EEEB253_chap1 to chap9_20140911

ELECTROMAGNETIC

FIELDS & WAVES

EEEB 253

CHAPTER 1

BEEE/BEPE

College of Engineering

2Electromagnetic spectrum

Ker Pin Jern Universiti Tenaga Nasional, College of Engineering, Dept. of Electrical Power

3Chapter 1: Vector algebra

1. Scalar and vector

2. Unit vector

3. Position vector

4. Vector multiplication

Electromagnetics (EM) A branch of physics or electrical engineering where

electric and magnetic phenomena are studied

Applications:- microwave, antenna, electric machine, radar, remote sensing etc.

Eg. EM energy change vegetable taste by reducing its acidicity


41.1: Scalar and vector

EM concepts can be understood through vector analysis

Need to have strong vector analysis

Scalar quantity that only has magnitude

Vector quantity that has both magnitude and direction, identified

by an arrow above the symbols (or bold in typing)

EM theory study of a particular field

Field is a function that specifies a particular quantity everywhere in

a region. Eg. Gravitational field


51.2: Unit vector

Magnitude of a vector A is A or |A|

A unit vector aA, is a vector with magnitude of unity and direction A

Vector A can be written as or

Magnitude of a vector

Unit vector is given by

Vector addition / subtraction (revision)


6Examples / practice exercises

PE 1.3 (position vector)

Answers / solution will be done during lecture



Example (unit vector)

What is the magnitude of unit vector ac???


81.3: Position vector

Position vector (rp) also called the radius vector of a point P is the distance directed from the origin to point P

Point (3, 4, 5) has a position vector

Distance vector displacement from one point to

another point

Prac. Exercise 1.1


91.4: Vector multiplication

2 types of vector multiplication

Dot product

2 vectors are orthogonal or perpendicular if

Dot product commutative law

Dot product Distributive law

Dot product perpendicular vector

Dot product parallel vector

Cross product , an is the unit vector normal to the plane containing vectors A and B


101.4: Vector multiplication

Cross product anticommutative

Not associative

Distributive

Special cases

Example 1.4 find the angle using dot product / cross product


ELECTROMAGNETIC

FIELDS & WAVES

EEEB 253

CHAPTER 2

BEEE/BEPE


12Chapter 2: Coordinate system

1. Cartesian coordinate

2. Cylindrical coordinate / transformation

3. Spherical coordinate / transformation

Physical quantities in EM are functions of time and space

To define all points accurately, use coordinate system

Orthogonal coordinate system the ones in which the coordinates are mutually

perpendicular. Eg. Cartesian, spherical, cylindrical, conical etc.


13

2.1: Cartesian coordinate / 2.2 Circular cylindrical coordinate

A point P can be represented by (x, y, z)

Cartesian coordinate = rectangular coordinate

Vector A

Cylindrical coordinate convenient for problems

with cylindrical symmetry

is the radial distance from the z-axis; is called the azimuthalangle measured from the x-axis on the xy-plane

z is the same as in Cartesian coordinate

Vector A can be written as

Magnitude of vector; Unit vectors are all perpendicular


142.2 Circular cylindrical coordinate

Relationship between cylindrical and Cartesian coordinates

Relationship between and

In matrix form

The unit vectors are related as below


152.3 Spherical coordinate

Most appropriate when dealing with problems

having spherical symmetry

A point can be represented by

r is the distance from the origin to the point

is the angle between z-axis and position

vector P (Colatitude angle)

is measured from the x-axis (same as Azimuthal angle)

Vector A can be expressed as and

The magnitude is

The unit vectors are all mutually orthogonal


162.3 Spherical coordinate

Relationship between spherical and Cartesian coordinates

The unit vectors are related as follows:-

Components of vector

The magnitude MUST stay the same

after transformation to check answer



Example 2.1

At point P,

Exclude Spherical!



PE 2.1



PE 2.2


ELECTROMAGNETIC

FIELDS & WAVES

EEEB 253

CHAPTER 3

BEEE/BEPE


22Chapter 3: Vector calculus

1. Line integral

2. Surface and volume integral

3. Del, Divergence, Curl and Laplacian

Deals with integration and differentiation of vectors (Calculus)

Learning the mathematical techniques in this chapter, which will be useful for the

EM applications in subsequent chapters

Differential normal area


233.1: Line integral

Integration concept extension to an integrand which is a vector

Vector field A; Line integral is the integral of the tangential component of A along curve L

We can define the integral as

For a closed path abca, closed contour integral

Example 3.2


24PE 3.2


253.2: Surface and volume integral

Vector A continuous in a region containing a smooth surface S

Surface integral or flux of A through Sor

At any point on S, an is the unit vector normal to the surface S

For a closed surface that defining a volume,

The volume integral , scalar over volume v

Problem 3.2 (d)


26

3.3: Del, Divergence, Curl, and Lapacian

The Del operator , also known as gradient operator

This operator is useful in defining:

in Cartesian coordinate

In cylindrical coordinate, using transformation

In spherical coordinate,

Divergence of A at any given point P, is the outward flux per unit volume as the volume shrinks about P

Net outflow of flux of a vector field A from a closed surface is obtained

from the surface integral


27


Divergence of A can be written as , is the volume

enclosed by the closed surface where the P is located

Imagine divergence as a measure of how much the fields diverge or emanate from a point P

+ve at a source point diverge

-ve at a sink point converge

Neither sink nor source Zero

Divergence of A at point P in a Cartesian system

In cylindrical coordinate

In spherical coordinate


28


Properties of the divergence of a vector field:-

It is a scalar product; Divergence of a scalar makes no sense;

;

Divergence theorem:- the total outward flux of a vector field A through a closed surface S is the same as the volume integral of the divergence

of A

Example 3.6


29PE 3.7


30

3.3: Del, Divergence, Curl, and Lapacian Curl of vector A is an axial or rotational vector whose magnitude is

the maximum circulation of A per unit area as the area tends to zero and

whose direction is the normal direction of the area when the area is

oriented so as to make the circulation maximum

is the area bounded by curve L, is the unit vector normal to the

surface , determined by the right hand rule

is independent on the coordinate system

Cartesian

Cylindrical


31

3.3: Del, Divergence, Curl, and Lapacian Properties of the curl:-

Stokes theorem

Example 3.8 Determine the curl vectors


32

3.3: Del, Divergence, Curl, and Lapacian Example 3.8

Example 3.9


33



34

3.3: Del, Divergence, Curl, and Lapacian Laplacian operator is the composite of the gradient and divergence

Definition:-

Laplacian of a scalar field is another scalar field

In Cartesian:-

In cylindrical:-

In spherical:-

A scalar field vector V is said to be harmonic in a given region if its

Laplacian vanishes in that region. i.e.

Laplacian of a vector NOT the divergence of the gradient of A but the gradient of the divergence of A

In Cartesian only, it becomes


35



36


P.E. 3.11


37Chapter 3 Conclusion

Line integral, surface and volume integrals

Del operator used in divergence, curl and Laplacian

Divergence of a vector results in a scalar quantity; total outward

flux through a closed surface (Dot)

Curl of a vector results in a rotational vector (Cross)

Laplacian of a scalar field results in another scalar; divergence of

a gradient of a vector

Tutorial questions: 3.1(a), 3.2(a, b), 3.3(a, b), 3.4, 3.10(a, b), 3.15, 3.16(a,

b), 3.17(a, b), 3.19, 3.24(a, b), 3.29(a, b), 3.31


ELECTROMAGNETIC

FIELDS & WAVES

EEEB 253

CHAPTER 4

BEEE/BEPE


39Chapter 4: Electrostatic fields


1. Coulombs law

2. Electric fields

3. Electric flux density

4. Gausss law

5. Electric potential

6. Electric energy

Static electric field time invariant; E field produced by static charge distribution

E field application Touch pad, capacitive keyboard, LCD

Industry E field used for paint spraying, electrodeposition, measure moisture

content of crops, speed baking of bread and smoking of meat

Coulombs law & Gausss law both for calculating electric field

404.1: Coulombs law

Coulombs law:- deals with the force exerted by a point charge on

another point charge

The force F between two point charges Q1 and Q2 is:-

Along the line joining them

Directly proportional to the product Q1Q2 of the charges

Inversely proportional to the square of the distance R between them

With a proportionality constant, it can be expressed as

Coulombs constant, k, expressed in terms of the permittivity of free space (in Farad per meter)

F12 force exerted on Q2 due to Q1, if Q1 and Q2 are located at position vectors r1 and r2 respectively.


414.1: Coulombs law

Using the formula for unit vector, or

A few points to note:-

Opposing force with same magnitude

Like charges repel, opposite charges attract

Distance R is much larger than the size of the charges i.e. point charges

All the charges must be in static i.e. not moving

Signs of the charges must be taken into account

For cases with more than two charges,

Electric field intensity (electric field strength), E force per unit charge when placed in the electric field

E is in the direction of F in N/C or V/m

For E due to many chargesKer Pin Jern Universiti Tenaga Nasional, College of Engineering, Dept. of Electrical Power

424.1: Coulombs law

Example 4.1

P.E. 4.1


43

4.2: Electric fields due to continuous charge distribution

More realistic cases charges distributed along a line, surface or

volume with line , surface and volume charge densities

The charge element dQ and the total charge Q

The electric field intensity = sum of the field contributed by the point

charges making up the charge distribution


444.3: Electric flux density

Electric field intensity is dependent on the medium (free space)

Introduce a new vector field D that is independent on medium

Electric flux , measured in SI unit Coulomb

Vector field D called the electric flux density; SI unit coulomb/m2

Can obtain D from E. Eg. For infinite sheet of charge

For a volume charge distribution,

Example 4.7


454.3: Electric flux density

P.E. 4.7


464.4: Gausss law

Gausss law states that the total electric flux through any closed

surface is equal to the total charge enclosed by the surface. i.e.

Using the divergence theorem, , therefore

1st of the 4 Maxwells equations the volume charge density is equal to

the divergence of the electric flux density

Gausss Law (GL) is an alternative statement of Coulombs Law

(CL);proper application of divergence theorem to CL results in GL

GL provides an easy way to determine E and D for symmetrical charge

distribution


474.4: Gausss law

Total net flux leaving surface v1 = 5 nC

Charges 20 nC and 15 nC do not affect

because net = 0; in = out

For surface v2, net flux = 0

Procedures to apply Gausss law to for calculation

Determine whether symmetric charge distribution exists

A closed surface (called the Gaussian surface) is chosen such that D is normal or tangential to the surface

For a point charge Q, choose a spherical surface

D is everywhere normal to the surface


484.4: Gausss law

For an infinite line charge, choose a cylindrical surf.

to satisfy the symmetry condition

GL:-

For top and bottom surface ZERO;

For an infinite sheet of charge, choose a rectangular

box that is cut symmetrically by the sheet

GL:-

If the top and bottom have an area A,


494.4: Gausss law

For a uniformly charged sphere, spherical surf.

For r a,

GL:- For r => a,


504.4: Gausss law

Try eg.4.8; P.E. 4.8

P.E. 4.9 (XX Need)


514.5: Electric potential

Recap:- To calculate E, can use CL for all cases; GL only suits highly

symmetric charge distribution

Another way use electric scalar potential V; easier to handle scalar than vector quantity

Case Moving a point charge Q from point A to B in an electric field E . From CL, , work done

-ve sign: indicates work done by external agent

Potential energy per unit charge potential difference between point A and B ; A is the initial point, B is the final point

VAB ve:- loss in PE when moving Q from A to B, work done by field

VAB +ve:- gain in PE when moving Q from A to B, external agent performs the work



VAB is independent of the path taken

VAB is measured in Joules/C, commonly called as volt (V)

Equations:-

Always determine electric potential relative to another potential

Calculation for point charge relative to infinity

Using superposition principle

Note:- if electric field is known, use



Example 4.10

P.E. 4.10



Try xample 4.11

P.E. 4.11


554.6: Electric energy

To calculate the energy present in an assembly of charges, need to first

determine the amount of work necessary to assemble them

Wish to move Q1,2,3 to the shaded region

Q1 no work required, E field = 0,no charge

Putting the charges in reverse order

Hence

; Generally,


564.6: Electric energy

P.E. 4.14

Try to calculate the total work done using the formula

Compare the answers!


ELECTROMAGNETIC

FIELDS & WAVES

EEEB 253

CHAPTER 5

BEEE/BEPE


58

Chapter 5: Electric fields in materials


1. Properties of materials

2. Conductors

3. Dielectrics

4. Continuity equation

5. Boundary conditions

Previous chapter consider electric field in free space

Electric field in other mediums / materials most of the equations are similar as in

last chapter with little modification

Materials classified based on electrical properties conductor and non-conductor

(insulator or dielectric)

595.1: Properties of material

Materials are categorized based on conductivity in the unit of mhos/m

or Siemens/m (S/m)

Conductivity is usually dependent on the temperature and frequency

High conductivity conductor / metal; low conductivity

insulator; in between semiconductor

Conductivity increases with decreasing temperature

Superconductor extremely high / infinite conductivity at very low

temperature (0-4 Kelvin) what degree celcius??


605.2: Conductors

Conductors many freely moving charges

An isolated conductor (top figure); external Efield is

applied+ve charges are pushed along Efield and -vecharges move in opposite direction (charge migration

happens very quickly)

2 things are done by the free charges:-

They accumulate on the surface of the conductor (induced

surface charges)

Induced charges set up an internal induced field

A conductor is an equipotential body potential is the

same everywhere


615.2: Conductors

According to GL, if E=0, then charge density=0

Under static condition,

What happens when the ends / terminals of a conductor are

maintained at a certain potential difference?

E=0??the conductor is no longer isolated but wired to a source of electromotive force (battery)

Disrupt the electrostatic equilibrium by forcing free charges to move

There is an Efield to have current flows ; Efield, +ve charge,

current have the same direction; electrons flow in opposite direction

Electrons movement is opposed by a damping force Resistance


625.2: Conductors

To obtain the resistance, assume the conductor has

a cross section area, S; current density,

Ohms law, ;

Resistivity of material,

For conductor with non-uniform cross section,

Using power and energy equation,

P.E. 5.3


635.3: Dielectrics

Charges in dielectric are bounded but can displace if a sufficiently

large external force is applied

When an Efield is applied, +ve charge is displaced in the direction of

Efield and ve charge is displaced in the opposite direction of Efield

Dipole separation of +ve and ve charges; Dipole is created the dielectric is said to be polarized

In a polarized state, the electrons are distorted; Distorted charge

distribution = original distribution + dipole moment ( ), d is the

distance vector from Q to +Q

Total dipole moment


645.3: Dielectrics

Calculate the field due to a polarized dielectric

Net effect of the dielectric on the electric field is to increase D by an

amount P. Polarization P will vary with E, usually as

is the electric susceptibility of material a measure of how

susceptible a dielectric is to electric fields

Dielectric constant

Permittivity of dielectric vs. permittivity of free space

Dielectric constant / relative permittivity - ratio of to ; may

change at high frequencies > 1 GHz


655.3: Dielectrics

Try eg. 5.7; In class: P.E. 5.7


665.4: Continuity equation

Principle of charge conservation time rate of decrease of charge =

net outward current flow through the closed surface of a volume

is the total charge enclosed by the surface

Using Divergence theorem, ; Also,

Hence, Current continuity equation

For steady current, or i.e. total charge entering a

volume = total charge leaving the volume

Consider the effect of introducing charge at some interior point using

Ohms law and Gausss law

Relaxation time or rearrangement time = - the time it takes a

charge placed in the interior of a material to drop to e-1 = 36.8% of its initial value


675.4: Continuity equation

Good conductor short Tr; Good dielectric long Tr

For copper, , what is Tr?

For fused quartz,


685.5: Boundary conditions

We considered Efield only in one medium

When it involves more than one medium, the conditions that the field

must satisfy at the interface separating the medium is called the

Boundary Conditions

Consider 3 cases 1) dielectric 1 and dielectric 2; 2) Conductor and

dielectric; 3) Conductor and free space

Use Maxwells equations:-

Decompose electric field intensity into 2 orthogonal components

tangential and normal components of E


ELECTROMAGNETIC

FIELDS & WAVES

EEEB 253

CHAPTER 6

BEEE/BEPE


70

Chapter 6: Electrostatic boundary value problems


1. Poissons equation

2. Laplaces equation

3. Procedures for solving Poissons

and Laplaces equations

4. Capacitance

Previous chapters charge distribution is known, use GL or CL to determine Efield

If the potential difference is known, use

Practical electrostatic problems:- only electrostatic conditions (Charge or potential

difference) at some boundaries are known need to find E and V throughout the

region Boundary value problems

Need to use Poissons or Laplaces equations derived from GL

71

6.1: Poissons equation // 6.2 Laplaces equation

Poissons equation

Special case for a charge-free region (Laplaces eqn)

Hence, the Laplaces equations in different coordinate systems are:-

Poissons equations in different coordinate systems are obtained by

replacing 0 with

Laplaces equations:- useful to solve electrostatic problems involving

a set of conductors maintained at different potentials. Eg. Capacitors,

vacuum tube diodes


72

6.3: Procedures for solving Poissons and Laplaces equations

Followings are the general procedures:-


736.4: Capacitance

Capacitor must have 2 or more conductors carrying equal but

opposite charge; separated by free space or dielectric

Capacitance ratio of the magnitude of the charge on one of the plate

to the potential difference between the plates

Unit Farad (F); indicate how much charge can be stored

2 methods to calculate capacitance of 2 conductors

1) choose a suitable coordinate; 2) Plates carrying charge +Q and Q;

3) Calculate E using CL or GL, calculate ; 4)


746.4: Capacitance

Parallel plate capacitor

Assume carrying charge +Q and Q; hence

Using Laplaces eqn. (Example 6.11)


756.4: Capacitance

Using Laplaces eqn. (Example 6.11)


766.4: Capacitance

Coaxial capacitor coaxial cable Length, L; inner radius, a; outer radius, b Using GL for a < < b, ,

Capacitance of a coaxial cable Using Laplaces eqn. (Example 6.8 with full cylinder) Coaxial = full cylinder (inner and outer surfaces)


776.4: Capacitance

Using Laplaces eqn. (Example 6.8 with full cylinder)

Q = s x surface area = s x 2piL C = Q/Vo = 2piL / ln (b/a)


786.5: Resistance

Resistance

1) Choose a suitable coordinate system; 2)Assume Vo as the potential difference between the conductor terminals; 3) Solve Laplaces

equation to obtain V, then determine E and I from ; 4)

Obtain R from V0/I

Example 6.8


796.5: Resistance


806.5: Resistance


816.5: Resistance

Example 6.9


ELECTROMAGNETIC

FIELDS & WAVES

EEEB 253

CHAPTER 7

BEEE/BEPE


83Chapter 7: Magnetostatic fields


1. Biot-Savarts law

2. Amperes circuit law

3. Magnetic scalar and vector potentials

Previous chapters concentrate on the electric field (E and D); this chapter study the static magnetic field (B and H) vs.

A few similarities between magnetic and electric fields Most of the equations derived for Efield can be readily used for Mfield

This chapter considers Mfield in free space due to the direct current

Development of motors, transformers, microphones, compasses, etc.

Biot-Savarts law CL and Amperes law GL

847.1: Biot-Savarts law

Biot-Savarts law states that the magnetic field intensity dH produced at a point P, by the differential current, I dl, is proportional to the product I dl and the sine of the angle betweenthe element and the line joining P to the element and is inversely proportional to the square of the distance R between P and the element

Use right-hand rule to determine the direction of dH Represent the directions of I and H with cross and dot Efield different charge distribution; Mfield different

current distribution



Example of line current straight current carrying conductor with length AB.

,

Hence,

This expression is applicable to any straight line. For an infinitely long current carrying wire, point A (0, 0, -infinity), B(0, 0, infinity)

P.E. 7.1



How to determine each parameter?? Angles (1, 2) follow the flow of current starting of current is 1

and end of current is 2 a UNIT VECTOR of the flow of current a UNIT VECTOR pointing from the filament to the point a = a x a



Example 7.2


897.2: Amperes circuit law

Amperes circuit law states that the line integral of the tangential component of H around a closed path is the same as the net current Ienc enclosed by the path comparison

A special case of the Biot-Savarts law AL can be used to obtain Honly when symmetrical current distribution exists

Stokes theorem , also, Therefore, 3rd Maxwells equation Infinite line current to determine H at point P Assume an Amperian path through P


90

7.3: Magnetic scalar and vector potential

Relate electric field intensity to potential Magnetic flux density B is related to magnetic field intensity H Permeability of freespace Magnetic scalar potential Vm (in amperes), Vector magnetic potential, A (in Wb/m) Example 7.7

Try to solve using Stokes theoremKer Pin Jern Universiti Tenaga Nasional, College of Engineering, Dept. of Electrical Power

91

7.3: Magnetic scalar and vector potential

Example 7.7 Try to solve using Stokes theorem


ELECTROMAGNETIC

FIELDS & WAVES

EEEB 253

CHAPTER 8

BEEE/BEPE


93

Chapter 8: Magnetic forces and magnetic materials


1. Force due to magnetic fields

2. Magnetic torque and moment

3. Magnetization

4. Inductances and inductors

5. Magnetic energy

Study the force exerted by the magnetic field on a charged particle, current element

and loop

Consider the magnetic field in different material media

948.1: Force due to magnetic field

3 ways a force can be experienced in a magnetic field:- 1) moving charged particle in a B field; 2) current element in an external Bfield; 3) between two current element

Magnetic force experienced by a charge Q moving with a velocity uin a magnetic field B ; Fm perpendicular to u and B

Fm cannot perform work as it is normal to the velocity For a moving charge in E and B fields Lorentz force

equation; With a mass m, Force on a current element current = flow of many charges for a closed path L, Note:- the B field produced by the current element does not exert

magnetic force on itselfKer Pin Jern Universiti Tenaga Nasional, College of Engineering, Dept. of Electrical Power

958.1: Force due to magnetic field

Example 8.3


968.2: Magnetic torque and moment

Very important to understand the concept of a current loop experiencing a torque in a magnetic field in order to understand d.c. motors and generators

The torque on a loop is the vector product of the force and the moment arm ,r , unit N.m

A rectangular loop with length l and width w Under a uniform B field Along sides 12 and 34: dl parallel to B; Fm = 0 Therefore, , but both forces acting at different point If the normal to the plane makes an angle with B,

or Ker Pin Jern Universiti Tenaga Nasional, College of Engineering, Dept. of Electrical Power

978.2: Magnetic torque and moment

The magnetic dipole moment:- is the product of the current and area of the loop in the direction normal to the loop

Hence, from When do we get maximum torque? When will it be minimum? Example 8.5


988.3: Magnetization

Magnetization, M in amperes/meter (A/m) is the magnetic dipole moment per unit volume

Without an external B field, the sum of magnetic moments is zero due to random orientation; when Bfield is applied, the magnetic moments of the electrons align themselves with B

A material is said to be magnetized if M is not zero Bound volume current density or magnetization volume

current density Bound surface current density , is the unit

vector normal to the surface , M depends linearly on H such that

where is the magnetic susceptibility - dimensionless Ker Pin Jern Universiti Tenaga Nasional, College of Engineering, Dept. of Electrical Power

998.3: Magnetization

Hence,

is the permeability of the material in Henrys/m (H/m) is the relative permeability


1008.4: Inductances and inductors

Circuit carrying current I magnetic field B causes flux With N identical turns, flux linkages Flux linkages is proportional to the current L is the proportionality constant, called Inductance Circuit that has inductance is called inductor Inductance=ratio of magnetic flux linkage to current Unit Henry (H): 1H = 1Wb/A Usually called self-inductance since the flux linkages are produced

by the circuit itself Capacitance measure of how much electric energy stored;

Inductance measure of how much magnetic energy storedKer Pin Jern Universiti Tenaga Nasional, College of Engineering, Dept. of Electrical Power

1018.4: Inductances and inductors

Magnetic energy stored in an inductor or Typical examples of inductor toroids, solenoids, coaxial

transmission line; parallel-wire transmission line To find the self-inductance:-

In an inductor such as coaxial or parallel wire transmission lines, inductance produced by the flux internal to the conductor is called the internal inductance; inductance produced by flux external to it is called the external inductance


1028.5: Magnetic energy

To express the magnetic energy in terms of B and H

Example 8.11


1038.5: Magnetic energy

Example 8.11 What about L/length??


ELECTROMAGNETIC

FIELDS & WAVES

EEEB 253

CHAPTER 9

BEEE/BEPE


105Chapter 9: Maxwells equation


1. Faradays law

2. Transformer emf

3. Motional emf

Stationary charge electrostatic fields; Steady currents magnetostatic fields

Time-varying current Electromagnetic fields (and waves)

2 major concepts electromotive force based on Faradays experiment;

displacement current resulted from Maxwells hypothesis

Maxwells equations:- summarize the laws of electromagnetism

1069.1: Faradays law

1831, Faraday and Henry discovered that a time-varying magnetic field would produce an electric current

Static field no current; time-varying field produce an induced voltage (called the electromotive force, emf)

Faradays law:-

Lenzs law: the ve signthe induced emf acts in such a way that opposes the flux producing it; or the induced magnetic field by the induced current will oppose the original field


1079.2: Transformer emf

Consider a single turn (N=1) circuit, In terms of E and B, , flux is replaced by

where S is the surface area enclosed by the closed path L 3 ways to cause the variation of flux with time:-

Stationary loop in a time-varying B field Time-varying loop area in a static B field Time-varying loop area in a time-varying B field

1st case: , emf is induced by a time-varying current (producing a time-varying B field) Transformer emf

Applying Stokes theorem, Maxwells eqn. for time-varying field;


1089.3: Motional emf

2nd case: moving loop in static B field; Recall:-force exerted on a charge moving at a velocity in a B field

Define the motional electric field Emf induced called the motional emf Type of emf found in motors and generators Example: voltage is generated when the coil

rotates within the magnetic field Rod moving between a pair of rails Perpendicular,



Be careful when using

Example 9.1

Transformer emf



Continue Eg. 9.1 Motional emf

It is easier to use



Problem 9.6



Problem 9.11


EEEB253_chap1 to chap9_20140911

Documents

Transcript of EEEB253_chap1 to chap9_20140911