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EEE241: Fundamentals of Electromagnetics
Introductory Concepts, Vector Fields and Coordinate Systems
Instructor: Dragica Vasileska
Outline
• Class Description
• Introductory Concepts
• Vector Fields
• Coordinate Systems
Class Description
Prerequisites by Topic:– University physics– Complex numbers– Partial differentiation– Multiple Integrals– Vector Analysis– Fourier Series
Class Description
• Prerequisites: EEE 202; MAT 267, 274 (or 275), MAT 272; PHY 131, 132
• Computer Usage: Students are assumed to be versed in the use MathCAD or MATLAB to perform scientific computing such as numerical calculations, plotting of functions and performing integrations. Students will develop and visualize solutions to moderately complicated field problems using these tools.
• Textbook: Cheng, Field and Wave Electromagnetics.
Class Description
• Grading:
Midterm #1 25%
Midterm #2 25%Final 25%
Homework 25%
Class Description
Why Study Electromagnetics?
Examples of Electromagnetic Applications
Examples of Electromagnetic Applications, Cont’d
Examples of Electromagnetic Applications, Cont’d
Examples of Electromagnetic Applications, Cont’d
Examples of Electromagnetic Applications, Cont’d
Research Areas of Electromagnetics
• Antenas• Microwaves• Computational Electromagnetics• Electromagnetic Scattering• Electromagnetic Propagation• Radars• Optics• etc …
Why is Electromagnetics Difficult?
What is Electromagnetics?
What is a charge q?
Fundamental Laws of Electromagnetics
Steps in Studying Electromagnetics
SI (International System) of Units
Units Derived From the Fundamental Units
Fundamental Electromagnetic Field Quantities
Three Universal Constants
Fundamental Relationships
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
Example of a Scalar Field
26
Scalar Fields
e.g. Temperature: Every location has associated value (number with units)
27
Scalar Fields - Contours
• Colors represent surface temperature• Contour lines show constant
temperatures
28
Fields are 3D
•T = T(x,y,z)
•Hard to visualize Work in 2D
29
Vector FieldsVector (magnitude, direction) at every point
in space
Example: Velocity vector field - jet stream
Vector Fields Explained
Examples of Vector Fields
Examples of Vector Fields
Examples of Vector Fields
VECTOR REPRESENTATION
3 PRIMARY COORDINATE SYSTEMS:
• RECTANGULAR
• CYLINDRICAL
• SPHERICAL
Choice is based on symmetry of problem
Examples:
Sheets - RECTANGULAR
Wires/Cables - CYLINDRICAL
Spheres - SPHERICAL
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
zP(x,y,z)
θ
z
rx y
z
P(r, θ, z)
θ
Φ
r
z
yx
P(r, θ, Φ)
Page 108
Rectangular Coordinates
-Parabolic Cylindrical Coordinates (u,v,z)-Paraboloidal Coordinates (u, v, Φ)-Elliptic Cylindrical Coordinates (u, v, z)-Prolate Spheroidal Coordinates (ξ, η, φ)-Oblate Spheroidal Coordinates (ξ, η, φ)-Bipolar Coordinates (u,v,z)-Toroidal Coordinates (u, v, Φ)-Conical Coordinates (λ, μ, ν)-Confocal Ellipsoidal Coordinate (λ, μ, ν)-Confocal Paraboloidal Coordinate (λ, μ, ν)
Parabolic Cylindrical Coordinates
Paraboloidal Coordinates
Elliptic Cylindrical Coordinates
Prolate Spheroidal Coordinates
Oblate Spheroidal Coordinates
Bipolar Coordinates
Toroidal Coordinates
Conical Coordinates
Confocal Ellipsoidal Coordinate
Confocal Paraboloidal Coordinate
Cartesian CoordinatesP(x,y,z)
Spherical CoordinatesP(r, θ, Φ)
Cylindrical CoordinatesP(r, θ, z)
x
y
zP(x,y,z)
θ
z
rx y
z
P(r, θ, z)
θ
Φ
r
z
yx
P(r, θ, Φ)
Coordinate Transformation
• Cartesian to Cylindrical(x, y, z) to (r,θ,Φ)
(r,θ,Φ) to (x, y, z)
• Cartesian to CylindricalVectoral Transformation
Coordinate Transformation
Coordinate Transformation
• Cartesian to Spherical(x, y, z) to (r,θ,Φ)
(r,θ,Φ) to (x, y, z)
• Cartesian to Spherical Vectoral Transformation
Coordinate Transformation
Page 109
x
y
z
Z plane
y planex plane
xyz
x1
y1
z1
Ax
Ay
Unit vector properties
0ˆˆˆˆˆˆ
1ˆˆˆˆˆˆ
xzzyyx
zzyyxx
yxz
xzy
zyx
ˆˆˆ
ˆˆˆ
ˆˆˆ
Vector Representation
Unit (Base) vectors
A unit vector aA along A is a vector whose magnitude is unity
A
Aa
zyx AzAyAxA ˆˆˆ
Page 109
x
y
z
Z plane
y planex plane
222zyx AAAAAA
xyz
x1
y1
z1
Ax
Ay
Az
Vector representation
Magnitude of A
Position vector A
),,( 111 zyxA
111 ˆˆˆ zzyyxx
Vector Representation
x
y
z
Ax
Ay
AzA
B
Dot product:
zzyyxx BABABABA
Cross product:
zyx
zyx
BBB
AAA
zyx
BA
ˆˆˆ
Back
Cartesian Coordinates
Page 108
Multiplication of vectors
• Two different interactions (what’s 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 A·B = 0– The vector A is zero
– The vector B is zero = 90°
ABBABA cos||||
A
B
– Vector or cross product :
• n is the unit vector along the normal to the plane containing A and B and its positive direction is determined as the right-hand screw rule
• the magnitude of the vector product of A and B is equal to the area of the parallelogram formed by A and B
• if there is a force F acting at a point P with position vector r relative to an origin O, the moment of a force F about 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 :
ABBA
ABBA
Distribution law :
CABACBA )(
CABACBA )(
Associative law :
))(( DCBADBCA
CBABCA )(
CBACBA )(
CBACBA )()(
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
kji
kkjjii
0
zyx
zyx
zzyyxx
zyx
zyx
BBB
AAA
kji
BA
BABABABA
kBjBiBB
kAjAiAA
Scalar triple product CBA
The magnitude of is the volume of the parallelepiped with edges parallel to A, B, and C.
CBA
A
BC
AB
],,[ CBABACACBACBCBACBA
Vector triple product CBA
The vector is perpendicular to the plane of A and B. When the further vectorproduct with C is taken, the resulting vector must be perpendicular to and hence in the plane of A and B :
BA
A
BC
AB
BA
nBmACBA )( where m and n are scalar constants to be determined.
0)( BnCAmCCBACACn
BCm
BACABCCBA )()()( Since this equation is validfor any vectors A, B, and CLet A = i, B = C = j:
1
CBABCACBA
ACBBCACBA
)()()(
)()()(
x
z
y
VECTOR REPRESENTATION: UNIT VECTORS
yaxa
zaUnit Vector
Representation for Rectangular
Coordinate System
xaThe 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
r
z
P
x
z
y
VECTOR REPRESENTATION: UNIT VECTORS
Cylindrical Coordinate System
za
a
ra
The Unit Vectors imply :
za
Points in the direction of increasing r
Points in the direction of increasing
Points in the direction of increasing z
ra
a
BaseVectors
A1
ρ radial distance in x-y plane
Φ azimuth angle measured from the positive x-axis
Z
r0
20
z
Cylindrical Coordinates
ˆˆˆ
,ˆˆˆ
,ˆˆˆ
z
z
z
zAzAAAaA ˆˆˆˆ
Pages 109-112Back
( ρ, Φ, z)
Vector representation
222zAAAAAA
Magnitude of A
Position vector A
Base vector properties
11 ˆˆ zz
Dot product:
zzrr BABABABA
Cross product:
zr
zr
BBB
AAA
zr
BA
ˆˆˆ
B A
Back
Cylindrical Coordinates
Pages 109-111
VECTOR REPRESENTATION: UNIT VECTORS
Spherical Coordinate System
r
P
x
z
y
a
a
ra
The Unit Vectors imply :
Points in the direction of increasing r
Points in the direction of increasing
Points in the direction of increasing
ra
aa
ˆˆˆ,ˆˆˆ,ˆˆˆ RRR
Spherical Coordinates
Pages 113-115Back
(R, θ, Φ)
AAARA Rˆˆˆ
Vector representation
222 AAAAAA R
Magnitude of A
Position vector A
1ˆRR
Base vector properties
Dot product:
BABABABA RR
Cross product:
BBB
AAA
R
BA
R
R
ˆˆˆ
Back
B A
Spherical Coordinates
Pages 113-114
zr aaa ˆˆˆ aaar ˆˆˆ zyx aaa ˆˆˆ
RECTANGULAR Coordinate Systems
CYLINDRICAL Coordinate Systems
SPHERICAL Coordinate Systems
NOTE THE ORDER!
r,, z r,,
Note: We do not emphasize transformations between coordinate systems
VECTOR REPRESENTATION: UNIT VECTORS
Summary
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”
Cartesian Coordinates
Differential quantities:
Differential distance:
Differential surface:
Differential Volume:
dzzdyydxxld ˆˆˆ
dxdyzsd
dxdzysd
dydzxsd
z
y
x
ˆ
ˆ
ˆ
dxdydzdv
Page 109
Cylindrical Coordinates:
Distance = r d
x
y
d
r
Differential Distances:
( dr, rd, dz )
Cylindrical Coordinates:
Differential Distances: ( dρ, rd, dz )
zadzadadld ˆˆˆ
zz addsd
adzdsd
adzdsd
ˆ
ˆ
ˆ
Differential Surfaces:
Differential Volume:
Spherical Coordinates:
Distance = r sin d
x
y
d
r sin
Differential Distances:
( dr, rd, r sind )
r
P
x
z
y
Spherical Coordinates
Differential quantities:
Length:
Area:
Volume:
dRRddRR
dldldlRld R
sinˆˆˆ
ˆˆˆ
RdRddldlsd
dRdRdldlsd
ddRRdldlRsd
R
R
R
ˆˆ
sinˆˆ
sinˆˆ 2
ddRdRdv sin2
dRdl
Rddl
dRdlR
sin
Pages 113-115Back
Representation of differential length dl in coordinate systems:
zyx adzadyadxld ˆˆˆ
zr adzadradrld ˆˆˆ
adrardadrld r ˆsinˆˆ
rectangular
cylindrical
spherical
METRIC COEFFICIENTS
Example
• For the object on the right calculate:
• (a) The distance BC• (b) The distance CD• (c) The surface area ABCD• (d) The surface area ABO• (e) The surface area A OFD• (f) The volume ABDCFO
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 = constant
In this course, area & surface integrals will be on similar types of surfaces e.g. r =constant or = constant or = constant et c….
Representation of differential surface element:
zadydxsd ˆ
Vector is NORMAL to surface
SURFACE NORMAL
DIFFERENTIALS FOR INTEGRALS
Example of Line differentials
or or
Example of Surface differentials
zadydxsd ˆradzrdsd ˆ
or
Example of Volume differentials dzdydxdv
xadxld ˆ
radrld ˆ
ardld ˆ
zz
yx
yxr
ˆˆ
cosˆsinˆˆ
sinˆcosˆˆ
zz
yx
yxr
AA
AAA
AAA
cossin
sincos
Back
Cartesian to Cylindrical Transformation
zz
xy
yxr
)/(tan 1
22
Page 115