emtl1

7/27/2019 emtL1

1/80

EEE241:

Fundamentals of

ElectromagneticsIntroductory Concepts, Vector

Fields and Coordinate Systems

Instructor: Dragica Vasileska

7/27/2019 emtL1

2/80

Outline

Class Description

Introductory Concepts

Vector Fields Coordinate Systems

7/27/2019 emtL1

3/80

Class Description

Prerequisites by Topic:

University physics

Complex numbers

Partial differentiation

Multiple Integrals

Vector Analysis

Fourier Series

7/27/2019 emtL1

4/80

Class Description

Prerequisites: EEE 202; MAT 267, 274 (or275), MAT 272; PHY 131, 132

Computer Usage: Students are assumed to be

versed in the use MathCAD or MATLAB toperform scientific computing such as numericalcalculations, plotting of functions and performingintegrations. Students will develop and visualizesolutions to moderately complicated fieldproblems using these tools.

Textbook: Cheng, Field and WaveElectromagnetics.

7/27/2019 emtL1

5/80

Class Description

Grading:

Midterm #1 25%

Midterm #2 25%

Final 25%

Homework 25%

7/27/2019 emtL1

6/80

Class Description

7/27/2019 emtL1

7/80

Why Study Electromagnetics?

7/27/2019 emtL1

8/80

Examples of Electromagnetic

Applications

7/27/2019 emtL1

9/80


Applications, Contd

7/27/2019 emtL1

10/80


Applications, Contd

7/27/2019 emtL1

11/80


Applications, Contd

7/27/2019 emtL1

12/80


Applications, Contd

7/27/2019 emtL1

13/80

Research Areas of

Electromagnetics

Antenas

Microwaves

Computational Electromagnetics

Electromagnetic Scattering

Electromagnetic Propagation

Radars

Optics

etc

7/27/2019 emtL1

14/80

Why is Electromagnetics

Difficult?

7/27/2019 emtL1

15/80

What is Electromagnetics?

7/27/2019 emtL1

16/80

What is a charge q?

7/27/2019 emtL1

17/80

Fundamental Laws of

Electromagnetics

7/27/2019 emtL1

18/80

Steps in Studying Electromagnetics

7/27/2019 emtL1

19/80

SI (International System) of

Units

7/27/2019 emtL1

20/80

Units Derived From the

Fundamental Units

7/27/2019 emtL1

21/80

Fundamental Electromagnetic Field

Quantities

7/27/2019 emtL1

22/80

Three Universal Constants

7/27/2019 emtL1

23/80

Fundamental Relationships

7/27/2019 emtL1

24/80

Scalar and Vector Fields

A scalar field is a function that gives us

a single value of some variable for

every point in space. Examples: voltage, current, energy,

temperature

A vector is a quantity which has both a

magnitude and a direction in space. Examples: velocity, momentum, acceleration

and force

7/27/2019 emtL1

25/80

Example of a Scalar Field

7/27/2019 emtL1

26/80

26

Scalar Fields

e.g. Temperature: Every location has

associated value (number with units)

7/27/2019 emtL1

27/80

27

Scalar Fields - Contours

Colors represent surface temperature

Contour lines show constant temperatures

7/27/2019 emtL1

28/80

28

Fields are 3D

T = T(x,y,z)

Hard to visualizeWork in 2D

7/27/2019 emtL1

29/80

29

Vector FieldsVector (magnitude, direction) at every point

in space

Example: Velocity vector field - jet stream

7/27/2019 emtL1

30/80

Vector Fields Explained

7/27/2019 emtL1

31/80

Examples of Vector Fields

7/27/2019 emtL1

32/80


7/27/2019 emtL1

33/80


7/27/2019 emtL1

34/80

VECTOR REPRESENTATION

3 PRIMARY COORDINATE SYSTEMS:

RECTANGULAR

CYLINDRICAL

SPHERICAL

Choice is based on

symmetry of problem

Examples:

Sheets - RECTANGULAR

Wires/Cables - CYLINDRICAL

Spheres - SPHERICAL

7/27/2019 emtL1

35/80

Orthogonal Coordinate Systems: (coordinates mutually perpendicular)

Spherical Coordinates

Cylindrical Coordinates

Cartesian Coordinates

P (x,y,z)

P (r, , )

P (r, , z)

x

y

zP(x,y,z)

z

rx

y

z

P(r, , z)

r

z

yx

P(r, , )

Page 108

Rectangular Coordinates

7/27/2019 emtL1

36/80

-Parabolic Cylindrical Coordinates (u,v,z)

-Paraboloidal Coordinates (u, v, )-Elliptic Cylindrical Coordinates (u, v, z)

-Prolate Spheroidal Coordinates (, , )

-Oblate Spheroidal Coordinates (, , )-Bipolar Coordinates (u,v,z)

-Toroidal Coordinates (u, v, )

-Conical Coordinates (, , )

-Confocal Ellipsoidal Coordinate (, , )

-Confocal Paraboloidal Coordinate (, , )

7/27/2019 emtL1

37/80

Parabolic Cylindrical Coordinates

7/27/2019 emtL1

38/80

Paraboloidal Coordinates

7/27/2019 emtL1

39/80

Elliptic Cylindrical Coordinates

7/27/2019 emtL1

40/80

Prolate Spheroidal Coordinates

7/27/2019 emtL1

41/80

Oblate Spheroidal Coordinates

7/27/2019 emtL1

42/80

Bipolar Coordinates

7/27/2019 emtL1

43/80

Toroidal Coordinates

7/27/2019 emtL1

44/80

Conical Coordinates

7/27/2019 emtL1

45/80

Confocal Ellipsoidal Coordinate

7/27/2019 emtL1

46/80

Confocal Paraboloidal Coordinate

7/27/2019 emtL1

47/80

Cartesian CoordinatesP(x,y,z)


P(r, , )


P(r, , z)

x

y

z

P(x,y,z)

z

rx

y

z

P(r, , z)

r

z

yx

P(r, , )

7/27/2019 emtL1

48/80

Coordinate Transformation

Cartesian to Cylindrical

(x, y, z) to (r,,)

(r,,) to (x, y, z)

7/27/2019 emtL1

49/80

Cartesian to CylindricalVectoral Transformation


7/27/2019 emtL1

50/80


Cartesian to Spherical

(x, y, z) to (r,,)

(r,,) to (x, y, z)

7/27/2019 emtL1

51/80

Cartesian to Spherical

Vectoral Transformation


V t R t ti

7/27/2019 emtL1

52/80

Page 109

x

y

z

Z plane

x plane

xyz

x1

y1

z1

AxAy

Unit vector properties

0

1

xzzyyx

zzyyxx

yxz

xzyzyx

Vector Representation

Unit (Base) vectors

A unit vector aA along A is a vector

whose magnitude is unity

A

Aa

V t R t ti

7/27/2019 emtL1

53/80

zyx AzAyAxA

Page 109

x

y

z

Z plane

x plane

222

zyxAAAAAA

xyz

x1

y1

z1

AxAy

Az

Vector representation

Magnitude of A

Position vector A

),,( 111 zyxA

111 zzyyxx

Vector Representation

7/27/2019 emtL1

54/80

x

y

z

Ax Ay

AzA

B

Dot product:

zzyyxx BABABABA

Cross product:

zyx

zyx

BBB

AAA

zyx

BA

Back


Page 108

7/27/2019 emtL1

55/80

Multiplication of vectors

Two different interactions (whats the

difference?)

Scalar or dot product :

the calculation giving the work done by a force during a

displacement

work and hence energy are scalar quantities which arise

from the multiplication of two vectors

if AB = 0 The vector A is zero

The vector B is zero

= 90

ABBABA cos||||

A

B

7/27/2019 emtL1

56/80

Vector or cross product :

n is the unit vector along the normal to the plane

containing A and B and its positive direction isdetermined as the right-hand screw rule

the magnitude of the vector product ofA and B is

equal to the area of the parallelogram formed byA

and B

if there is a force Facting at a point Pwith position

vectorrrelative to an origin O, the moment of a force

Fabout O is defined by :

if A x B = 0

The vector A is zero

The vector B is zero

= 0

nsin|||| BABA

A

B

ABBA

FrL

Commutative law :

7/27/2019 emtL1

57/80

Commutative law :

ABBA

ABBA

Distribution law :

CABACBA )(

CABACBA )(

Associative law :))(( DCBADBCA

CBABCA )(

CBACBA )(

CBACBA )()(

7/27/2019 emtL1

58/80

Unit vector relationships

It is frequently useful to resolve vectors into components

along the axial directions in terms of the unit vectors i,j,

and k.

1

0

kkjjii

ikkjji

jik

ikj

kjikkjjii

0

zyx

zyx

zzyyxx

zyx

zyx

BBB

AAA

kji

BA

BABABABA

kBjBiBB

kAjAiAA

S l t i l d t CBA

7/27/2019 emtL1

59/80

Scalar triple product CBA

The magnitude of is the volume of the parallelepiped with edges parallel to

A, B, and C.

CBA

A

BC

A B

],,[ CBABACACBACBCBACBA

V t t i l d t CBA

7/27/2019 emtL1

60/80

Vector triple product CBA

The vector is perpendicular to the plane of A and B. When the further vector

product with C is taken, the resulting vector must be perpendicular to and

hence in the plane of A and B :

BA

A

BC

A B

BA

nBmACBA )( where m and n are scalar constants to be determined.

0)( BnCAmCCBAC ACn

BCm

BACABCCBA )()()(

Since this equation is valid

for any vectors A, B, and C

Let A = i, B = C =j:

1

CBABCACBA

ACBBCACBA

)()()(

)()()(

7/27/2019 emtL1

61/80

x

z

y

VECTOR REPRESENTATION: UNIT VECTORS

yaxa

za

Unit Vector

Representationfor Rectangular

Coordinate

System

xa

The Unit Vectors imply :

ya

za

Points in the direction of increasing x

Points in the direction of increasing y

Points in the direction of increasing z

Rectangular Coordinate System


7/27/2019 emtL1

62/80

r

f

z

P

x

z

y


Cylindrical Coordinate System

za

fa

ra


za

Points in the direction of increasing r

Points in the direction of increasing j

Points in the direction of increasing z

ra

fa


7/27/2019 emtL1

63/80

Base

Vectors

A1

radial distance in x-y plane

azimuth angle measured from the positive

x-axis

Z

r0

20

z

y

,

,

z

z

z

zAzAAAaA

Pages 109-112Back

( , , z)


222

zAAAAAA

Magnitude of A

Position vector A

Base vector properties

11 zz


7/27/2019 emtL1

64/80

Dot product:

zzrr BABABABA ff

Cross product:

zr

zr

BBB

AAA

zr

BA

f

f

f

BA

Back

y

Pages 109-111


7/27/2019 emtL1

65/80


Spherical Coordinate System

r

f

P

x

z

y

a

fa

ra


Points in the direction of increasing r

Points in the direction of increasing j

Points in the direction of increasing ra

fa

a


7/27/2019 emtL1

66/80

,, RRR


Pages 113-115Back

(R, , )

f f AAARA R


222

f AAAAAA R

Magnitude of A

Position vector A

1RR

Base vector properties


7/27/2019 emtL1

67/80

Dot product:

ff BABABABA RR

Cross product:

f

f

f

BBB

AAA

R

BA

R

R

Back

BA

Pages 113-114


7/27/2019 emtL1

68/80

zr aaa

f f aaar

zyx aaa

RECTANGULAR

Coordinate

Systems

CYLINDRICAL

Coordinate

Systems

SPHERICAL

Coordinate

Systems

NOTE THE ORDER!

r,f, z r, ,f

Note: We do not emphasize transformations between coordinate systems


Summary

METRIC COEFFICIENTS

7/27/2019 emtL1

69/80

METRIC COEFFICIENTS

1. Rectangular Coordinates:

When you move a small amount in x-direction, the distance is dx

In a similar fashion, you generate dy and dz

Unit is in meters


7/27/2019 emtL1

70/80


Differential quantities:

Differential distance:

Differential surface:

Differential Volume:

dzzdyydxxld

dxdyzsd

dxdzysd

dydzxsd

z

y

x

dxdydzdv

Page 109

C li d i l C di t

7/27/2019 emtL1

71/80

Cylindrical Coordinates:

Distance = r df

x

y

dfr

Differential Distances:

( dr, rdf, dz )

C li d i l C di t

7/27/2019 emtL1

72/80

Cylindrical Coordinates:

Differential Distances: ( d, rdf, dz )zadzadadld ff

zz addsd

adzdsd

adzdsd

f

f

ff

Differential Surfaces:

Differential Volume:

Spherical Coordinates:

7/27/2019 emtL1

73/80

Spherical Coordinates:

Distance = r sin df

x

y

dfr sin

Differential Distances:

( dr, rd, r sin df )

r

f

P

x

z

y


dRdl

7/27/2019 emtL1

74/80

Differential quantities:

Length:

Area:

Volume:

dRRddRR

dldldlRld R

sin

RdRddldlsd

dRdRdldlsd

ddRRdldlRsd

R

R

R

sin

sin 2

ddRdRdv sin2

dRdl

Rddl

dRdlR

sin

Pages 113-115Back

METRIC COEFFICIENTS

7/27/2019 emtL1

75/80

Representation of differential length dl in coordinate systems:

zyx adzadyadxld

zr adzadradrld ff

f f adrardadrld r sin

rectangular

cylindrical

spherical

METRIC COEFFICIENTS

7/27/2019 emtL1

76/80

Example

For the object on the rightcalculate:

(a) The distance BC

(b) The distance CD

(c) The surface areaABCD

(d) The surface area ABO

(e) The surface areaA OFD

(f) The volumeABDCFO

AREA INTEGRALS

7/27/2019 emtL1

77/80

AREA INTEGRALS

integration over 2 delta distances

dx

dy

Example:

x

y

2

6

3 7

AREA = 7

3

6

2

dxdy = 16

Note that: z = con stant

In this course, area & surface integrals will be

on similar types of surfaces e.g. r =constantorf = constant or = constant et c.

SURFACE NORMAL

7/27/2019 emtL1

78/80

Representation of differential surface element:

zadydxsd

Vector is NORMAL

to su r face

SURFACE NORMAL

DIFFERENTIALS FOR INTEGRALS

7/27/2019 emtL1

79/80

DIFFERENTIALS FOR INTEGRALS

Example of Line differentials

or or

Example of Surface differentials

zadydxsd

radzrdsd f

or

Example of Volume differentials dzdydxdv

xadxld

radrld

ff ardld

Cartesian to Cylindrical Transformation

7/27/2019 emtL1

80/80

zz

yx

yxr

cossin

sincos

fff

ff

zz

yx

yxr

AA

AAAAAA

ffff

f cossinsincos

zz

xy

yxr

)/(tan 1

22

f

emtl1

Documents