This document provides an introduction to wave theory and propagation. It outlines topics that will be covered, including introductory concepts, vector fields, and coordinate systems. Prerequisites for the class are also listed. The document discusses why electromagnetics is studied and gives examples of electromagnetic applications. Research areas in electromagnetics are identified. Fundamental concepts such as scalar and vector fields, units of measurement, and coordinate systems are explained.
19. 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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20. Example of a Scalar Field
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22. 22
Scalar Fields - Contours
• Colors represent surface temperature
• Contour lines show constant temperaturesAwab Sir (www.awabsir.com)
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23. 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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26. 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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27. 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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29. 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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30. 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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31. 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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32. 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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33. r
f
z
P
x
z
y
VECTOR REPRESENTATION: UNIT VECTORS
Cylindrical Coordinate System
zaˆ
faˆ
raˆ
The Unit Vectors imply :
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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34. VECTOR REPRESENTATION: UNIT VECTORS
Spherical Coordinate System
r
f
P
x
z
y
q
qaˆ
faˆ
raˆ
The Unit Vectors imply :
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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35. 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
VECTOR REPRESENTATION: UNIT VECTORS
Summary
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36. 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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38. METRIC COEFFICIENTS
2. Cylindrical Coordinates:
Distance = r df
x
y
df
r
Differential Distances:
( dr, rdf, dz )
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39. 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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