x-rays (n=3->2 is L-alpha,

n=4->2 is L-beta, etc.

X-ray production typically involves bombarding a metal target in an X-ray tube

with high speed electrons which have been accelerated by tens to hundreds of kilovolts of

potential. The bombarding electrons can eject electrons from the inner shells of the atoms

of the metal target. Those vacancies will be quickly filled by electrons dropping down

from higher levels, emitting X-rays with sharply defined frequencies associated with the

difference between the atomic energy levels of the target atoms.

The frequencies of the characteristic X-rays can be predicted from the Bohr

model. Moseley measured the frequencies of the characteristic x-rays from a large

fraction of the elements of the periodic table and produces a plot of them which is now

called a "Moseley plot".

Characteristic X-rays are used for the investigation of crystal structure by X-ray

diffraction. Crystal lattice dimensions may be determined with the use of Bragg's law in

a Bragg spectrometer.

Moseley’s law and its importance

The English physicist Henry Moseley (1887-1915) found, by bombarding high

speed electrons on a metallic anode, that the frequencies of the emitted X-ray spectra

were characteristic of the material of the anode. The spectra were called characteristic X-

rays.

He interpreted the results with the aid of the Bohr theory, and found that the

wavelengths λ of the X-rays were related to the electric charge Z of the nucleus.

According to him, there was the following relation between the two values (Moseley’s

law; 1912).

1/λ = c(Z - s)2

----- (1)

Where,

c and s are constants applicable to all elements and Z is an integer.

When elements are arranged in line according to their position in the periodic

table, the Z value of each element increases one by one.

Moseley correctly interpreted that the Z values corresponded to the charge

possessed by the nuclei. Z is none other than the atomic number.

Importance of Moseley’s law:

Atomic no. is more important than Atomic weight as it is equals to charge of

nucleus.

Difference between Ni, Co, Te & I etc., is explained when periodic table was

constructed with atomic no.

Moseley predicted the existence of elements with atomic no. 43, 61, 72 & 75.

Thus, X-ray spectrum analysis new elements can be discovered.

Absorption of X-Ray

When the x-rays hit a sample, the oscillating electric field of the electromagnetic

radiation interacts with the electrons bound in an atom. Either the radiation will be

scattered by these

A narrow parallel monochromatic x-ray beam of intensity I0

passing through a

sample of thickness x will get a reduced intensity I according to the expression:

ln (I0

/I) = μ x ------- (1)

Where μ is the linear absorption coefficient, which depends on the types of atoms

and the density ρ of the material.

At certain energies where the absorption increases drastically and gives rise to an

absorption edge. Each such edge occurs when the energy of the incident photons is just

sufficient to cause excitation of a core electron of the absorbing atom to a continuum

state, i.e. to produce a photoelectron.

Thus, the energies of the absorbed radiation at these edges correspond to the

binding energies of electrons in the K, L, M, etc.., shells of the absorbing elements. The

absorption edges are labeled in the order of increasing energy, K, LI, L

II, L

III, M

I,….,

corresponding to the excitation of an electron from the 1s(2

S½), 2s(

2

S½), 2p(

2

P½),

2p(2

P3/2

), 3s(2

S½), … orbitals (states), respectively.

Compton effect

Arthur H. Compton observed the scattering of x-rays from electrons in a carbon

target and found scattered x-rays with a longer wavelength than those incidents upon the

target. The shift of the wavelength increased with scattering angle according to the

Compton formula:

Compton explained and modeled the data by assuming a particle (photon) nature

for light and applying conservation of energy and conservation of momentum to the

collision between the photon and the electron.

In figure, the electron is initially at rest with incident photon of wavelength and

momentum p; scattered photon with longer wavelength f and momentum p and

recoiling electron with momentum P. The direction of the scattered photon makes an

angle φ with that of the incident photon, and the angle between p and p is also φ.

called Compton wavelength.

Compton scattering cannot be understood on the basis classical electromagnetic

theory. On the basis of classical principles, the scattering mechanism is induced by

motion of electrons in the material, caused by the incident radiation. This motion must

have the same frequency as that of incident wave because of forced vibration, and so the

scattered wave radiated by the oscillating charges should have the same frequency. There

is no way the frequency can shift by this mechanism.

Applications of X-Rays

X-rays are used in industrial, medical, pure science research and X-ray

crystallography etc…

X-rays are used to detect defects in radio valves.

X-rays are used to detect cracks in structures.

X-rays are used to analyses the structures of alloys and other composite bodies by

diffraction of X-rays.

They are also used to study are structure of materials like rubber, cellulose, plastic,

fibres etc…

X-rays are used in analysis of crystal structure and structure of complex organic

molecule.

They are also used in determining the atomic number and identification of various

chemical elements.

X-rays are used to detect fractures and formation of stones in human body.

X-rays can destroy abnormal internal tissues.

They are also being used for tumor treatment and for this purpose hard X-rays are

used.

X-rays are also used in X-ray crystallography for Laue method, Rotating crystal

method, Powder method, etc….

nmmc

hc

00243.0

1-1 Architectural Acoustics

Architectural Acoustics

Syllabus :

Classification of sound : Loudness, Weber-Fechner law,

Absorption coefficient, Reverberation, Sabine’s formula, Factors

affecting acoustics of building and their remedies

Introduction :

Sound is always produced by some vibrating body. The vibrating

body generates mechanical waves and these waves spreads in the

surrounding medium. We are aware that these waves propagate in the

form of a series of compressions and rarefactions in air or the surrounding

medium. When reached upto the human ear drum it causes a sensation of

hearing. As far as architectural acoustics are concerned, we are interested

the combined effect of sound waves which creates a sense of sound on

human ear.

Some important characteristics

(1) The propagation of sound requires the presence of an elastic

medium.

(2) Sound can not travel through vacuum

(3) The compression and rarefactions due to a sound modulate the

normal atmospheric pressure with small pressure changes occuring

regularly above and below it.

(4) The velocity of sound depends on the nature and temperature of the

medium.

1-2 Architectural Acoustics

1.1 Classification of Sound :

Based upon frequency of sound waves, it can be classified into its

three main categories.

(a) Audible waves : Sound waves with frequency in the range of

20Hz to 20KHz.

(b) Infrasonic waves : Sound waves below audible range i.e. below

20Hz.

(c) Ultrasonic waves : Sound waves above audible range i.e.

20KHz.

1.1.1 Characteristics of Musical Sound :

Musical sounds & Noise sound:-

Musical sound are distinguished from noises in that they are

composed of regular, uniform vibrations, while noises are irregular

and disordered vibrations. One musical tone is distinguished from

another on the basis of pitch, intensity, or loudness and quality, or

timbre.

Noise sound that Produce Jarring effect on the ear is called Noise

sound.Noice sound make unpleasent to hear .Example are sound

produce by flying aeroplane,road traffic,cracker etc

Pitch describes how high or low a tone is and depends upon the

rapidity with which a sounding body vibrates, i.e. upon the frequency

of vibration. The higher the frequency of vibration, the higher the tone;

the pitch of a siren gets higher and higher as the frequency of vibration

increases. The apparent change in the pitch of a sound as a source

approaches or moves away from an observer is described by the Doppler

effect.

The intensity or loudness of a sound depends upon the extent to

which the sounding body vibrates, i.e. the amplitude of vibration. A

1-3 Architectural Acoustics

sound is louder as the amplitude of vibration is greater, and the intensity

decreases as the distance from the source increases. Loudness is measured

in units called decibels.

Timber is the Quality of the sound which Enable us to

distinguish between two sound having the same loudness & pitch.The

sound waves given off by different vibrating bodies differ in quality, or

timbre. A note from a saxophone, for instance, differs from a note of the

same pitch and intensity produced by a violin or a xylophone; similarly

vibrating reeds, columns of air, and strings all differ. Quality is dependent

on the number and relative intensity of overtones produced by the

vibrating body (see harmonic), and these in turn depend upon the nature

of the vibrating body.

1.2 Important Terms Used :

In the study of sound waves we come across various terms like Pitch

(This law does not hold good near the upper and lower limits of

audiability), Timber which basically deals with the quality of the sound

waves and source. At the same time for technical assessment, we make

use of important parameters like intensity and loudness.

1.2.1 Weber Fechner Law :

This law has it roots hidden in psychology and proved scientifically

according to which : The loudness of sound sensed by ear is directly

proportional to logarithm of its intensity.

According to Weber-Fechner law :

Suppose the loudness is S for intensity I and S0 for intensity I0,

S = K log10 I

S0 = K log10 I0

The intensity level L is the difference in loudness.

L = S – S0

1-4 Architectural Acoustics

= K log10 I – K log10 I0 = K log10 I

I0

take, K = 1

L = log10 I

I0 …(1.1)

Intensity and loudness are the two words which are similar but with

slight difference.

Table 1.1

Sr.

No.

Intensity Loudness

1. Defined as the quantity of energy

propagating through a unit area per

unit time, in the direction of

propagation being perpendicular to

the area (unit : watt/m2).

It is just an aural

sensation and it a

physiological

phenomenon rather than a

physical one.

2. It refers to the external or the

objective measurement.

It refers to an internal or

subjective aspect.

3. It is a physical quantity. Merely a degree of

sensation.

Loudness ‘S’ increases with intensity ‘I’ as per the following

relation*

or S log I …(1.2)

dS

dI =

K

I …(1.3)

Where K is proportionality constant

Here ds

dI is called the sensitiveness of the ear.

In practice, it is the relative intensity that is important and not the

absolute value. Hence the intensity of sound is often measured as the

ratio to a standard intensity I0. The intensity level is I / I0.

1-5 Architectural Acoustics

The standard intensity taken is I0 = 10–12

watts / m2. (It is an

arbitrarily selected value. It is an intensity that can just be heard

at frequency 1 kHz)

1.2.2 Bel :

As discussed in art 1.2.1, whenever the intensity of sound increases

by a factor of 10, the increase in the intensity is said to be 1 bel (A unit named after Alexander Graham Bell, the inventor of telephone)

Therefore dynamic range of audibility of the human ear is 12 bels or

120 dB. When the intensity increases by a factor of 100.1

, the

increase in intensity is 0.1 bel or 1dB.

From Equation 1.1

L = log10 I

I0

in decibel

L = 10 log10 I

I0

For the intensity level change = 1 dB

1 = 10 log10 I

I0

I

I0 = 1.26 …(1.4)

If I = I0,

L = 10 log 1 = 0

This represents the threshold of audibility.

It means that intensity level alters by 1dB when intensity of sound

changes by 26%

1-6 Architectural Acoustics

Table 1.2 : Intensity levels of different sounds

Sr. No. Sound Intensity level (in db)

(1) Threshold of hearing 0

(2) Rustle of leaves 10

(3) Whisper 15 – 20

(4) Normal conversation 60 – 65

(5) Heavy traffic 70 – 80

(6) Thunder 100 – 110

(7) Painful sound 130 and above

1.2.3 Phon :

The intensity levels given in the above Table 1.2 refer to the

loudness in decibels with the assumption that the threshold of

audibility is the same irrespective of the pitch (Pitch is a subjective

sensation perceived when a tone of a given frequency is sounded. It

enables us to classify a note as high or low and to distinguish a shrill

sound from a flat sound of the same intensity on the same instrument.) of the sound.

However, the sensitivity of the ear and the threshold audibility vary

over wide ranges of frequency and intensity.

Hence the intensity level will be different at different frequencies

even for the same value of I0.

For measuring the intensity level a different unit called phon is used.

The measure of loudness in phons of any sound is equal to the

intensity level in decibels of an equally loud pure tone of frequency

1000 Hz.

Hence Phon scale and decibel scale agree for a frequency of 1000 Hz

but the two values differ at other frequencies.

Suppose the intensity level of a note of frequency 480 Hz is to be

determined. A standard source of frequency 1000 Hz is sounded and

the intensity of the standard source is adjusted so that it is equal to

1-7 Architectural Acoustics

the loudness of the given note of frequency

480 Hz.

The intensity level of the standard source in decibels is numerically

equal to the loudness of the given source in phons.

Ex. 1.1 : Calculate the change in intensity level when the intensity of

sound increases 100 times its original intensity.

Soln. :

Given :

Initial intensity = I0

Final intensity = I

I

I0 = 100

Increase in intensity level = L

L = 10 log10 I

I0 (in dB)

L = 10 log10 100 = 20 dB …Ans.

Ex. 1.2 : Find the intensity level in phons if 3000 Hz with intensity level

of 70 dB produces the same loudness as a standard source of

frequency 1000 Hz at a intensity level 67 dB.

Soln. :

As the 3000 Hz source has the same loudness of standard source of

1000 Hz with 67 db, the intensity level of the note of frequency 3000 Hz

is 67 phons. …Ans.

1.3 Architectural Acoustics :

Lets try to understand what exactly acoustics of a hall means.

Consider the following cases :

(a) Imagine a hall, it is easy for any one to understand that sound

produced at a point will reach the other point directly as well as after

reflections from walls, roof etc. The intensity of the sound depends

1-8 Architectural Acoustics

on the distance covered by sound on different paths. These sounds

are generally out of phase and due to interference the distribution of

intensity in the room is not uniform.

(b) It is also important to consider a possibility that the different

frequency sounds of a musical instrument may interfere differently at

some point and quality of music may become unpleasent.

(c) It is known that sound persists for some time due to multiple

reflections, even when the original sound has ceased. During this

time if any other syllable is received, superimposition of these two

will affect audiability as both will remain indistinct. If this takes

place during a speech, a confusion will be created.

(d) Concentration of sound taking place at any part of the hall.

The above mentioned points are very common but needs a special

scientific attention. Prof. W.C. Sabine was the first person who took it

seriously.

1.4 Reverberation Time :

Reverberation means the prolonged reflection of sound from walls,

floor or roof of a hall. In simple language it is nothing but persistence of

sound even after the sources of the sound has stopped.

Reverberation time :

The time gap between the initial direct note and the reflected

note upto a minimum audibility level is called reverberation time.

More precisely, the interval of time taken by a sustained or

continuous sound to fall to an intensity level equal to one millionth

of its original value. (i.e. fall by 60 db in loudness) is called

reverberation time.

In a good auditorium it is necessary to keep the reverberation time as

small as possible. The intensity of the sound as received by listener

is shown graphically in Fig. 1.1.

1-9 Architectural Acoustics

Fig. 1.1

When a source emits sound, the waves spread out and the listener is

aware of the commencement of sound when the direct waves reach

his ears. Subsequently the listener receives sound energy due to

reflected waves also. If the note is continuously sounded, the

intensity of sound at the listener’s ears gradually increases. After

sometime, a balance is reached between the energy emitted per

second by the source and energy lost or dissipated by walls or other

materials.

The resultant energy attains an average steady value and to the

listener the intensity of sound appears to be steady and constant.

This is represented by a portion BC of the curve ABCD.

If at C, the source stops emitting sound, the intensity of sound falls

exponentially as shown by the curve CD.

Fig. 1.2

When intensity of sound falls below the minimum audibility level,

the listener will not get the sound.

When a series of notes are produced in an auditorium each note will

give rise to its own intensity curve with respect to time. The curve

for these notes are shown in Fig. 1.2.

1-10 Architectural Acoustics

In order to maintain distinctness in speech it is necessary that :

(a) Each separate note should give sufficient intensity of sound in

every part of the auditorium.

(b) Each note should die down rapidly before the maximum

average intensity due to the next note is heared by the listener.

1.5 Absorption :

When a sound wave strikes a surface there are three possibilities.

(a) Part of energy is absorbed

(b) Part of it is transmitted

(c) Remaining energy is reflected

The effectiveness of surface in absorbing sound energy is expressed

by absorption coefficient denoted by a.

a = Sound energy absorbed by the surface

Total sound energy incident on the surface …(1.5)

For the comparison of relative efficiencies of different absorbing

material, it is necessary to select a standard or reference.

Sabine selected a unit area of open window, as standard. For any

open window the sound falling on it completely passes out no

reflection, and more importantly no absorption.

Hence open window is an ideal absorber of the sound. The

absorption coefficient is measured in open window unit.

(OWU) or Sabine :

The absorption coefficient of a material is defined as the reciprocal

of its area which absorbs the same sound energy as absorbed by unit

area of open window.

Effective absorbing area A of the surface having total area S and

absorption coefficient ‘a’ is given by

A = a S …(1.6)

1-11 Architectural Acoustics

If the a1, a2, a3, …. , an are the absorption coefficients for each

reflecting surface and S1, S2, S3, …. Sn are the corresponding areas,

then the average value of absorption co-efficient is

a + 89 = a1 S1 + a2 S2 + a3 S3 + …… + an Sn

S1 + S2 + S3 + …. + Sn

=

n

i = 1 ai Si

S …(1.7)

Where S is total surface area.

1.6 Sabine’s Formula :

Prof. W.C. Sabine observed the concept of reverberation time for

varieties of conditions like empty room, furnished room, small room,

auditorium etc.

He concluded the following,

(a) Reverberation time depends upon reflectivity of sound form

various surfaces available in side the hall. If the reflection is

good, reverberation time of the hall will be longer as sound take

more time to die out.

(b) Reverberation time depends upon volume of the hall.

i.e. T V

(c) Reverberation time depends upon coefficient of absorption of

various surfaces present in the hall. For shorter reverberation,

absorption should be more.

(d) As absorption coefficient is found to be increased with increase

in frequency, reverberation time decreases with frequency.

Reverberation time T V

A

where, V = Volume of hall

A = Absorption

1-12 Architectural Acoustics

or T = K V

A

where, K = Proportionality constant

It has been further observed that is all the parameters are taken in SI

then, proportionality constant is found to be 0.161.

T = 0.161 V

A …(1.8)

Equation (1.8) is Sabine’s formula.

Absorption A given in Equation (1.8) represents overall absorption

which is given as

A =

n

i = 1 a S = a1 S1 + a2 S2 + …… + an Sn

Ex. 1.3 : For an empty assembly hall of size 20 15 10 cubic meter

with absorption coefficient 0.106 . Calculate reverberation

time.

Soln. :

Given :

(i) Size of the room = 20 15 10

= 3000 cubic meter

(ii) a = 0.106

Formula T = 0.161 V

A

= 0.161 V

aS

Here S = Total surface area of the hall is given by

2 (20 15 + 15 10 + 20 10)

= 1300 sqm

Reverberation time T = 0.161 3000

0.106 1300

1-13 Architectural Acoustics

Reverberation time = 3.5 sec ...Ans.

1.7 Determination of Absorption Coefficient :*(only for reference)

Step 1 : Using a source of sound inside the hall, reverberation time is

measured with the help of chronograph without inserting any test

material (whose co-efficient of absorption is to be calculated). Let

the reverberation time be T1,

T1 = 0.161 V

A

= 0.161 V

aS

1

T1 =

aS

0.161 V …(1.9)

Step 2 : Now consider a material like curtain or stage screen whose

co-efficient of absorption is to be found out suspended inside the

room and reverberation time T2 is obtained. Since the material is

suspended in hall, surface area from both the side are to be

considered.

1

T2 =

0.161 V

aS + 2a2 S2

where a2 = Co-efficient of absorption of the material under

investigation

S2 = Surface of the material (since both the sides are used,

it is multiplied by 2)

1

T2 =

aS + 2a2 S2

0.161 V …(1.10)

From Equation (1.9) and (1.10)

1

T2 –

1

T1 =

1

0.161

2a2 S2

V

2a2 S2 = 0.161 V 1

T2 –

1

T1

1-14 Architectural Acoustics

a2 = 0.161 V

2 S2

1

T2 –

1

T1 …(1.11)

All the quantities on RHS are known, co-efficient of absorption of an

absorbing material which is suspended in hall with both the surfaces open

can be calculated.

Table 1.3 : Absorption coefficients of some materials

Material Absorption coefficient per m2 at 500 Hz

Open window 1.0

Stage curtain 0.2

Common plaster 0.3

Carpet 0.4

Heavy curtain 0.5

Perforated cellulose fiber tiles 0.85

1.8 Conditions for Good Acoustic :

As already introduced in art 1.3, a lecture hall or auditorium should

satisfy the following conditions in order to be acoustically good.

(a) The initial sound from the source should be of adequate intensity.

(b) The sound should spread evenly with proper loudness every where is

the hall

(c) The sound of speech or music should be clear and words of or

musical notes must be distinctly audible to all.

(d) All undesired or extraneous noise must be reduced to the extent that

it will not interfere with normal hearing of speech or hearing.

(e) Any distortion due to shape and size must be absent.

1.9 Methods of Design for Good Acoustics :

In order to make acoustically correct hall following points may be

considered. These are merely the guidelines, depending upon specific

requirement a justified step be taken.

1-15 Architectural Acoustics

(a) Selection of proper site :

Avoid noisy places like railway tack, roads with heavy traffic,

airports, industrial vicinity for auditorium.

(b) Volume :

Size of the hall/ auditorium should be such that it remains

optimum.

Small halls leads to irregular distribution of sound because of

formation of standing waves.

Too big halls may also create a weaker intensity and larger

reverberation time which is a very serious issue.

(c) Shape :

It is one of the most important parameter to be considered for

acoustically correct hall.

As the reflections are created by roof and side walls, they

should be designed in such a way that echos are not allowed to

generate.

In place of parallel walls, splayed side walls are preferred.

Curved surface on walls, ceilings or floor produce concentration

of sound into particular region and absence of sound in other

regions.

Hence curved surface must be designed with proper care.

(d) Use of absorbents :

Once the construction of hall is completed certain errors are

found or the hall requires further correction as far as acoustics

are concerned. For this use of absorbents is very common.

As the reflections from rear wall are of no use. It must be

covered with absorbents, so as the ceiling.

1-16 Architectural Acoustics

False ceiling provided in large halls solves this problem

effectively. The floor needs to be covered with carpet so as

unwanted reflections and the noise created by audience is

suppressed.

1-17 Architectural Acoustics

(e) Reverberation :

Reverberation time must be maintained in such a that it does not

remain too short or too large i.e. nearly 0.5 seconds for lecture

hall, around 1.2 for concerts hall and around 2 for cinema halls.

Proper use of absorbing materials, sufficient people as audience,

presence of open windows presence of furniture etc are the

major components which can decide the reverberation time.

Calculated use of such components will be helpful to either

increase or decrease the reverberation time.

(f) Echelon effect :

Fig. 1.3 : Echelon effect

A set of railings or staircase or any regular spacing of reflected

surfaces may produce a musical note due to regular succession

of echoes of the original sound to listener.

This makes original sound to appear confused. Either one

should avoid use of such surfaces or keep them covered with

thick carpet.

1.10 Solved Problems :

Ex. 1.4 : Calculate the change in intensity level when intensity level

increases by 106 times its original intensity.

Soln. :

1-18 Architectural Acoustics

Given :

Initial intensity = I0

Final intensity = I

I

I0 = 10

6

Increase in intensity level in dB

L = 10 log10 I

I0 = 10 log10 (10

6)

L = 60 dB …Ans.

Ex. 1.5 : A room has dimensions 6 4 5 meters calculate :

(a) the mean free path of the sound waves in the room

(b) the number of reflections made per second by the sound

wave with the

walls of the room

Given : Velocity of sound in air = 350 m/sec

Soln. :

(a) The mean free path of sound waves is defined as the average

distance travelled by sound wave through air between any two

consecutive encounters with the walls of the room. Jaeger had

calculated as

l = 4V

S =

4 (Volume of the room)

Total surface area

Here V = 6 4 5 = 120 m3

S = 2 [6 4 + 4 5 + 5 6] = 148 m2

l = 4 120

148 = 3.243 m …Ans.

Number of reflections made per second

n = Velocity of sound

Mean free path

1-19 Architectural Acoustics

n = 350

3.243 = 107.9 …Ans.

Ex. 1.6 : The sound from a drill gives a noise level 90 dB at a point

short distance from it. What is the noise level at this point if

four such drills are working simultaneously at the same

distance from the point ?

Soln. : Acoustic intensity level is given by

L = 10 log10 I

I0 dB …(1)

Reference to I0 in watts / m2

Let I1 be the intensity level due to one drill and I2 be the intensity

level due to four such drills.

I2

I1 = 4 …(2)

Consider one drill on

L1 = 10 log I1

I0 dB …(3)

In second case with four drills on

L2 = 10 log I2

I0 dB …(4)

Increase in noise level (in dB)

L2 – L1 = 10 log I2

I0 – log

I1

I0

= 10 log I2

I1

but I2

I1 = 4

L2 – L1 = 10 log 4 = 6.021 dB

Final intensity level

1-20 Architectural Acoustics

= L1 + 6.021 = 90 + 6.021

Final intensity level = 96.021 dB …Ans.

Ex. 1.7 : Calculate the increase in the acoustic intensity level in dB.

When the sound is doubled.

Soln. :

Intensity level in dB is

L = 10 log I

I0

Let the intensity level in case 1 be I1 and the in case 2 be I2

For case – 1

L1 = 10 log I1

I0 dB

For case – 2

L2 = 10 log I2

I0 dB

Change in intensity level in dB

L2 – L1 = 10 log I2

I0 – log

I1

I0

= 10 log I2

I1

but I2

I1 = 2 (given)

L2 – L1 = 10 log 2

= 10 (0.3010)

L2 – L1 = 3.01 dB …Ans.

Ex. 1.8 : An air conditioner unit operates at a sound intensity level of 70

dB. If it is operated in room with an existing sound intensity

level of 80 dB, what will be the resultant intensity level.(4 Marks)

1-21 Architectural Acoustics

Soln. :

Here for case – 1

Intensity level is 70 dB

70 = 10 log L1 = 10 log I1

I0

I1

I0 = Antilog 7.0

or I1 = 107 I0 watts/m

2 …(1)

Similarly for Case – 2, intensity level is 80 dB.

80 = 10 log L2 = 10 log I2

I0

I2

I0 = Antilog 8.0

I2 = 1 108 I0 watts/m

2 …(2)

Resultant intensity

I = I1 + I2

= 107 I0 + 10

8 I0

= I0 (1.1 108)

Resultant intensity level in dB

L = 10 log I

I0

= 10 log 1.1 10

8 I0

I0 = 10 log (1.1 10

8)

= 80.41 dB

Resultant intensity level (in dB) is 80.41 …Ans.

Ex. 1.9 : The noise form an aeroplane engine 100 m from an observer is

40 dB in intensity. What will be the intensity when the

aeroplane flies overhead at an altitude of 2 km ?

1-22 Architectural Acoustics

Soln. : Intensity of sound is given by formula

I = P

4 R2

Where P = Acoustic pressure level

R = Radial distance

Here, for case – 1

I1 = P

4 R2

1

And for case – 2

I2 = P

4 R2

2

I2

I1 =

R2

1

R2

2

Now R1 = 100 m, R2 = 2000 m (given)

I2

I1 =

1002

20002 =

1

400

or I1

I2 = 400 …(1)

For the case – 1, intensity level in dB is given by

L1 = 10 log I1

I0

…(2)

and for case – 2

L2 = 10 log I1

I0

…(3)

as intensity level is suppose to decrease, we will take L1 – L2

L1 – L2 = 10 log I1

I0 – log

I2

I0

1-23 Architectural Acoustics

= 10 log I1

I2

= 10 log 400 = 26.021 dB

as L1 = 40 dB given

L2 = L1 – (L1 – L2)

L2 = 40 – 26.021 = 13.97 dB …Ans.

Ex. 1.10 : A hall of volume 5500 m3 is found to have a reverberation

time of 2.3 sec. The sound absorbing surface of the hall has an

area of 750 m2. Calculate the average absorption coefficient.

Soln. :

Given : V = 5500 m3

T = 2.3 sec

S = 750 m2

Let absorption coefficient be a

Using Sabine’s formula

T = 00.161 V

aS

a = 0.161 V

ST

= 0.161 5500

750 2.3

a = 0.513 …Ans.

Ex. 1.11 : For an empty hall of size 20 12 12 cubic meter, the

reverberation time is

2.5 sec. Calculate the average absorption co-efficient of the

hall. What area of the floor should be covered by carpet so as

to reduce the reverberation time to 2.0 sec. Given that

absorption co-efficient of carpet is 0.5.

Soln. :

1-24 Architectural Acoustics

(a) Reverberation time

T1 = 0.161 V

aS …(1)

aS = 0.161 V

T1

= 0.161 (20 12 12)

2.5

= 185.47

Now total surface area of the hall,

S = 2 (20 12 + 12 12 + 20 12)

= 1248 m2

a = 185.47

1248 = 0.1486 …Ans.

(b) By using the carpet of surface area S1 whose absorption coefficient is

0.5, the reverberation time is reduced to 2.0 sec.

Let T2 = 2.0 sec

Carpet surface = S1

Co-efficient of absorption of carpet ac = 0.5

Writing Sabine’s formula

T2 = 0.161 V

aS + aC S1– aS1 …(2)

(Here Total surface area = S, now if carpet is used of area S1, the

area covered by the material with co-efficient of absorption a is a (S – S1)

= aS – aS1)

From Equation (1)

1

T1 =

aS

0.161 V …(3)

1-25 Architectural Acoustics

From Equation (2)

1

T1 =

aS + aC S1– aS1

0.161 V …(4)

1

T2 –

1

T1 =

1

0.161 V [aC S1 – aS1]

= S1 (aC – a)

0.161 V

S1 = 0.161 V

aC – a

1

T2 –

1

T1

Substituting various value

S1 = 0.161 (20 12 12)

0.5 – 0.1486

1

2 –

1

2.5

= 131.95 m2

Carpet area required to reduce reverberation time up to 2.0 sec is

131.95 m2 …Ans.

Ex. 1.12 : Calculate the reverberation time for the seminar hall with

(a) No one inside.

(b) 50 persons inside

(c) Full capacity of audience.

Given that

Sr.

No.

Surface Area Absorption

co-efficient

1. Carpet covering entire

floor (10 12)

sqm

0.06

2. False ceiling (10 12)

sqm

0.03

3. Cushioned seats 100 Nos 1.00

4. Walls covered with

absorbent

346 sqm 0.2

1-26 Architectural Acoustics

Sr.

No.

Surface Area Absorption

co-efficient

5. Audience occupying

seats

– 0.46 /

person

6. Wooden door (3 2) sqm 0.2

Soln. :

Let us calculate total absorption in the hall in case – 1 i.e. for empty hall

(1) Absorption due to carpet 120 0.06 = 7.2

(2) Absorption due to false ceiling 120 0.03 = 3.6

(3) Absorption due to seats 100 1 = 100

(4) Walls covered with absorbent 346 0.2 = 69.2

(5) Wooden door 6 0.2 = 1.2

aS = 181.2 …(1)

Now Area of floor = Area of ceiling = (l b)

= 120 sq.m

Area of wall + Area of door = 346 + 6 = 352

= 2 [(b h) + (l h)]

as l b = 120 m2

let us take l = 12 m, b = 10 m

352 = 2 [(10 h) + (12 h)]

h = 8 m …(2)

hence volume V = 12 10 8 = 960 m3 …(3)

Case 1 :

For empty hall

Reverberation time T1 = 0.161 V

aS

= 0.161 960

181.2

1-27 Architectural Acoustics

T1 = 0.85 sec …Ans.

Case 2 :

With occupancy of 50 persons.

Absorption = aS + 50 (0.46)

Reverberation time T2 = 0.161 V

aS + 50 (0.46)

= 0.161 960

181.2 + 23

T2 = 0.757 sec …Ans.

Case 3 :

With full occupancy. i.e. 100 persons here, the absorption is = aS +

100 (0.46)

Reverberation time T3 = 0.161 V

aS + 100 (0.46)

T3 = 0.68 sec …Ans.

1.11 Solved Examples :

Ex. 1.11.1 :The volume of room is 600 m3. The wall area of the room is 220 m

2, the floor area is 120 m

2 and the ceiling area is 120 m

2. The average sound absorption coefficient, (a) for the walls is 0.03 (b) for the ceiling is 0.8 (c) the floor it is 0.06. Calculate the average sound

absorption coefficient and the reverberation time.

Soln. :

Given :

Let S1 = 220 m2 a1 = 0.03

S2 = 120 m2

a2 = 0.8

S3 = 120 m2 a3 = 0.06

The average sound absorption coefficient is

a = a1 S1 + a2 S2 + a3 S3

S1 + S2 +S3

1-28 Architectural Acoustics

= 220 0.03 + 120 0.8 + 0.06 120

220 + 120 + 120 =

0.238

a = 0.24 …Ans.

Total sound absorption of the room = aS

= 0.24 460

= 110.4 Sabine

Reverberation time, using Sabine’s formula

T = 0.161 V

aS =

0.161 600

110.4

T = 0.875 sec. …Ans.

Ex. 1.11.2 :What is the resultant sound level when a 70 dB sound is added to a 80 dB

sound ? (4 Marks)

Soln. : Increase in intensity level = L = 70 dB

Say, resultant intensity increased by x times the original intensity

Hence, L = 10 log10 x Io

Io dB

70 = 10 log10 (x )

7 = log10 x

or x = 107

So, Resultant sound level is increased 107 times the original

intensity.

1-29 Architectural Acoustics

example :

Q. 1 A class room has dimensions 20 15 5 m3. The reverberation

time is 3.5 sec. Calculate the total absorption of its surface and

average absorption co-efficient.

Ans. :(0.07)

Q. 2 The reverberation time is found to be 1.5 sec for an empty hall and

itis found to be

1 sec when a curtain of 20 m2 is suspended at the center of the hall.

If the dimensions of the hall are 10 8 6 m3, calculate co-

efficient of absorption of curtain.

Ans. :(0.64)

Q. 3 For an empty assembly hall of size 20 15 10 m3, the

reverberation time is 3.5 sec. Calculate the average absorption co-

efficient of the hall. What area of the wall should be covered by the

curtain so as to reduce the reverberation time to 2.5 sec. Given the

absorption co-efficient of the curtain cloth is 0.5.

Ans. :(0.106, 140.12 m2)

ULTRASONIC WAVE

Introduction :

The term ultrasonics applies to sound waves that vibrate at a frequency higher than

the frequency that can be heard by the human ear (or higher than about 20,000

hertz).

Sound is transmitted from one place to another by means of waves. The

character of any wave can be described by identifying two related properties:

its wavelength (lambda, λ) or its frequency (f). The unit used to measure the

frequency of any wave is hertz. One hertz is defined as the passage of a single

wave per second.

Ultrasonics, then, deals with sound waves that pass a given point at least 20,000

times per second. Since ultrasonic waves vibrate very rapidly, additional units also

are used to indicate their frequency. The kilohertz (kHz), for example, can be

used to measure sound waves vibrating at the rate of 1,000 times per second, and

the unit megahertz (MHz) stands for a million vibrations per second. Some

ultrasonic

devices have been constructed that produce waves with frequencies of more than

a billion hertz.

PROPERTIES OF ULTRASONIC WAVES

(1) They have a high energy content.

(2) Just like ordinary sound waves, ultrasonic waves get reflected, refracted and

absorbed.

(3) They can be transmitted over large distances with no appreciable loss of

energy.

(4) If an arrangement is made to form stationary waves of ultrasonics in a liquid,

it serves as a diffraction grating. It is called an acoustic grating.

(5) They produce intense heating effect when passed through a substance.

Ultrasonic Production :

There are three methods for producing Ultrasonic waves. They are:

(i) Mechanical generator or Galton’s whistle.

(ii) Magnetostriction generator.

(iii) Piezo-electric generator.

Magnetostriction method:

Principle:

“When a magnetic field is applied parallel to the length of a ferromagnetic rod

made of material such as iron or nickel, a small elongation or contraction

occurs in its length. This is known as magnetostriction. The change in length

depends on the intensity of the applied magnetic field and nature of the

ferromagnetic material. The change in length is independent of the direction

of the field. “

The change in length (increase or decrease) produced in the rod depends upon the

strength of the magnetic field, the nature of the materials and is independent of the

direction of the magnetic field applied.

Construction:-

The experimental arrangement is shown in Figure

XY is a rod of ferromagnetic materials like iron or nickel. The rod is

clamped in the middle.

The alternating magnetic field is generated by electronic oscillator.

The coil L1 wound on the right hand portion of the rod along with a variable

capacitor C.

This forms the resonant circuit of the collector tuned oscillator. The

frequency of oscillator is controlled by the variable capacitor.

The coil L2 wound on the left hand portion of the rod is connected to the

base circuit. The coil L2 acts as feed –back loop.

Working:-

When High Tension (H.T) battery is switched on, the collector circuit

oscillates with a frequency,

f =

This alternating current flowing through the coil L1 produces an alternating

magnetic field along the length of the rod. The result is that the rod starts

vibrating due to magnetostrictive effect.

The frequency of vibration of the rod is given by

n =

where l = length of the rod

Y = Young’s modulus of the rod material and

=density of rod material

• The capacitor C is adjusted so that the frequency of the oscillatory circuit is

equal to natural frequency of the rod and thus resonance takes plate.

• Now the rod vibrates longitudinally with maximum amplitude and generates

ultrasonic waves of high frequency from its ends.

Condition for resonance:

Frequency of the oscillatory circuit = Frequency of the vibrating rod

Merits: 1. Magnetostrictive materials are easily available and inexpensive.

2. Oscillatory circuit is simple to construct.

3. Large output power can be generated.

Limitations 1. It can produce frequencies upto 3 MHz only.

2. It is not possible to get a constant single frequency, because rod depends on

temperature and the degree of magnetization.

3. As the frequency is inversely proportional to the length of the vibrating rod, to

increase the frequency, the length of the rod should be decreased which is

practically impossible.

piezo electric ossilator

Introduction

Can all the crystals exhibit piezoelectric effect? What is special about the

piezoelectric crystal?

Is the piezoelectric effect direction dependent?

Learning Objectives On completion of this chapter you will be able to:

1. define piezoelectric effect

2. define inverse piezoelectric effect

3. know what type of crystals will exhibit piezoelectric effect

4. Understand the working of piezoelectric generator

Piezoelectric effect:

When crystals like quartz or tourmaline are stressed along any pair of opposite

faces, electric charges of opposite polarity are induced in the opposite faces

perpendicular to the stress. This is known as Piezoelectric effect.

Inverse piezoelectric effect:

When an alternating e.m.f is applied to the opposite faces of a quartz or tourmaline

crystal it undergoes contraction and expansion alternatively in the perpendicular

direction. This is known as inverse piezoelectric effect. This is made use of in the

piezoelectric generator.

Piezoelectric generator:

CIRCUIT:-

CONSTRUCTION:-

The quartz crystal is placed between two metal plates A and B.

The plates are connected to the primary (L3) of a transformer which is

inductively coupled to the electronics oscillator.

The electronic oscillator circuit is a base tuned oscillator circuit.

The coils L1 and L2 of oscillator circuit are taken from the secondary of a

transformer T.

The collector coil L2 is inductively coupled to base coil L1.

The coil L1 and variable capacitor C1 form the tank circuit of the oscillator.

Working:-

When H.T. battery is switched on, the oscillator produces high frequency

alternating voltages with a frequency.

Due to the transformer action, an oscillatory e.m.f. is induced in the coil L3.

This high frequency alternating voltages are fed on the plates A and B.

Inverse piezo-electric effect takes place and the crystal contracts and

expands alternatively.The crystal is set into mechanical vibrations.

The frequency of the vibration is given by

n =

where P = 1,2,3,4 … etc. for fundamental, first over tone, second over tone etc.,

Y = Young’s modulus of the crystal and

ρ = density of the crystal.

The variable condenser C1 is adjusted such that the frequency of the applied

AC voltage is equal to the natural frequency of the quartz crystal, and thus

resonance takes place.

The vibrating crystal produces longitudinal ultrasonic waves of large

amplitude.

Advantages Ultrasonic frequencies as high as 5 x 108Hz or 500 MHz can be obtained

with this arrangement.

The output of this oscillator is very high.

It is not affected by temperature and humidity.

Disadvantages The cost of piezo electric quartz is very high

The cutting and shaping of quartz crystal are very complex.

DETECTION OF ULTRASONIC WAVES

Ultrasonic waves propagated through a medium can be detected in a number of

ways. Some of the methods employed are as follows:

(1) Kundt’s tube method:

Ultrasonic waves can be detected with the help of Kundt’s tube. At the nodes,

lycopodium powder collects in the form of heaps. The average distance between

two adjacent heaps is equal to half the wavelength. This method cannot be used if

the wavelength of ultrasonic waves is very small i.e., less than few mm. In the

caseof a liquid medium, instead of lycopodium powder, powdered coke is used to

detect the position of nodes.

(2) Sensitive flame method: A narrow sensitive flame is moved along the medium. At the positions

of antinodes, the flame is steady. At thepositions of nodes, the flame flickers

because there is a change in pressure. In this way, positions of nodes and antinodes

2

P Y

l

can be found out in the medium. The average distance between the two adjacent

nodes is equal to half the wavelength. If the value of the frequency of ultrasonic

wave is known, the velocity of ultrasonic wave propagated through

the medium can be calculated.

(3) Thermal detectors: This is the most commonly used method of detection of ultrasonic waves. In this

method, a fine platinum wire is used. This wire is moved through the medium. At

the position of nodes, due to alternate compressions ad rarefactions, adiabatic

changes in temperature takes place. The resistance of the platinum wire changes

with respect to time. This can be detected with the help of Callendar and

Garrifith’s bridge arrangement. At the position of the antinodes, the temperature

remains constant. This will be indicated by the undisturbed balanced position of

the bridge.

(4) Quartz crystal method: This method is based on the principle of Piezo-electric effect. When one pair of the

opposite faces of a quartz crystal is exposed to the ultrasonic waves, the other pairs

of opposite faces developed opposite charges. These charges are amplified and

detected using an electronic circuit.

ACOUSTING GRATING

Principle:

“When ultrasonic waves are passed through a liquid, the density of the liquid

varies layer by layer due to the variation in pressure and hence the liquid will

act as a diffraction grating, so called acoustic grating. Under this condition,

when a monochromatic source of light is passed through the acoustical

grating, the light gets diffracted. Then, by using the condition for diffraction,

the velocity of ultrasonic waves can be determined.”

This method is based on the fact that ultrasonic waves which consist of alternate

compressions and rarefactions changes the density of the medium through which

they pass.

This leads to a periodic variation of refractive index of the liquid, such a liquid

column is subjected to ultrasonic waves constitutes an acoustical grating. If

monochromatic light is passed through the waves the liquid causes the diffraction

of light.

Figure shows the experimental arrangement, standing ultrasonic waves are

produced in a liquid contained in a glass tube. The density and so the refractive

index of the liquid is maximum at the nodal point and minimum at antinodal

points. Hence the nodal area acts as opaque region, while antinodal area acts as

transparent region for light. The liquid column thus resembles the rules grating.

The grating period d equal to /λ/2 and is given by

d sine θ=mλ

λ= wavelength of monochromatic light beam

m = order of minima.

An acoustic diffraction grating produced by a liquid column subjected to

ultrasonic waves.

Experimental set up of acoustic grating

Applications of Ultrasonic Waves in Engineering

(1)Detection of flaws in metals(Non Destructive Testing –NDT)

Principle

Ultrasonic waves are used to detect the presence of flaws or defects in the

form of cracks, blowholes porosity etc., in the internal structure of a

material

By sending out ultrasonic beam and by measuring the time interval of the

reflected beam, flaws in the metal block can be determined.

Experimental setup

It consists of an ultrasonic frequency generator and a cathode ray oscilloscope

(CRO),transmitting transducer(A), receiving transducer(B) and an amplifier.

Working

In flaws, there is a change of medium and this produces reflection of

ultrasonic at the cavities or cracks.

The reflected beam (echoes) is recorded by using cathode ray oscilloscope.

The time interval between initial and flaw echoes depends on the range of

flaw.

By examining echoes on CRO, flaws can be detected and their sizes can be

estimated.

(2) Ultrasonic Drilling

Ultrasonics are used for making holes in very hard materials like glass,

diamond etc.

For this purpose, a suitable drilling tool bit is fixed at the end of a powerful

ultrasonic generator.

Some slurry (a thin paste of carborundum powder and water) is made to flow

between the bit and the plate in which the hole is to be made

Ultrasonic generator causes the tool bit to move up and down very quickly

and the slurry particles below the bit just remove some material from the

plate.

This process continues and a hole is drilled in the plate.

(3) Ultrasonic welding

The properties of some metals change on heating and therefore, such metals

cannot be welded by electric or gas welding.

In such cases,the metallic sheets are welded together at room temperature by

using ultrasonic waves.

(4) Ultrasonic soldering

Metals like aluminium cannot be directly soldered.However, it is possible to

solder such metals by ultrasonic waves.

An ultrasonic soldering iron consists of an ultrasonic generator having a tip

fixed at its end which can be heated by an electrical heating element.

The tip of the soldering iron melts solder on the aluminium and the

ultrasonic vibrator removes the aluminium oxide layer.

The solder thus gets fastened to clear metal without any difficulty.

(5) Ultrasonic cutting and machining

Ultrasonic waves are used for cutting and machining.

(6) Ultrasonic cleaning

It is the most cheap technique employed for cleaning various parts of the

machine, electronic assembles, armatures, watches etc., which cannot be

easily cleaned by other methods.

(7) SONAR

SONAR is a technique which stands for Sound Navigation and Ranging.

It uses ultrasonics for the detection and identification of under water objects.

The method consists of sending a powerful beam of ultrasonics in the

suspected direction in water.

By noting the time interval between the emission and receipt of beam after

reflection, the distance of the object can be easily calculated.

The change in frequency of the echo signal due to the Dopper effect helps to

determine the velocity of the body and its direction.

Measuring the time interval (t) between the transmitted pulses and the

received pulse,

the distance between the transmitter and the remote object is determined

using the formula., where v is the velocity of sound in sea water.

The same principle is used to find the depth of the sea.

Applications of SONAR

Sonar is used in the location of shipwrecks and submarines on the bottom

of the sea.

It is used for fish-finding application .

It is used for seismic survey.

Applications of Ultrasonics

(1)Diagnostic sonography

Medical sonography (ultrasonography) is an ultrasound-based diagnostic

medical imaging technique used to visualize muscles, tendons, and many

internal organs, their size, structure and any pathological lesions.

Obstetric ultrasound is primarily used to:

• Date the pregnancy

• Check the location of the placenta

• Check for the number of fetuses

• Check for physical abnormities

• Check the sex of the baby

• Check for fetal movement, breathing, and heartbeat.

(2)Ultrasound therapeutic applications

More power ultrasound sources may be used to clean teeth in dental hygiene

or generate local heating in biological tissue, e.g. in occupational therapy,

physical therapy and cancer treatment.

Extracorporeal shock wave lithotripsy uses a powerful focused ultrasound

source to break up kidney stones.

It can also used in Ultrasonic blood Flow mete

(3) Ultrasound in research

Scientists often use in research, for instant to break up high molecular

weight polymers, thus creating new plastic materials.

Indeed, ultrasound also makes it possible to determine the molecular weight

of liquid polymers, and to conduct other forms of investigation on the

physical properties of materials.

Ultrasonic can also speed up certain chemical reactions. Hence it has gained

application in agriculture, that seeds subjected to ultrasound may germinate

more rapidly and produce higher yields.

C = 0.0254 PF

2.

Calculate the frequency to which a piezo electric oscillator circuit should be

tuned so that a piezo electric crystal of thickness 0.1 cm vibrates in

fundamental mode to generate ultrasonic waves. (Young’s Modulus and

density of material of crystal are 80 Gpa and 2654 kg / m3)

Solution:

Given,

Thickness of quartz crystal t=0.1cm.=0.1×10-2

Young modulus Y=80GPa=80×109 n/m

2

Density of crystal =2654Kg/m3

Frequency F=1/2t√Y/ρ

=1/2×0.1×10-2

√80×109 /2654

=509.28/2×10-3

=2.7451×106Hz.

EXAMPLE:-1.

Calculate the capacitance to produce ultrasonic waves of 106 Hz with

anInductance of 1 Henry.

f = 1/2∏√LC

106 = 1/2*3.14*√1*C

C = 1/(2*3.14*106)

2

= 0.0254*10-12

F