csc 126 2017/2018 1 [email protected] 126 2017/2018 2 [email protected] logic circuit...

CSC 126 2017/2018 1 [email protected]

CSC 126 PHYSICS FOR COMPUTING SYSTEMS

Course Instructor: Dr. Justus Simiyu

Department of Physics, UoN

Course Objective

This is a Physics course for beginners in Bachelor of Science in Computer Science course for the

University of Nairobi. The main objective of the course is to introduce the learner to the basic

principles of Physics applied in computing, specifically electricity, magnetism and semiconductor

Physics. The course is divided into two main parts one being Basic Physics and the second

Semiconductor Physics. In the first section, the learner is introduced to the relationship of Physics

with Computer Science touching on the Physics principles behind the operation of computers at

both hardware and software domain. In the second section, the learner is introduced to

Semiconductor Physics as the building block to basic computing platform of operation such as the

ALU, communication as well as interfacing among others. Upon successful completion of this

course, the learner is expected to be equipped with the knowledge about the physics of these

devices, how they work and how they are integrated in the whole system. This will also equip the

learner with knowledge and ability to implement routines, carry out maintenance and more

importantly have proper judgment on the action to be taken in case of failure or malfunction of

the systems.

Specific Objectives

The specific objectives are as per the topics covered in the entire syllabus.

Introduction

This section introduces the learner to the conception of the devices to be studied later in the

course. The learner is expected to understand the accepted model describing the atomic structure

and atomic interaction that brings about bonding to form a whole system of semiconductor

materials.

Semiconductor p-n junction devices (diodes)

This section introduces the learner to the band model that is used to describe conductivity and

classify materials in terms of conductivity. The section narrows down to semiconductors where

the learner is expected to distinguish between metals, semiconductors and insulators. Since

almost all the active devices used in computers are based on the basic diode characteristics, this

section deals in detail with these characteristics and the various applications according to the

characteristics.

Transistors

After having studied p-n junctions in the previous section, this chapter goes a step further where

several diodes are put together to form transistor devices. In this section, the learner is expected to

know the type of transistors (Bipolar and Field effect transistors), their characteristics and the

configuration models (i.e. common base, common emitter and common collector for the case of

bipolar junction transistors). The learner is also expected to gain knowledge on where and for

what requirements the transistor modes are used. Various biasing circuits are also dealt with in

detail in this section. This addresses the issue of determining practically bias points of the

transistors and can be ascertained as labelled by transistor manufactures and how they are

integrated in the systems.


Logic circuit applications

This section utilizes the knowledge acquired on transistors to apply in computer logic operations.

Since logic is the backbone of system operations of computers’ integrated circuitry, it is

important that the learner gets informed about the various logic items that are used. The logic

items dealt with in this section are AND, OR, NAND, NOR, INVERTER, EXCLUSIVE OR and

INHIBIT (ENABLE) operations and highlights the areas they are applied in the integrated

circuitry.

COURSE OUTLINE

PART ONE: BASIC PHYSICS

1. Electricity

Electrostatics: charge force, electrics fields, Coulomb’s law, Gauss’s law, electrics

potential, electric energy, capacitance

2. DC circuits

Current, Ohms law, resistance, DC circuits, Kirchhoff’s laws, voltage sources, circuit

analysis

3. Magnetism

Magnetics properties of matter, magnetic fileds, Biot-savart law, force law, applications

4. Electromagnetism

Faradays law, Lenz’s law, induced emf, inductance, applications

5. Waves

Wave equation, elastic waves in solids, progressive waves, interference, and reflection of

waves.

6. Optics

Reflection and refraction at planar and a planar surfaces, fibre optics, optical instruments.

PART TWO: SEMICONDUCTOR ELECTRONICS

A. INTRODUCTION

The atomic structure, Atomic and molecular bonds. Ionic bonding. Covalent bonding.

Metallic bonds. Insulators and semiconductors.

B. PHYSICS OF SEMICONDUCTOR MATERIALS

Band model.

Intrinsic semiconductors. Conduction by electrons and holes. Carrier concentration.

Extrinsic semiconductors.

Photoconduction and photovoltaic effects.

The p-n junction. I-V characteristics.

Diodes: resistance; Zener, tunnel, photo and light emitting diodes. Diode circuits.

Transistors: bipolar junction transistor. Common base, common emitter and common

collector: configuration and their characteristics. Transistor circuits. The transistor as a

switch. Logic circuit applications. Field effect transistor.

Pre-requisites: High School Physics

Delivery

Lectures, tutorials, supervised laboratories.


Reference books

1. Physics for Scientist and Engineers, by Serway

2. University Physics by Pierce

3. Principles of Physics by Ohanian

4. Physics by Halliday and Resnick

5. Advanced Level Physics by Nelkon and Parker

6. Elementary semiconductor Physics by Writ, HC. Cho. QC 11 .w93

7. Handbook of semiconductor electronics by Hunter, L.P 3rd Ed. TK 7872 .S4H8 1970

8. Introduction to physics of semiconductor devices by Roulston D. Cho Qc 611 .R86

9. Physics of semiconductor devices by SM Sze. TK 7871 85 .S988 1981

10. Integrated electronics by Millman and Halkias

© First prepared 2004, reprinted 2005, 2006, 2010, revised 2011, 2012, 2016, 2017, 2018


PART ONE: BASIC PHYSICS

I Electrostatics

1 Properties of Electric Charges

Simple experiments used to demonstrate the existence of electric forces and charges are for

example:

Running a comb through hair on dry day, the comb starts to attract bits of papers.

Inflated balloon is rubbed with wool. The balloon adheres to the wall or the ceiling of a

room

When materials behave like this, they are said to be electrified or to have become electrically

charged

During wet day, excessive amount of moisture can lead to a leakage of charge from the

electrified body to the earth by various conducting path

Benjamin Franklin (1706 - 1790) performed a systematic series of experiments and found out

that there are two kinds of electric charges which he named positive (+ve) and negative (-ve)

Figure 1.1 (a) a negatively charged rubber rod suspended by a thread is attracted to a

positively charged glass rod. (b) a negatively charged rubber rod is repelled by nother

negatively charged rubber rod

When a rubber rod is rubbed with fur and then suspended by a non- metallic thread, when a glass

rod rubbed with silk is brought near the rubber rod, the rubber rod is attracted to glass rod.

Likewise, if two charged rubber/glass rods are brought near each other, the force between them is

repulsive.

This experiments shows that, there are two different state of electrification.

Conclusion


Like charges repel one another and unlike charges attract one another. Using convection

suggested by Franklin, the electric charges on glass rod is called positive and that on

rubber rod is called negative.

Electric charge is always conserved. The body becomes charged due to transfer of

charged from one body to another.

In 1909, Robert Millikan (1868 - 1953) confirmed that electric charge always occurs as some

integral multiple of some fundamental unit of charge e. In modern terms, charge q is said to be

quantized, q is the standard symbol of charge. Electric charge exists as discrete packets, thus we

can write q=Ne, where N is an integer .

Charles Coulomb (1736 - 1806), using torsion balance confirmed that electric force between two

small charged spheres is proportional to the inverse square of their separation, that is,

2

1F

r (1.1)

2 Insulators and conductors

i. Conductors are materials in which electric charges move quite freely, eg. Copper,

aluminum and silver.

ii. Insulators are materials that do not ready transport charge, e.g glass, rubber and Lucite.

When insulators are charged by rubbing, charge is unable to move to other regions of the

material. A conductor on the other hand, when charged the charge is distributed over the

entire surface.

iii. Semiconductors, their electrical properties are somewhere between those of insulators

and those of conductors eg. Silicon and germanium. When a conductor is connected to

Earth by means of a conducting wire or pipe it is said to be grounded.

3 Coulomb’s Law

Coulomb’s law states that:

i. Electric force is inversely proportional to the square of separation, r between the two

particles and directed along the line joining the particles

ii. The force is proportional to the product of the charges q1 and q2 on the two particles

iii. The force is attractive if the charges are of opposite sign and repulsive if the charges have

the same sign.

1 2

2

ek q qF

r (1.2)

Where ke is called coulombs constant = 9 2 28.99 10 N.m / C , the constant ke is also written

1

4e

o

k

(1.3)

where o called permittivity of free space and has a value 12 2 28.85 10 C / N.mo .

When the current in a wire is 1A, the amount of charge that flows past a given point ion

the wire in 1sec is 1C. The smallest unit of charge known in nature is the charge of an

electron or proton. The charge of an electron or proton has

magnitude191.602 10 Ce . Therefore, 1C of charge is equal to the charge of

186.3 10 electrons.


Charge and mass of the electron, proton and neutron

Particle Charge, C Mass, kg

Electron (e) 191.602 10 319.11 10

Proton (p) 191.602 10 271.67 10

Neutron (n) 0 271.67 10

Figure 1. 2. Two point charges separated by a distance r exert a force on each other given

by Coulomb's law.

The electric force exerted on q2 due to a charge q1 written F21 can be written 1 221 2

ek q qF

r r,

where r is a unit vector directed from q1 to q2 as shown in the figure above. Coulomb’s law obeys

Newton’s third law; the electric force exerted on q1 by q2 is equal in magnitude to the force

exerted on q2 by q1 and in the opposite direction, i.e F12 = -F21. If q1q2 have the same sign, the

product q1q2 is positive and the force of interaction is repulsive; likewise, if the product is

negative, the force of interaction is attractive. If we have a system of charges, the resultant force

is the vector sum given by F1=F12+F13+F14+………..F1N, ie, principle of superposition applies in

electrostatic and it is an experimentally observed fact.

Example.

Consider three point charges located at the corners of a

triangle, as shown in the figure, where

1 3 25.0 , 2.0q q C q C and a = 0.10m. Find the

resultant force q3.

Solution


The force exerted on q3 by q2 is F32. The resultant force F3 exerted on q3 is the vector sum

F31+F32. The magnitude 3 2

32 2

ek q qF

a =

9 2 2 6 6

2

8.99 10 / (5.0 10 ) (2.0 10 )

(0.10 )

Nm C C C

m

=9.0N

Note that since q3 and q2 have opposite signs, F32 is to the left as shown in figure. The magnitude

of the force exerted on q3 by q1 is 3 1

31 2( 2 )

ek q qF

a

=

9 2 2 6 6

2

8.99 10 / (5.0 10 ) (5.0 10 )

2(0.10)

Nm C C C = 11N. Force F31 is repulsive and makes

an angle of 45o with the x axis, therefore the x and y component of F31 are equal with magnitude

given by 31 cos45oF =7.9N. The force F32 is in the negative x direction. Hence, the x and y

component of the resultant force on q3 are Fx = F31x+F32 = 7.9N – 9.0N = -1.1N. Fy = F31y = 7.9N

which can be expressed as vectors as F3 = (-1.1i+7.9j)N.

Example

Three charges lie along the x axis as shown in

the figure. The positive charge 1 15q C is at

x = 2.00m and positive charge 2 6.0q C is at

the origin. Where a negative charge, q3 must be

placed on the x axis such that the resultant force

on it is zero.

Solution

Q3 is positive and both q2 and q1 are positive,

hence the forces F31 and F32. If we let x be the coordinates of q3 then the forces

3 1

31 22.0

ek q qF

x

and 3 2

32 2

ek q qF

x in order for the resultant force on q3 to be zero, F32 must be equal to and

opposite F31. Since ek and q3 are common to both sides we solve for x and

3 2 3 1

222.0

e ek q q k q q

x x

2 1

2 2 4 4

q q

x x x

2 2

1 2 4 4q x q x x

6 2 6 215 10 6 10 4 4C x C x x

6

2 2

6

6 104 4

15 10

Cx x x

C

= 20.4 4 4x x

2 20.4 1.6 1.6x x x this equation reduces to

26 16 16 0x x

-0.775 or 0.775

4 The Electric Field


The electric field vector E at a point in space is defined as the electric force F acting on a positive

test charge placed at that point divided by the magnitude of the test charge qo.

oq

FE (1.4)

E is the field produced by some charge external to the test charge and not the field

produced by the test charge, E has unit N/C

Since the electric field at the position of the test charge is defined by

oq

FE we find that, at the

position of qo the electric charge by q is

2

ek q

rE r (1.4)

r is a unit vector directed from q towards qo

The total electric field due to a group of charges equals the vector sum of the electric

fields of all the charges e.g to calculate the electric field at point P due to a group of point

charges, first calculate electric field vectors at P individually using equation 1.5 then add

them vectorially.

Thus, the electric field of a group of charges (excluding the test charge qo) can be expressed as

2

ie

i i

qk

r E r (1.5)

ri is the distance from the ith charge to the point P (the location of the test charge) and ri is a unit

vector directed from qi towards P.

Example

Find the electric force on a proton placed in an electric field of 42.0 10 N/C directed along the positive x axis.

Solution

Charge of proton 191.6 10 Ce , the electric force on it is

F eE

19 4 151.6 10 2.0 10 / 3.2 10F C N C N

Example

A charge 1 7q C is located at the origin and the second

charge 2 5.0q C is located on the x axis 0.30 m from the

origin. Find the electric field at the point P, which has

coordinates (0,0.4)m

Solution

The field E1 is due to 7.0 C charge and E2 is due to 5.0 C charge, their magnitudes are

9 2 2 6

1

1 22

1

5

8.99 10 . / 7.0 10

0.40

3.9 10 /

eN m C Ck q

Er m

N C


2

9 2 2 6

2

2 2

2

5

8.99 10 . / 5.0 10

0.5

1.8 10 /

eN m C Ck q

Er m

N C

The vector E1 has only y component. The vector E2 has an x component given by

2 2

3cos

5 E E and negative y component given by

2

4sin sin

5 E . Hence the vectors

can be expressed as 5

1 3.9 10 /E jN C 5 5

2 1.1 10 1.4 10 /E i j N C . The resultant

field E at P is the superposition of E1 and E2, E = E1 + E2

5 51.1 10 2.46 10 /i j N C

E

E has the magnitude 2 2 51.1 2.46 2.69 10 /E N C and makes an angle of 66o

with positive x axis.

Electric Field Lines

The electric lines for a point charge

i. For a positive point charge, the lines are radially outward

ii. For a negative point charge, the lines are radially inwards

Rules for drawing electric field lines for any charge distribution

i. The lines must begin on positive charges and terminate on negative charges, although if the

net charge is not zero, the lines may begin or terminate at infinity

ii. The number of lines drawn leaving a positive charge or approaching a negative charge is

proportional to the magnitude of the charge

iii. No two field lines can cross or touch

5 Motion of Charged Particles in a Uniform Electric Field

When a particle of charge q and mass m is placed in an electric field E, the electric force exerted

on the charge is qE. If this is the only force exerted on the charge, then Newton’s second law

applied to the charge gives q ma F E , acceleration of the particle is therefore

qE

am

(1.6)

If E is uniform (that is constant in magnitude and direction) the acceleration is a constant of the

motion. If the charge is positive, the acceleration is in the direction of the electric field. If the

charge is negative the acceleration is in the direction opposite the electric field.

Example

A positive point charge q of mass m is released from rest in a uniform electric

field E directed along the x axis as shown in figure on the left. Describe its

motion.

Solution

qEa

m and is constant. The motion of charge is simple linear motion along the

x axis thus 21

2o o ox x v t at v v at

+

+

+

+

+

+

-

-

-

-

--x

vv=0

+ +

q

E


2 2 2 ( )o ov v a x x taking 0 and 0ox v gives 2 21

2 2

qEx at t

m but v at therefore

qE

v at tm

and 2 2

2qE

v ax xm

. The kinetic energy of the charge after it has

moved a distance x is 21 1 2

2 2

qEk mv m x qEx

m

.

6 Gauss Law

Electric Flux

Electric flux is represented by the number of electric field lines

penetrating some surface. Consider an electric filed that is

uniform in both magnitude and direction as in figure 1. The

electric field lines penetrate a rectangular surface of area A,

which is perpendicular to the field. Recall that the number of

lines per unit area is proportional to the magnitude of the

electric field. (NB. When surface is being penetrated it encloses

some net charge, the net number of lines that go through the surface is proportional to the net

charge within the surface. The number of lines counted is independent of the shape of the surface

enclosing the charge). Therefore the number of lines penetrating the surface is proportional; to the

product EA. The product of the electric field strength E and a surface area A perpendicular to the

field is called the electric flux, .

EA units N.m/C (1.7)

If the surface under consideration is not perpendicular to the field, the number of lines (in other

words, the flux) through it must be less than that

given by equation 1.7. Consider figure 2 on the left.

The normal to the surface of area A is at an angle

to the uniform electric filed. NB the number of lines

that cross this area is equal to the number that

crosses the projected area A’, which is perpendicular

to the field, from the figure, the areas are related as

' cosA A . Since the flux through the area A

equals the flux through A’, we conclude that the flux

through A is

cosEA (1.8)

From equation 2, flux through a surface of fixed area has the maximum value, EA, when the

surface is perpendicular to the field (in other words, when the normal to the surface is parallel to

the filed, that is, 0o ); the flux is zero when the surface is parallel to the field (when the

normal to the surface is perpendicular to the filed, that is, 90 ).

The above definition applies over small element of area.

EA

Figure 1

normal

E

A

A'=Acos

Figure 2


II Current and Resistance

1. Electric Current

Consider charge moving perpendicular to surface of area A as shown in figure 1. Current is the

rate at which charge flows through this surface. If ΔQ is the amount

of charge that passes through this area in a time interval Δt, the

average current avI is

av

QI

t

(2.1)

If the rate at which charge flows varies in time, the current also varies in time and we define the

instantaneous current I as the differential limit of equation 1.1

dQ

Idt

(2.2)

Consider current in a conductor of cross sectional area A. The

volume of an element of the conductor of length Δx is AΔx. If

n is the number of mobile charge carriers per unit volume then

the number of mobile carriers in the volume element is nAΔx.

Therefore the charge ΔQ in this element is

number of charges charge per particle = (nA x)qQ (2.3)

If the charge carriers move with a speed dV , the distance they move in a time Δt is tVx d

Therefore qtnAVQ d )( . If we divide through by Δt, the current in the conductor is given by

(nA )d

d

V t qQI nAV q

t t

(2.4)

Example: Drift speed in a copper wire

A copper wire of cross-sectional area 3.0x10-6

m2 carries a current of 10.0A. Find the drift speed

of the electrons in this wire. Density of copper is 8.95g/cm2

3

209.7

/95.8

5.63cm

cmg

gmV

Cu = 63.5g/mol

236.02 10AN atoms

2322 3

3

6.02 108.48 10 /

7.09

x electronsn electrons cm

cm

22 3 6 3 3(8.48 10 / )(10 / )electrons cm cm m

28 38.48 10 /electrons m

+

+

++

+

A

IFigure 1

q

vd

xFigure 2


28 3 19 6 2

10.0 /

(8.48 10 ) (1.6 10 ) (3.0 10 )d

I C sV

nqA x m x C x m

42.46 10 /m s

2. Resistance and Ohm’s Law

Consider a conductor of cross sectional area A carrying current I. The current density J in the

conductor is defined to be the current per unit area. Since the current AnqVI d , the current

density is

2 units A/md

IJ nqV

A (2.5)

A current density J and an electric field E are established in a conductor when potential

difference is maintained across the conductor. If the potential difference is constant, the current is

also constant. Very often, the current density is proportional to the electric field and we express

J E (2.6)

Materials that obey equation 2.6 are said to follow Ohm’s law named after George Simon Ohm

(1787-1854). More specifically, Ohm’s law states that: the ratio of the current density to the

electric field is a constant, , that is independent of the electric field producing the current.

Materials that obey Ohm’s law and hence demonstrate this linear behavior between E and J are

said to be Ohmic. Potential difference is related to the electric field through the relationship

V lE where l is the distance of separation where electric field E is acting. Therefore

V

l J E (2.7)

Since J = I/A, the potential difference can be expressed as

l l

V IA

J (2.8)

The quantity l

Ais called the resistance R of the conductor, hence

units V/A=l V

RA I

(2.9)

The inverse of conductivity is resistivity

I

(2.10)

Hence

l

RA

(2.11)

Example: The resistance of a conductor

Calculate the resistance of an aluminum cylinder that is 10.0 cm long and has a cross sectional

area of 2.00x10-4

m2. Repeat the calculation for a glass cylinder of resistivity 3.0x10

10 m.

Solution

R 8 5

4 2

0.100(2.82 10 ) 1.41 10

2.00 10

l mm

A m

10 13

4 2

0.100(3.0 10 ) 1.5 10

2.00 10

l mR m

A m


Resistance varies with temperature if a wire has a resistance Ro at a temperature To then its

resistance R at a temperature T is

)( oooo TTRRR where α is the temperature coefficient of resistance of the

material of the wire and it varies with temperature and so its application is over short ranges.

Units K-1

or -1Co

A similar relation applies for resistivity

oT - To o

3. Electrical Energy and Power

Consider the circuit shown in figure 3. Imagine a positive charge

ΔQ moving around the circuit from point a through the battery and

resistor and back to a. As the charge moves from a to b its potential

energy increases by an amount VΔQ (where V is potential at b).

However as the charge moves from c to d through the resistor, it

loses this electrical potential energy as it undergoes collision with

atoms in the resistor.

Electrical power delivered by an energy source as it carries a charge

q through potential V in time t is

but work VQ Q

Power Itime t t

(2.12)

Hence P =VI units in Watts.

The power loss in a resistor is found by replacing V in VI by IR, or by replacing I in VI by V/R to

obtain

22Power loss in resistor

VVI I R

R (2.13)

The heat generated in a resistor per second is equal to the power loss in the resistor

RIVI 2 (2.14)

Convenient conversion:

min/4.427461

min9.56341.11

239.011

BtuWhp

BtuhpkW

scalsJW

DC – Circuits

(a) Resistors in Series

R

a

b

c

d

+

-

I

Figure 3

R1 R2 R3

V1 V2 V3

I

a b c d

Resistors in series

Figure 4


For resistors in series

1 2 3..........eq nR R R R R

ad ab bc cdV V V V

ab bc cdI I I I

(b) Resistors in parallel

For resistors in parallel

1 2 3

1 1 1 1 1........

eq nR R R R R

ab abV V

1 2 3I I I I

III Magnetism

R1

R2

R3

I

a b

Resistors in parallel

Figure 5

I

Vab

I2

I1

I3


Metallic materials that attract ion filings have a property of magnetism and the materials are

called magnets. Magnetic materials have the following properties

1. Have two poles (north and south)

2. Like poles repel each other while unlike poles attract each other.

3. Magnetic material will rest freely in the north south orientation

Magnetism is used in storage media for computers.

1. Magnetic Fields

When a charged particle is moving through a magnetic field, a magnetic force acts on it. It is

found experimentally that the strength of the magnetic force F on the particle is proportional to

the magnitude of the charge q, the magnitude of the velocity v, the strength of the external

magnetic field B, and the sine of the angle θ between the direction of v and the direction of B. The

force is given by

(3.1)

This force has its maximum value when the charge moves in a direction perpendicular to the

magnetic field lines, decreases in value at other angles, and becomes zero when the particle

moves along the field lines. Further, the electric force is directed parallel to the electric field

while the magnetic force on a moving charge is directed perpendicular to the magnetic field.

If F is in newtons, q in coulombs, and v in meters per second, then the SI unit of magnetic field is

the tesla (T), also called the weber (Wb) per square meter (1 T = 1 Wb/m2. From the equation

above, it can be seen that the force on a charged particle moving in a magnetic field has its

maximum value when the particle’s motion is perpendicular to the magnetic field, corresponding

to θ = 90°, so that sin θ = 1. The magnitude of this maximum force has the value

Fmax = qvB (3.2)

Also F is zero when is parallel to (corresponding to θ = 0° or 180°), so no magnetic force is

exerted on a charged particle when it moves in the direction of the magnetic field or opposite the

field.

Experiments show that the direction of the magnetic force is always perpendicular to both and ,

as shown in Figure for a positively charged particle. Which is determined by the right-hand rule

number 1:

Figure 3.1: Right hand rule number 1

1. Point the fingers of your right hand in the direction of the velocity v.

2. Curl the fingers in the direction of the magnetic field , moving through the smallest angle

3. Your thumb is now pointing in the direction of the magnetic force exerted on a positive charge.


2. Magnetic Force on a Current-Carrying Conductor

If a magnetic field exerts a force on a single charged particle when it moves through a magnetic

field, it should be no surprise that magnetic forces are exerted on a current-carrying wire, as well.

This follows from the fact that the current is a collection of many charged particles in motion;

hence, the resultant force on the wire is due to the sum of the individual forces on the charged

particles. The force on the particles is transmitted to the ―bulk‖ of the wire through collisions with

the atoms making up the wire. Some explanation is in order concerning notation in many of the

figures.

3. Motion of a Charged Particle in a Magnetic Field

Consider the case of a positively charged particle moving in a uniform magnetic field so that the

direction of the particle’s velocity is perpendicular to the field.

Figure 3.2: motion of a charged particle in a magnetic field

The label Bin and the crosses indicate that B is directed into the page. Application of the right-

hand rule at point P shows that the direction of the magnetic force F at that location is upward.

The force causes the particle to alter its direction of travel and to follow a curved path.

Application of the right-hand rule at any point shows that the magnetic force is always directed

toward the center of the circular path; therefore, the magnetic force causes a centripetal

acceleration, which changes only the direction of v and not its magnitude. Because F produces

the centripetal acceleration, we can equate its magnitude, qvB in this case, to the mass of the

particle multiplied by the centripetal acceleration v 2/r. From Newton’s second law, we find that

(3.4)

which gives

(3.5)

This equation says that the radius of the path is proportional to the momentum mv of the particle

and is inversely proportional to the charge and the magnetic field. The equation is often called the

cyclotron equation, because it’s used in the design of these instruments (popularly known as atom

smashers).

4. Magnetic Field of a Long, Straight Wire and Ampère’s Law

The direction of the magnetic field B in a long wire is found by right-hand rule number 2:


Figure 3.3: Right hand rule no. 2

Point the thumb of your right hand along a wire in the direction of positive current, as in Figure.

The fingers then naturally curl in the direction of the magnetic field B. When the current is

reversed, B also reverses. These observations lead to a mathematical expression for the strength

of the magnetic field due to the current I in a long, straight wire:

(3.6)

5. Ampère’s Law and a Long, Straight Wire

Equation 3.6 enables one to calculate the magnetic field due to a long, straight wire carrying a

current. This statement, known as Ampère’s circuital law, can be written

(3.7)

where BE is the component of parallel to the segment of length ∆l and means that we take the sum

over all the products BE∆l around the closed path. Ampère’s law is the fundamental law

describing how electric currents create magnetic fields in the surrounding empty space.

6. Magnetic Force Between Two Parallel Conductors

As shown earlier, a magnetic force acts on a current-carrying conductor when the conductor is

placed in an external magnetic field. Because a conductor carrying a current creates a magnetic

field around itself, it is easy to understand that two current-carrying wires placed close together

exert magnetic forces on each other. Consider two long, straight, parallel wires separated by the

distance d and carrying currents I1 and I2 in the same direction, as shown below

Figure 3.4: two current carrying conductors

Wire 1 is directly above wire 2. What’s the magnetic force on one wire due to a magnetic field set

up by the other wire? In this calculation, we are finding the force on wire 1 due to the magnetic

field of wire 2. The current I2, sets up magnetic field B2 at wire 1. The direction of B2 is

perpendicular to the wire, as shown in the figure. From


it is found that the magnitude of this magnetic field

is

(3.8)

From the magnitude of the magnetic force on wire 1 in the presence of field

due to I2 is

(3.9)

We can rewrite this relationship in terms of the force per unit length:

(3.10)

The direction of F1 is downward, toward wire 2, as indicated by right-hand rule number 1. This

calculation is completely symmetric, which means that the force on wire 2 is equal to and

opposite to F1, as expected from Newton’s third law of action–reaction. We have shown that

parallel conductors carrying currents in the same direction attract each other If a conductor

carries a steady current of 1 A, then the quantity of charge that flows through any cross section in

1s is 1C.


IV Alternating Current Circuits and Electromagnetic Waves

Introduction

Every time we turn on a television set, a stereo system, a pc or laptop or any of a multitude of

other electric appliances, we call on alternating currents (AC) to provide the power to operate

them. These equipments mentioned above have circuits that comprise among other items,

resistors, capacitors and inductors (these are known as the basic circuit devices). When these

devices encounter and ac supply, they behave differently under different conditions. We begin

our study of AC circuits by examining the characteristics of a circuit containing a source of power

and one other circuit element: a resistor, a capacitor, or an inductor. Then we examine what

happens when these elements are connected in combination with each other. The discussion is

limited to simple series configurations of the three kinds of elements. The section ends with a

discussion of electromagnetic waves, which are composed of fluctuating electric and magnetic

fields. Electromagnetic waves in the form of visible light enable us to view the world around us;

infrared waves warm our environment; radio-frequency waves carry our television and radio

programs, as well as information about processes in the core of our galaxy. X-rays allow us to

perceive structures hidden inside our bodies, and study properties of distant, collapsed stars.

1. Resistor Circuit

An AC circuit consists of combinations of circuit elements and an AC generator or an AC source,

which provides the alternating current. We have seen that the output of an AC generator is

sinusoidal and varies with time according to

(4.1)

where ∆v is the instantaneous voltage, ∆Vmax is the maximum voltage of the AC generator, and f

is the frequency at which the voltage changes, measured in hertz (Hz). Consider a simple circuit

consisting of a resistor and an AC source (Figure 4.1). The current and the voltage across the

resistor are shown in Figure (b).

Fig 4.1. (a) Simple resistor circuit (b) the corresponding current and voltage profile (c) phasor

diagram

Assuming there are no losses in the circuit, the voltage drop across the resistor (vR) should be

same as the supply voltage (∆v). With the voltage drop across R given by iR, current through the

resistor can be found as

ftIftR

V

R

viR 2sin2sin max

max

(4.2)

Comparing this equation with ∆v shows that in a resistor circuit v and i are in phase, (it is also

said that a resistor is a linear device due to this property). The behavior is shown in figure (b). In

the figure, between points a and b, the current is decreasing in magnitude but is still in the

∆i

R

∆vR

(a) (b) (c)


positive direction. At point b, the current is momentarily zero; it then begins to increase in the

opposite (negative) direction between points b and c. At point c, the current has reached its

maximum value in the negative direction. The current and voltage are in step with each other

because they vary identically with time. Because the current and the voltage reach their

maximum values at the same time, they are said to be in phase.

2. Capacitor Circuit

Still the input voltage varies as

(4.3)

And is connected as shown in figure 2(a) below.

Fig. 4. 2: Series capacitor circuit (b) the corresponding current and voltage profile (c) phasor

diagram

Assuming there are no losses in the circuit, the voltage drop across the capacitor (∆vC) should be

same as the supply voltage (∆v) (i.e. ∆v = ∆vC). It is known that when a capacitor acquires

charge, it raises its potential proportionally, i.e

Q = C∆vC (4.4)

where ∆vc is the voltage drop across the capacitor and C is the capacitance of the capacitor. ∆vC is

then given by

∆vC = Q/C (4.5)

To find current flowing through the capacitor, we find the relationship between charge (Q) and

current i: (i = dQ/dt) and remembering that ∆v = ∆vC we have

ftVfCftVdt

dC

dt

vdCi c

c 2cos2)2sin()(

maxmax

(4.6)

This is plotted on figure (b) as a cosine profile with C∆Vmax = Imax and it is compared with the

input voltage ∆v.

From the figure, when an alternating voltage is applied across a capacitor, the voltage reaches its

maximum value one-quarter of a cycle after the current reaches its maximum value. It is said that

the voltage across a capacitor always lags the current by 90°. The impeding effect of a

capacitor on the current in an AC circuit is expressed in terms of a factor called the capacitive

reactance XC, defined as

fCX c

2

1 (4.7)

Which reduces equation to

∆i

C

∆vC


ftIftX

Vi

c

c 2cos2cos maxmax

(4.8)

This is similar to Ohm’s law for a resistor with a difference being the capacitative reactance being

equivalent to resistance in the resistor circuit.

3. Inductors Circuit

Now consider an AC circuit consisting only of an inductor connected to the terminals of an AC

source, as in figure… The changing current output of the generator produces a back emf that

impedes the current in the circuit according to Fraday’s law of induction.

Fig. 4.3: (a) Series inductor circuit (b) the corresponding current and voltage profile (c) phasor

diagram

The magnitude of this back emf is

(4.9)

Where L is the coefficient of inductance.

Assuming there are no losses in the circuit, then the voltage drop across the inductor (∆vL) is

equal to the voltage supply (∆v), i.e

∆vL = ∆v,

Current through the inductor is given by

ftfL

VftdtV

Ltv

Li LL

2cos

22sin

11 maxmax (4.10)

Which is plotted in figure (b) with LXfL 2 known as the inductive reactance of the inductor

circuit. The equation therefore becomes

ftIftX

Vi

L

L 2cos2cos maxmax (4.11)

The above equation is similar to ohm’s law for the resistor with the difference being XL in the

inductor circuit. The inductive reactance increases with increasing frequency and increasing

inductance. Contrast these facts with capacitors, where increasing frequency or capacitance

decreases the capacitive reactance.

∆iL

∆vL

(c)


4. The RLC Series Circuit

Consider what happens when R, C and L are combined as shown below.

Fig. 4.4: (a) RLC Series circuit, phasor diagram for (b) the three elements (c) resultant voltage

drop for the three elements.

The current in the circuit is the same at all points in the circuit at any instant and varies

sinusoidally with time, as

ftIic 2sinmax (4.12)

Earlier, it was shown that the voltage across each element may or may not be in phase with the

current. The instantaneous voltages across the three elements, shown in figure (b), have the

following phase relations to the instantaneous current:

1. The instantaneous voltage ∆vR across the resistor is in phase with the instantaneous current.

2. The instantaneous voltage ∆vL across the inductor leads the current by 90°.

3. The instantaneous voltage∆vC across the capacitor lags the current by 90°.

The net instantaneous voltage ∆v supplied by the AC source equals the sum of the instantaneous

voltages across the separate elements: ∆v = vR + ∆vC + ∆vL. These are added vectorially using the

phasor diagram (b). The resultant of the vector addition is given in figure (c) and mathematically

is given by

22

max CLR VVVV (4.13)

In the above equation, all voltages are maximum values. The result for the maximum voltage

∆Vmax in the above equation reinforces the fact that the voltages across the resistor, capacitor,

and inductor are not in phase, so one cannot simply add them to get the voltage across the

combination of element, or the source voltage.

The above equation can be written in terma of the current flowing to form the equivalent Ohm’s

law relationship by letting ∆VR = ImaxR, ∆VC = ImaxXC and ∆VL = ImaxXL where Imax is the

maximum current in the circuit leading to

22

maxmax CL XXRIV (4.14)

Giving

22

CL XXRZ which is known as the series impedance of the circuit.

This reduces the equation for ∆Vmax to be

∆VRmax = ImaxZ (4.15)

I which is equivalent to Ohm’s law.

V

∆vR

∆vL - ∆vc

∆i ∆vR

∆vL

∆vC

∆i

(a) (b) (c)


A summary of the discussed properties is given in table below

5. Applications

The Antenna

A current varying circuit produces a changing magnetic field and likewise a varying magnetic

field produces accompanying varying electric field (that is electricity and magnetism co-exists).

Energy stored in an LC circuit is continually transferred between the electric field of the capacitor

and the magnetic field of the inductor. However, this energy transfer continues for prolonged

periods of time only when the changes occur slowly. If the current alternates rapidly, the circuit

loses some of its energy in the form of electromagnetic waves. As a fact, electromagnetic waves

are radiated by any circuit carrying an alternating current. The fundamental mechanism

responsible for this radiation is the acceleration of a charged particle. Whenever a charged

particle accelerates it radiates energy. An alternating voltage applied to the wires of an antenna

forces electric charges in the antenna to oscillate. This is a common technique for accelerating

charged particles and is the source of the radio waves emitted by the broadcast antenna of a radio

station.


Figure 4. 5: Production of electromagnetic waves by the antenna

Figure 4.5 illustrates the production of an electromagnetic wave by oscillating electric charges in

an antenna. Two metal rods are connected to an AC source, which causes charges to oscillate

between the rods. The output voltage of the generator is sinusoidal. At t = 0, the upper rod is

given a maximum positive charge and the bottom rod an equal negative charge, as in Figure a.

The electric field near the antenna at this instant is also shown in the same figure. As the charges

oscillate, the rods become less charged, the field near the rods decreases in strength, and the

downward-directed maximum electric field produced at t = 0 moves away from the rod. When the

charges are neutralized, as in Figure b, the electric field has dropped to zero, after an interval

equal to one-quarter of the period of oscillation. Continuing in this fashion, the upper rod soon

obtains a maximum negative charge and the lower rod becomes positive, as in Figure c, resulting

in an electric field directed upward. This occurs after an interval equal to one-half the period of

oscillation. The oscillations continue as indicated in Figure d. Note that the electric field near the

antenna oscillates in phase with the charge distribution: the field points down when the upper rod

is positive and up when the upper rod is negative. Further, the magnitude of the field at any

instant depends on the amount of charge on the rods at that instant. As the charges continue to

oscillate (and accelerate) between the rods, the electric field set up by the charges moves away

from the antenna in all directions at the speed of light.

Other applications range from long-distance electric power transmission to wireless networks in

computer systems.

6. Properties of Electromagnetic Waves

1. Electromagnetic waves travel at the speed of light.

2. Electromagnetic waves are transverse waves, because the electric and magnetic fields are

perpendicular to the direction of propagation of the wave and to each other.

3. The ratio of the electric field to the magnetic field in an electromagnetic wave equals the speed

of light.

4. Electromagnetic waves carry both energy and momentum, which can be delivered to a surface.

7. The Electromagnetic Wave Spectrum

All electromagnetic waves travel in a vacuum with the speed of light, c. These waves transport

energy and momentum from some source to a receiver. In 1887, Hertz successfully generated and

detected the radio-frequency electromagnetic waves predicted by Maxwell. Maxwell himself had

recognized as electromagnetic waves both visible light and the infrared radiation discovered in

1800 by William Herschel. It is now known that other forms of electromagnetic waves exist that

are distinguished by their frequencies and wavelengths. Because all electromagnetic waves travel

through free space with a speed c, their frequency f and wavelength λ are related by


c = f λ

The various types of electromagnetic waves are presented in Figure 6 below

Figure 4.6: The electromagnetic wave spectrum

Radio waves are the result of charges accelerating through conducting wires. They are, used in

radio and television communication systems.

Microwaves (short-wavelength radio waves) have wavelengths ranging between about 1 mm and

30 cm and are generated by electronic devices. Their short wavelengths make them well suited

for the radar systems used in aircraft navigation and for the study of atomic and molecular

properties of matter. Microwave ovens are an interesting domestic application of these waves

Infrared waves (sometimes incorrectly called ―heat waves‖), produced by hot objects and

molecules, have wavelengths ranging from about 1 mm to the longest wavelength of visible light,

7 x 10-7

m. They are readily absorbed by most materials. The infrared energy absorbed by a

substance causes it to get warmer because the energy agitates the atoms of the object, increasing

their vibrational or translational motion. The result is a rise in temperature. Infrared radiation has

many practical and scientific applications, including physical therapy, infrared photography, and

the study of the vibrations of atoms.


Visible light, the most familiar form of electromagnetic waves, may be defined as the part of the

spectrum that is detected by the human eye. Light is produced by the rearrangement of electrons

in atoms and molecules. The wavelengths of visible light are classified as colors ranging from

violet λ ~ 4 x 10-7

m) to red (λ ~ 7 x 10-7

m). The eye’s sensitivity is a function of wavelength and

is greatest at a wavelength of about 5 .6 x 10-7

m (yellow green).

Ultraviolet (UV) light covers wavelengths ranging from about 4 x 10-7

m (400 nm) down to 6 x

10-10

m (0.6 nm). The Sun is an important source of ultraviolet light (which is the main cause of

suntans).

X-rays are electromagnetic waves with wavelengths from about 10-8

m (10 nm) down to 10-13

m

(10-4

nm). The most common source of x-rays is the acceleration of high-energy electrons

bombarding a metal target. X-rays are used as a diagnostic tool in medicine and as a treatment for

certain forms of cancer.

Gamma rays—electromagnetic waves emitted by radioactive nuclei—have wavelengths ranging

from about 10-10

m to less than 10-14

m. They are highly penetrating and cause serious damage

when absorbed by living tissues. Accordingly, those working near such radiation must be

protected by garments containing heavily absorbing materials, such as layers of lead. When

astronomers observe the same celestial object using detectors sensitive to different regions of the

electromagnetic spectrum, striking variations in the object’s features can be seen.

8. The Doppler Effect in Electromagnetic Waves

Doppler Effect is a phenomenon of apparent frequency of a wave observed when there is a

relative motion between the source of the wave and the observer. It is a phenomena that occurs

commonly in sound waves but it is also observed in electromagnetic waves. In Doppler effect, the

observed frequency of the wave is larger or smaller than the frequency emitted by the source of

the wave. Doppler Effect occurring in electromagnetic waves differs from that for sound waves in

two ways. First, in the case for sound waves, motion relative to the medium is most important,

because sound waves require a medium in which to propagate. In contrast, the medium of

propagation plays no role in the case for electromagnetic waves, because the waves require no

medium in which to propagate. Second, the speed of sound that appears in the equation for the

Doppler effect for sound depends on the reference frame in which it is measured. In contrast, the

speed of electromagnetic waves has the same value in all coordinate systems that are either at rest

or moving at constant velocity with respect to one another.

The single equation that describes the Doppler effect for electromagnetic waves is given by the

approximate expression

(4.16)

where fO is the observed frequency, fS is the frequency emitted by the source, c is the speed of

light in a vacuum, and u is the relative speed of the observer and source.

The positive sign in the equation must be used when the source and observer are moving toward

one another, while the negative sign must be used when they are moving away from each other.

Thus, we anticipate an increase in the observed frequency if the source and observer are

approaching each other and a decrease if the source and observer recede from each other.


V Wave Theory

Introduction

Wave motion is a form of disturbance that travels through the medium due to the repeated

periodic motion of particles of the medium about their mean positions. This disturbance is

transferred from one particle to the next, e.g water waves, sound. This also involves the transfer

of energy from one point to the other.

Characteristics of a wave

1. It’s a disturbance produced in the medium by repeated periodic motion of the particles of

the medium.

2. The wave travels forward while particles of the medium vibrate about their mean

positions

3. there is a regular phase change between various particles of the medium

4. Velocity of the wave is different from the velocity with which the particles of the

medium are vibrating about their mean positions.

Types of wave motion

1. Transverse wave: particles of the medium vibrate about their mean positions in the

direction perpendicular to the direction of propagation, e.g. light waves, water waves

etc

2. Longitudinal: particles of the medium vibrate about their mean position in the

direction of the wave, e.g sound wave

1. General wave equation

Since wave motion involves vibration of particles about the mean position, they are also a SHM

type of wave motion. The displacement of a particle P in SHM is given by

tay sin

Suppose another particle Q is at a distance x from P and the wave is traveling with velocity v in

positive x direction (Fig below).

Then the displacement of the particle Q will be given by

)sin( tay (5.1)

Where φ is the phase difference between the particles P and Q. The phase difference

corresponding to the path difference (wavelength, λ) is 2π, which leads to the relation

x

2 (5.2)

Where x is the path difference between P & Q. Hence the phase difference φ between P and Q is

x2 and (5.3)

Angular frequency ω is

P

Q

x

a


v

Tf

222 (5.4)

Hence equation (5.1) becomes

)22

sin(

xt

vay (5.5)

Or

(5.6)

This is the general wave equation for a traveling wave in positive x direction with velocity v.

2. Differential form of general wave equation

The general wave equation can also be represented in differential form. Differentiating the

equation (5.3) w.r.t. t gives

)(2

cos2

xvtav

dt

dy

(5.7a)

Further differentiation of equation (a) w.r.t. t gives

)(2

sin2

2

2

2

xvtav

dt

yd

(5.8)

To get the compression of the wave (in x space), we differentiate equation (5.2) w.r.t. x, i.e.

)(2

cos2

xvta

dx

dy

(5.9)

Further differentiation of equation (c) w.r.t. x will give the compression in terms of distance (in x-

direction), i.e

)(2

sin2

2

2

2

xvta

dx

yd

(5.10)

Comparing equations (5.7 and (5.9) we see that

dt

dy

vdx

dy 1 (5.11)

And from (5.8) and (5.10) we have

(5.12)

Which represents the differential form of a wave equation.

Particle velocity of a wave

)(2

sin xvtay

2

2

22

2 1

dt

yd

vdx

yd


If the velocity of a particle is denoted by u then

)(2

cos2

xvtav

dt

dyu

(5.13)

And va

u

2max , which implies that

Maximum particle velocity is 2πa/λ times the wave velocity (v).

Likewise maximum particle acceleration is given by

av

f

2

max

2

(Show this)

Distribution of velocity & Pressure in a wave

If a wave is progressive (i.e, new waves are continuously formed) then there is a continuous

transfer of energy in the direction of propagation of the wave.

Remember particle velocity is given by

)(2

cos2

xvtav

dt

dyu

(5.15)

The strain in the medium is given by dy/dx, i.e, if it is positive, it represents a region of

rarefaction and when negative, represents region of compression. For such a medium, the

modulus of elasticity is given by

)/( dxdy

dP

strainvolume

pressureinchangeK

And

dx

dyKdP .

Implying that dP is positive in regions of compression and –ve in rarefaction region

y

T

T

T

u

dP

t

Po

C R C

2πav/λ


3. Interference

When two waves are moving along a straight line in a medium, then every particle of the medium

is simultaneously acted upon by both of the waves. If the two waves arrive at a point in phase

(two crests or two troughs) superimpose and the resultant amplitude is equal to the sum of the

respective waves (this is the principle of superposition). If the waves arrive at a point when they

are completely out of phase (a crest of one falls over the trough of another, then the resultant

amplitude is equal to the difference of the individual amplitudes. This implies that at points where

the two waves meet in phase will give maximum amplitude (hence maximum sound intensity)

while where they meet out of phase gives minimum amplitude (minimum sound intensity). The

phenomenon described above is called interference.

Fig. (a) constructive and (b) destructive interference

Let two waves having amplitudes a1 and a2 be represented by equations

)(2

sin

)(2

sin

22

11

xvtay

and

xvtay

(5.16)

Where φ is the phase difference between the two waves after some time. The resultant

displacement will be

sin)(2

coscos)(2

sin

sin)(2

coscos)(2

sin)(2

sin

)(2

sin)(2

sin

221

221

2121

axvtaaxvt

xvtaxvtaxvta

xvtaxvtayyY

(5.17)

(using the trig identity Sin (A + B) = Sin A cos B + sin B cos A)

Letting a1 + a2cos φ = A cos θ, a2sin φ = A sin θ and using trigonometric identities leads to

combined

Wave 1

Wave 1

(a) (b)


)(2

sin

)(2

cossin)(2

sincos

xvtAY

xvtAxvtAY

(5.18)

This shows that the resultant wave has the same frequency but a different amplitude and phase

from the component wave trains.

Special cases:

1. when φ = 0, A = a1 + a2 and hence tan θ = 0 showing that the resultant is in phase

with the component

2. when φ = 180o, A = a1 – a2

From 1 and 2 above, it implies that when the phase difference between two waves is zero the

two waves reinforce each other. This also implies that the resultant has the same period and

its amplitude is the sum of the amplitudes of the component waves. However, if the two

waves have a phase difference of 180o, they destroy each other and the resultant amplitude is

the difference between the amplitudes of the component waves.

Conditions for interference of waves

1. the two wave trains must move in the same direction.

2. The two sources must give waves of same frequency and amplitude so that the positions

of maxima and minima are distinct

3. the two sources must be in phase, i.e must be coherent source


VI Nature of Light

1. Reflection of Light

When light ray traveling in a medium encounters a boundary leading into a second medium, part

or the entire incident ray is reflected back into the first medium. The reflected rays are parallel to

each other. Reflection of light from a smooth surface is called specular reflection. If the surface is

rough the reflected rays are not parallel but set into various directions. Reflection from rough

surface is known as diffuse reflection.

Figure a shows specular reflection and figure b shows

diffuse reflection

Experiments show that the angle of reflection equals the

angle of incidence

'

1 1

By convection, the angles of incidence and

reflection are measured from the normal

to the surface rather than from the surface itself

2. Refraction of Light

When a ray of light traveling through a transparent medium

encounters a boundary leading into another transparent

medium, part of the ray is reflected and part enters the

second medium. The ray that enters the second medium is

bend at the boundary and is said to be refracted. Thje

incident ray, the reflected ray, the normal and the refracted

ray all lie in the same plane. The angle of refraction, 2

depends on the properties of the two media and on the

angle of incidence through the relationship

2 2

1 1

sinconstant

sin

v

v

(1.5)

Where 2, 1v v speed of light in media 2 and 1 respectively.

The equation 1.1 is called Snell’s law after Willebroad

Snell (1591-1627) In France this law is called Descartes Law after Rene Descartes who is

credited for the discovery (1596-1650). The path of alight ray through refracting surface is

reversible from point A B.

3. Index of Refraction

The speed of light in any material is less than the speed in vacuum except near very strong

absorption bands.

speed of light in vacuum

speed of light in a medium

cn

v (6.1)


From this definition, index of refraction is dimensionless number usually

greater than unity, because v < c.

As light travels from one medium to another, its frequency does not

change but its wavelength does. Therefore, because the relationship

v f must be valid in both media and because 1 2f f f we see that

1 1 2 2 and v f v f (6.2)

Since 1 2v v it follows that 1 2

1 1 1

2 2 2

/

/

v c n

v c n

(6.3)

Which gives 1 1 2 2n n . Snell’s law can also be written as

1 1 2 2sin sinn n

Example

A light of wavelength 589nm traveling through air is incident on a

smooth flat slab of crown glass at an angle of 30 to the normal, find

angle of refraction

Solution

12 1

2

sin sinn

n

1

2

1.001.52n

n

2

1.00sin (sin30 ) 0.329

1.52

1

2 sin (0.329) 19.2

4. Dispersion and Prism

For given materials the index of refraction

varies with the wavelength of light passing

through the material. The index of refraction

decreases with increasing wavelength. The

various wavelengths are refracted at different

wavelengths. This phenomenon is known as

dispersion.

5. Huygens Principle

Huygens principle is a construction for using knowledge of an earlier wavefront to determine the

position of a new wavefront at some instant. In Huygen’s construction, all points on a given

wavefront are taken as point sources for the production of spherical secondary waves called

wavelets, which propagate outward with speeds characteristic of waves in that medium. After

some time has elapsed, the new position of the wavefront is the surface tangent to the wavelets.


6. Total Internal Reflection

Suppose that a ray of light passes from a material of higher

index of refraction to one of lower index as shown in the

figure on the left.. Part of the incident ray is refracted and

part is reflected at the interface. Because 2 must be larger

than 1 , it is possible to make 1 large enough so

that 2 90 , given by ray 4. This value for 1 is called the

critical angle. For 1 larger than this, no refracted ray can

exist; all the light is reflected. The condition for total

internal reflection is that 1 exceed the critical angle c

given by 21 2

1

sin sin90 or sinc c

nn n

n . Because the

sine of an angle can never be larger than unity, this relation confirms that total internal reflection

can occur only if 1 2n n .

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