Chapter 1 Electromagnetic Introduction and Vector Analysis

You Kok Yeow

SEE 2523 Theory Electromagnetic

Brief Flow Chart for Electromagnetic Study

2. Revision on Vector

1. Introduction the Electromagnetic Study

Basic Law of Vector

Vector Multiplication

3. Orthogonal Coordinate Systems

Length, Area and Volume in Cartesian Coordinates

Length, Area and Volume in Cylindrical Coordinates

Length, Area and Volume in Spherical Coordinates

Contents

Electromagnetic Fields

Static Fields

(Time Independence)

Dynamic Fields

(Time Dependence)

Electrostatics Magnetostatics Electric Fields Magnetic Fields

Phenomenon

Electromagnetic Wave

Electromagnetics (EM) is a branch of physics or electrical engineering in which electric and magnetic phenomena are studied

Brief Flow Chart for Electromagnetic Study

Basis of Electromagnetic Laws

Description Study Chapter Critical Experiment

Gauss’s Law

(electric fields)

Charge and Electric

fields

Electrostatic /

Electric fields

Charge repel and attract different charges.

Comply with the inverse square law.

Charge on an insulated conductor moves

outward surface.

Gauss’s Law

(magnetic

fields)

Magnetic fields Magnetostatic

Magnetic fields

Confirmed that only exist magnetic dipole.

(naturedly no magnetic monopole)

Faraday’s Law The electrical effect

of a changing

magnetic fields

Magnetic

induction

The bar magnet is pushed through a closed

loop of wire will produce a current in the loop.

Ampere’s Law The magnetic effect

of a changing

electric fields

Magnetic fields The current in the wire produces a magnetic

field close to the wire.

1. A vector A

has a

a) magnitude AA

b) direction specified by a unit vector a

2. A vector A

Aa

AaA

ˆ

ˆ

3. The unit vector a is given by

A

A

A

Aa

ˆ

Basic Law of Vector (1)

Magnitude A

ˆA aA

a

1

Example

The vector A

may be represented as:

zyx AzAyAxA ˆˆˆ

The magnitude of A

222

zyx AAA

AA

The direction of A

222

ˆˆˆ

ˆ

zyx

zyx

AAA

AzAyAx

A

Aa

6ˆ2ˆ3ˆ zyxA

7

623 222

A

857.0ˆ286.0ˆ429.0ˆ

7

6ˆ2ˆ3ˆˆ

zyx

zyxa

3

2

6

Basic Law of Vector (2)

Basic vector

Components of vector A

A

1. There are two types of vector multiplication:

a) Scalar (or dot) product

b) Vector (or cross) product

BA

BA

2. The dot product of two vectors A

and B

is expressed as:

cosABBA

where is the smaller angle between A

and

3. If zyx AAAA ,,

and

zyx BBBB ,,

zzyyxx BABABABA

Vector Multiplication (1)

4. Two vectors A

and B

are said to be orthogonal (perpendicular)

with each other, if 0BA

5. The cross product of two vectors A

and B

is expressed as:

naABBA ˆsin

where na is a unit vector normal to the plane containing A

and B

6. If zyx AAAA ,,

and

zyx BBBB ,,

zxyyxyzxxzxyzzy

zyx

zyx

zyx

aBABAaBABAaBABA

BBB

AAA

aaa

BA

ˆˆˆ

ˆˆˆ

Vector Multiplication (2)

Direction of

field lines

Current

BA

The cross product

Magnetic field

Current

Motion of

force

Vector Multiplication (3)

A

B

ˆna

Z Z Z

XX

X

Y Y Yz

x

y

z

ρ ρ

θ

ØØ

P (x, y, z) P (ρ, Ø, z) P (ρ, Ø, θ)

Cartesian Cylindrical Spherical

cosx

siny

zz

cosx

siny

cosz

222 zyx

x

y1tan

z

yx 22

1tan

zz

Orthogonal Coordinate Systems

Z

X

Y0

dz

dxdy

D

G

A B

C

F

EH

dl

dzdydxdv

dydxzSd ABCDˆ

dzdyxSd HCDEˆ

dxdzySd FBDEˆ

dzzdyydxxld ˆˆˆ

2222 dzdydxdl

P point is expanded from ( x, y, z ) to ( x+dx, y+dy, z+dz ).

x

z

y

Length, Area and Volume in Cartesian Coordinates

0

X

Z

P point is expanded from (ρ, Ø, z ) to (ρ+dρ, Ø +dØ, z+dz ).

Y

dρ

ρ dØ

dz

Ø

ρ

z

CA

B

D

E

F

G

dl

dzdddv

dzdSd ADEF

ddzSd AFGBˆ

dzzddld ˆˆˆ

22222 dzdddl

dzdSd ABCD

Length, Area and Volume in Cylindrical

Coordinates

0

Z

X

YØ

θ

ρ dθ

dρρ sinθ dØ

ρ

P point is expanded from (ρ, θ, Ø) to (ρ+dρ, θ+dθ, Ø+dØ ).

AB

C

D

E

FG

H

ddddv sin2

ddSd ABCD sinˆ 2

ddSd CDEH sinˆ

ddSd BCHGˆ

dddld sinˆˆˆ

2222222 sin ddddl

dl

Length, Area and Volume in Spherical

Coordinates

The electromagnetic (static) problem cases

CoordinateThe problems most often encountered

Electrostatic Cases Magnetostatic Cases

Cartesian - Any uniform charge distribution using a

scale of cartesian coordinates.

- The current flowing on the horizontal

infinite extent plane.

Cylindrical - Uniform charge distribution on cylindrical

conductor.

- Uniform charge distribution on the infinite

length of line.

- Uniform charge distribution on the infinite

extent horizontal plane.

- Uniform charge distribution on the

horizontal circle plane.

- Uniform charge distribution on the circle

line.

- The current flowing in circumference.

- The current flowing in an infinite

straight line.

- The current flows in the solenoid and

toroid.

Spherical - Uniform charge distribution on the surface

of a sphere.

- Uniform charge distribution of the point.

- Problem for the case of spheres less

found.

References

J. A. Edminister. Schaum’s outline of Theory and problems of electromagnetics, 2nd

Ed. New York: McGraw-Hill. 1993.

M. N. O. Sadiku. Elements of Electromagnetics, 3th Ed. U.K: Oxford University Press.2010.

F. T. Ulaby. Fundamentals of Applied Electromagnetics, Media ed. New Jersey:Prentice Hall. 2001.

Bhag Singh Guru and Hüseyin R. Hiziroglu. Electromagnetic Field TheoryFundamentals, 2nd Ed. U.K.: Cambridge University Press. 2009.

W. H. Hayt. Jr. Engineering Electromagnetics, 5th Ed. New York: McGraw-Hill. 2009.