1. EEE 161 Applied Electromagnetics
Dr. Milica Markovic
9/17/2018
Dr. Milica Markovic, EEE 161 Applied Electromagnetics

2. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
Chapter 1
Vectors
9/17/2018
Dr. Milica Markovic, EEE 161 Applied Electromagnetics

3. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
Scalars and Vectors
Scalars quantities are defined by magnitude only:
Temperature 75 deg. F
Mass 75kg
Vectors are defined by magnitude and direction:
Wind speed 75m/h in NW direction
9/17/2018
Dr. Milica Markovic, EEE 161 Applied Electromagnetics

4. Point in Cartesian Coordinate System
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

5. Unit Vectors in Cartesian Coordinates
X-direction
Y-direction
Z-direction
Unit vectors have magnitude of 1!
9/17/2018
Dr. Milica Markovic, EEE 161 Applied Electromagnetics

6. Position Vector in Cartesian Coordinates
Unit Vectors
Components
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

7. Example of Position Vector in Cartesian Coordinates
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

8. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
9/17/2018
Dr. Milica Markovic, EEE 161 Applied Electromagnetics

9. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
More on Vectors
Magnitude – length of the vector
Direction – Unit vector in the direction of vector A
Magnitude = 1
9/17/2018
Dr. Milica Markovic, EEE 161 Applied Electromagnetics

10. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
Addition of Vectors
Head to Tail Rule
Parallelogram Rule
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

11. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
Negative Vector
Negative Sign Changes Direction!
9/17/2018
Dr. Milica Markovic, EEE 161 Applied Electromagnetics

12. Subtraction of Vectors
First we change direction of vector B
Then we add A and –B up!
9/17/2018
Dr. Milica Markovic, EEE 161 Applied Electromagnetics

13. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
Distance Vector
Can be represented by two position vectors ,
Coordinates of points B and E
9/17/2018
Dr. Milica Markovic, EEE 161 Applied Electromagnetics

14. Distance Vector Magnitude and Unit Vector
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

15. Vector Multiplication
Scalar or Dot Product
Vector or Cross Product
Scalar Triple Product
Vector Triple Product
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

16. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
Scalar Product
Theta is the smaller angle between two vectors
Projection of vector B in the direction of vector A (the green line)
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

17. Scalar Product in Cartesian Coordinate System
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

18. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
5-min Practice
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

19. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
Vector Product
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

20. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

21. Vector Product in Cartesian Coordinate System
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

22. Properties of Cross Product
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

23. Direction of Vector Product
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

24. Coordinate Systems and vector calculus
Chapters 2 and 3
Coordinate Systems and vector calculus
9/17/2018
Dr. Milica Markovic, EEE 161 Applied Electromagnetics

25. Cartesian Coordinates
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

26. Differential Length - Cart Coord
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

27. Differential Surface – Cart Coord
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

28. Differential Volume – Cart Coord
Volume is base times height
9/17/2018
Dr. Milica Markovic, EEE 161 Applied Electromagnetics

29. Position Vector in Cylindrical Coordinates
Three coordinates r, θ and z.
Θ= 60deg
Position vector in Cylindrical Coordinates has only r and z directions!
9/17/2018
Dr. Milica Markovic, EEE 161 Applied Electromagnetics

30. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

31. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

32. Differential Length –Cyl Coord
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

33. Differential Surface – Cyl Coord
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

34. Differential Volume Cyl Coord
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

35. Magnitude Transformation Relations Cyl Coord – Cart Coord
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

36. Unit Vectors Transformation Relations Cyl-Cart
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

37. Cylindrical-Cartesian Coordinates
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

38. Position Vector in Spherical Coordinates
Three coordinates r, θ and Φ.
Position vector in Cylindrical Coordinates is only in the R direction!
9/17/2018
Dr. Milica Markovic, EEE 161 Applied Electromagnetics

39. Differential Length – Spherical Coord.
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

40. Differential Surface – Spher. Coord
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

41. Differential Volume- Spher Coord
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

42. Distance Between Two Points
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

43. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
Line Integral
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

44. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
Surface Integral
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

45. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
Volume Integral
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

46. Representation of Vector Fields
Vector Fields are usually represented by arrows.
The stronger the field at a point the longer the vector at the point.
2. The stronger the field in an area the higher the density of vectors in that area. All vectors have the same magnitude.
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

47. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
A Del Operator
Del operator is used to define
Gradient
Divergence
Laplacian
Curl.
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

48. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

49. Gradient of a Scalar Field
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

50. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

51. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

52. Directional Derivative
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

53. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
Flux of a vector
Weak
Strong
Number of vector lines “flowing” through a surface
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

54. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
Divergence of a Vector
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

55. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
Divergence Theorem
Volume integral thorough of divergence over a volume ~ this is usually easier to find.
Flux through a closed surface
9/17/2018
Dr. Milica Markovic, EEE 161 Applied Electromagnetics

56. Curl of a Vector = Rotation (Curling) of Field
Direction perpendicular to vector field.
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

57. Finding the direction of curl with paddle
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

58. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
Stoke’s Theorem
Surface integral of the curl of A over the surface bounded by S
Circulation of vector A
9/17/2018
Dr. Milica Markovic, EEE 161 Applied Electromagnetics

59. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
Laplacian of a Scalar
Divergence of Gradient
Scalar field is harmonic if:
(Laplace’s Equation)
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

60. Solenoidal or Divergenceless Field
Field has no source or sink.
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

61. Irrotational or Potential Field
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

62. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

63. Classification of Vector Fields
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics

64. Dr. Milica Markovic, EEE 161 Applied Electromagnetics
Why isn’t del D equal to zero
If the curl is zero is the field not spinning
Issue with D
If the curl and divergence are zero what’s happening
Is the curl of C positive or negative
Are you using the density or length notation
Can we write del cross A =magnitude del magn
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Dr. Milica Markovic, EEE 161 Applied Electromagnetics