Introduction to Wave Theory and Propagation: Vector Fields and Co-ordinate Systems

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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21
Scalar Fields
e.g. Temperature: Every location has
associated value (number with units)Awab Sir (www.awabsir.com)
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22
Scalar Fields - Contours
• Colors represent surface temperature
• Contour lines show constant temperaturesAwab Sir (www.awabsir.com)
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23
Vector Fields
Vector (magnitude, direction) at every point
in space
Example: Velocity vector field - jet streamAwab 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
1
222
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 aaar
ˆˆˆ
ffqq aAaAaAA rr
ˆˆˆ 

ffqq BABABABA rr 
 Dot Product
(SCALAR)
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x
z
y
VECTOR REPRESENTATION: UNIT VECTORS
yaˆxaˆ
zaˆ Unit Vector
Representation
for Rectangular
Coordinate
System
xaˆ
The Unit Vectors imply :
yaˆ
zaˆ
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
zaˆ
faˆ
raˆ
zaˆ
Points in the direction of increasing r
Points in the direction of increasing j
Points in the direction of increasing z
raˆ
faˆ
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Spherical Coordinate System
r
f
P
x
z
y
q
qaˆ
faˆ
raˆ
Points in the direction of increasing r
Points in the direction of increasing j
Points in the direction of increasing q
raˆ
faˆ
qaˆ
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 zr aaa ˆˆˆ f  fq aaar
ˆˆˆ 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 
Page 109
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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 and Propagation: Vector Fields and Co-ordinate Systems

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