1. EEE241: Fundamentals of Electromagnetics
Introductory Concepts, Vector Fields and Coordinate Systems
Instructor: Dragica Vasileska

2. Outline Class Description Introductory Concepts Vector Fields
Coordinate Systems

3. Class Description Prerequisites by Topic: University physics
Complex numbers
Partial differentiation
Multiple Integrals
Vector Analysis
Fourier Series

4. 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.

5. Class Description Grading: Midterm #1 25% Midterm #2 25% Final 25%
Homework 25%

6. Class Description

7. Why Study Electromagnetics?

8. Examples of Electromagnetic Applications

9. Examples of Electromagnetic Applications, Cont’d

10. Examples of Electromagnetic Applications, Cont’d

11. Examples of Electromagnetic Applications, Cont’d

12. Examples of Electromagnetic Applications, Cont’d

13. Research Areas of Electromagnetics
Antenas
Microwaves
Computational Electromagnetics
Electromagnetic Scattering
Electromagnetic Propagation
Radars
Optics
etc …

14. Why is Electromagnetics Difficult?

15. What is Electromagnetics?

16. What is a charge q?

17. Fundamental Laws of Electromagnetics

18. Steps in Studying Electromagnetics

19. SI (International System) of Units

20. Units Derived From the Fundamental Units

21. Fundamental Electromagnetic Field Quantities

22. Three Universal Constants

23. Fundamental Relationships

24. 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

25. Example of a Scalar Field

26. Scalar Fields
Week 01, Day 1
e.g. Temperature: Every location has associated value (number with units) 26
Class 01 26

27. Scalar Fields - Contours
Week 01, Day 1
Colors represent surface temperature
Contour lines show constant temperatures 27
Class 01 27

28. Fields are 3D T = T(x,y,z) Hard to visualize  Work in 2D 28
Week 01, Day 1
T = T(x,y,z)
Hard to visualize  Work in 2D 28
Class 01 28

29. Vector Fields Vector (magnitude, direction) at every point in space
Week 01, Day 1
Vector Fields
Vector (magnitude, direction) at every point in space
Example: Velocity vector field - jet stream 29
Class 01 29

30. Vector Fields Explained
Week 01, Day 1
Vector Fields Explained
Class 01 30

31. Examples of Vector Fields

32. Examples of Vector Fields

33. Examples of Vector Fields

34. VECTOR REPRESENTATION
3 PRIMARY COORDINATE SYSTEMS:
Choice is based on symmetry of problem
RECTANGULAR
CYLINDRICAL
SPHERICAL
Examples:
Sheets - RECTANGULAR
Wires/Cables - CYLINDRICAL
Spheres - SPHERICAL

35. Orthogonal Coordinate Systems: (coordinates mutually perpendicular)
Cartesian Coordinates z
P(x,y,z) y
Rectangular Coordinates
P (x,y,z) x z z
P(r, θ, z)
Cylindrical Coordinates
P (r, Θ, z) r y x θ z
Spherical Coordinates
P(r, θ, Φ) θ r
P (r, Θ, Φ) y x Φ
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36. 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)

37. Parabolic Cylindrical Coordinates

38. Paraboloidal Coordinates

39. Elliptic Cylindrical Coordinates

40. Prolate Spheroidal Coordinates

41. Oblate Spheroidal Coordinates

42. Bipolar Coordinates

43. Toroidal Coordinates

44. Conical Coordinates

45. Confocal Ellipsoidal Coordinate

46. Confocal Paraboloidal Coordinate

47. Cartesian Coordinates Cylindrical Coordinates Spherical Coordinatesz z
Cartesian Coordinates
P(x,y,z)
P(x,y,z)
P(r, θ, Φ) θ r y y x x Φ
Cylindrical Coordinates
P(r, θ, z)
Spherical Coordinates
P(r, θ, Φ) z z
P(r, θ, z) r y x θ

48. Coordinate Transformation
Cartesian to Cylindrical
(x, y, z) to (r,θ,Φ)
(r,θ,Φ) to (x, y, z)

49. Coordinate Transformation
Cartesian to Cylindrical
Vectoral Transformation

50. Coordinate Transformation
Cartesian to Spherical
(x, y, z) to (r,θ,Φ)
(r,θ,Φ) to (x, y, z)

51. Coordinate Transformation
Cartesian to Spherical
Vectoral Transformation

52. Vector Representationz z1
Z plane
Unit (Base) vectors
x plane
A unit vector aA along A is a vector whose magnitude is unity
y plane y1 y Ay Ax x1 x
Unit vector properties
Page 109

53. Vector Representationz
Vector representation z1
Z plane
Magnitude of A
x plane
y plane Az y1 y Ay Ax
Position vector A x1 x
Page 109

54. Cartesian Coordinatesz
Dot product: Az y
Cross product: Ay Ax x
Back
Page 108

55. 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° A  B

56. 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° A  B

57. Commutative law :
Distribution law :
Associative law :

58. 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.

59. Scalar triple product
The magnitude of is the volume of the parallelepiped with edges parallel to
A, B, and C.
AB C B A

60. Vector triple product
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 :
where m and n are scalar constants to be determined. A B C
AB
Since this equation is valid
for any vectors A, B, and C
Let A = i, B = C = j:

61. VECTOR REPRESENTATION: UNIT VECTORS
Rectangular Coordinate System x z y
Unit Vector Representation for Rectangular Coordinate System
The Unit Vectors imply :
Points in the direction of increasing x
Points in the direction of increasing y
Points in the direction of increasing z

62. VECTOR REPRESENTATION: UNIT VECTORS
Cylindrical Coordinate System r f z P x y
The Unit Vectors imply :
Points in the direction of increasing r
Points in the direction of increasing j
Points in the direction of increasing z

63. Cylindrical Coordinates ( ρ, Φ, z)
ρ radial distance in x-y plane
Φ azimuth angle measured from the positive
x-axis Z A1
Vector representation
Base
Vectors
Magnitude of A
Base vector properties
Position vector A
Back
Pages

64. Cylindrical Coordinates
Dot product: A B
Cross product:
Back
Pages

65. VECTOR REPRESENTATION: UNIT VECTORS
Spherical Coordinate System r f P x z y q
The Unit Vectors imply :
Points in the direction of increasing r
Points in the direction of increasing q
Points in the direction of increasing j

66. Spherical Coordinates
Vector representation
(R, θ, Φ)
Magnitude of A
Position vector A
Base vector properties
Back
Pages

67. Spherical Coordinates
Dot product: A B
Cross product:
Back
Pages

68. VECTOR REPRESENTATION: UNIT VECTORS
Summary
RECTANGULAR Coordinate Systems
CYLINDRICAL Coordinate Systems
SPHERICAL Coordinate Systems
NOTE THE ORDER!
r,f, z
r,q ,f
Note: We do not emphasize transformations between coordinate systems

69. METRIC COEFFICIENTS 1. Rectangular Coordinates:
Unit is in “meters”
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

70. Cartesian Coordinates
Differential quantities:
Differential distance:
Differential surface:
Differential Volume:
Page 109

71. Cylindrical Coordinates:
Differential Distances: x y df r
Distance = r df
( dr, rdf, dz )

72. Cylindrical Coordinates:
Differential Distances:
( dρ, rdf, dz )
Differential Surfaces:
Differential Volume:

73. Spherical Coordinates:
Differential Distances: x y df
r sinq
Distance = r sinq df
( dr, rdq, r sinq df ) r f P x z y q

74. Spherical Coordinates
Differential quantities:
Length:
Area:
Volume:
Back
Pages

75. METRIC COEFFICIENTS
Representation of differential length dl in coordinate systems:
rectangular
cylindrical
spherical

76. 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

77. AREA INTEGRALS integration over 2 “delta” distances Example: AREA =dx dy
Example: x y 2 6 3 7
AREA =
= 16
Note that: z = constant
In this course, area & surface integrals will be on similar types of surfaces e.g. r =constant or f = constant or q = constant et c….

78. Vector is NORMAL to surface
SURFACE NORMAL
Representation of differential surface element:
Vector is NORMAL to surface

79. DIFFERENTIALS FOR INTEGRALS
Example of Line differentials or or
Example of Surface differentials or
Example of Volume differentials

80. Cartesian to Cylindrical Transformation
Back
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