introduction to wave theory & propogation

Introduction to Wave Theory and

Propagation

Outline

• Introductory Concepts

• Vector Fields

• Co-ordinate Systems

Awab Sir (www.awabsir.com)

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Class Description

Prerequisites by Topic:

– University physics

– Complex numbers

– Partial differentiation

– Multiple Integrals

– Vector Analysis

– Fourier Series


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Why Study Electromagnetics?


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Examples of Electromagnetic

Applications


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Research Areas of

Electromagnetics

• Antenas

• Microwaves

• Computational Electromagnetics

• Electromagnetic Scattering

• Electromagnetic Propagation

• Radars

• Optics

• etc …


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Why is Electromagnetics

Difficult?


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What is Electromagnetics?


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What is a charge q?


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Fundamental Laws of

Electromagnetics


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Steps in Studying Electromagnetics


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SI (International System) of

Units


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Units Derived From the

Fundamental Units


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Fundamental Electromagnetic Field

Quantities


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Three Universal Constants


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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


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Example of a Scalar Field


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Scalar Fields

e.g. Temperature: Every location has

associated value (number with units) Awab Sir (www.awabsir.com)

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Scalar Fields - Contours

• Colors represent surface temperature

• Contour lines show constant temperatures Awab Sir (www.awabsir.com)

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Vector Fields Vector (magnitude, direction) at every point

in space

Example: Velocity vector field - jet stream Awab Sir (www.awabsir.com)

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Examples of Vector Fields


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VECTOR REPRESENTATION

3 PRIMARY COORDINATE SYSTEMS:

• RECTANGULAR

• CYLINDRICAL

• SPHERICAL

Choice is based on

symmetry of problem

Examples:

Sheets - RECTANGULAR

Wires/Cables - CYLINDRICAL

Spheres - SPHERICAL


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Orthogonal Coordinate Systems: (coordinates mutually perpendicular)

Spherical Coordinates

Cylindrical Coordinates

Cartesian Coordinates

P (x,y,z)

P (r, Θ, Φ)

P (r, Θ, z)

x

y

z P(x,y,z)

θ

z

r x

y

z

P(r, θ, z)

θ

Φ

r

z

y x

P(r, θ, Φ)

Page 108

Rectangular Coordinates


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P(x,y,z)

Spherical Coordinates

P(r, θ, Φ)

Cylindrical Coordinates

P(r, θ, z)

x

y

z P(x,y,z)

θ

z

r x

y

z

P(r, θ, z)

θ

Φ

r

z

y x

P(r, θ, Φ)


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VECTOR NOTATION

VECTOR NOTATION:

zzyyxx aAaAaAA ˆˆˆ Rectangular or

Cartesian

Coordinate

System

x

z

y

zzyyxx BABABABA

Dot Product

zyx

zyx

zyx

BBB

AAA

aaa

BA

ˆˆˆ

Cross Product

2

1222

zyx AAAA

Magnitude of vector

(SCALAR)

(VECTOR)


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VECTOR REPRESENTATION: CYLINDRICAL COORDINATES

Cylindrical representation uses: r ,f , z

zzrr aAaAaAA ˆˆˆ ff

zzrr BABABABA ff

UNIT VECTORS:

zr aaa ˆˆˆ

f

Dot Product (SCALAR)

r

f

z

P

x

z

y


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VECTOR REPRESENTATION: SPHERICAL COORDINATES

r

f

P

x

z

y

q

Spherical representation uses: r ,q , f UNIT VECTORS:

fq aaarˆˆˆ

ffqq aAaAaAA rrˆˆˆ

ffqq BABABABA rr

Dot Product (SCALAR)


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x

z

y

VECTOR REPRESENTATION: UNIT VECTORS

yaxa

za Unit Vector

Representation

for 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


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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


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Spherical Coordinate System

r

f

P

x

z

y

q

qa

fa

ra


Points in the direction of increasing r

Points in the direction of increasing j

Points in the direction of increasing q

ra

fa

qa


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zr aaa ˆˆˆ

f fq aaarˆˆˆ

zyx aaa ˆˆˆ

RECTANGULAR

Coordinate

Systems

CYLINDRICAL

Coordinate

Systems

SPHERICAL

Coordinate

Systems

NOTE THE ORDER!

r,f, z r,q ,f


Summary


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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”


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Differential quantities:

Length:

Area:

Volume:

dzzdyydxxld ˆˆˆ

dxdyzsd

dxdzysd

dydzxsd

z

y

x

ˆ

ˆ

ˆ

dxdydzdv

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METRIC COEFFICIENTS

2. Cylindrical Coordinates:

Distance = r df

x

y

df

r

Differential Distances:

( dr, rdf, dz )


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3. Spherical Coordinates:

Distance = r sinq df

x

y

df

r sinq

Differential Distances:

( dr, rdq, r sinq df )

r

f

P

x

z

y

q

METRIC COEFFICIENTS


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introduction to wave theory & propogation

Education

force awab sir

z z r x y z pr

scalar field awab sir

z r z y x pr

z x y z px

fundamental units awab

unitsawab sir

unit vectors summary