EEE 161 Applied Electromagnetics Dr Milica Markovic 10312021

EEE 161 Applied Electromagnetics Dr. Milica Markovic 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 1

Chapter 1 VECTORS 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 2

Scalars and Vectors • Scalars quantities are defined by magnitude only: – Temperature 75 deg. F – Mass 75 kg • Vectors are defined by magnitude and direction: – Wind speed 75 m/h in NW direction 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 3

Point in Cartesian Coordinate System 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 4

Unit Vectors in Cartesian Coordinates X-direction Y-direction Z-direction Unit vectors have magnitude of 1! 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 5

Position Vector in Cartesian Coordinates Unit Vectors Components 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 6

Example of Position Vector in Cartesian Coordinates 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 7

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More on Vectors • Magnitude – length of the vector • Direction – Unit vector in the direction of vector A Magnitude = 1 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 9

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Addition of Vectors • Head to Tail Rule • Parallelogram Rule 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 11

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Negative Vector Negative Sign Changes Direction! 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 13

Subtraction of Vectors • First we change direction of vector B • Then we add A and –B up! 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 14

Distance Vector Can be represented by two position vectors , . Coordinates of points B and E 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 15

Distance Vector Magnitude and Unit Vector 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 16

Vector Multiplication • • Scalar or Dot Product Vector or Cross Product Scalar Triple Product Vector Triple Product 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 17

Scalar Product • Theta is the smaller angle between two vectors • Projection of vector B in the direction of vector A (the green line) 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 18

Scalar Product in Cartesian Coordinate System 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 19

5 -min Practice 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 20

Vector Product 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 21

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Vector Product in Cartesian Coordinate System 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 23

Properties of Cross Product 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 24

Direction of Vector Product 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 25

Chapters 2 and 3 COORDINATE SYSTEMS AND VECTOR CALCULUS 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 26

Cartesian Coordinates 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 27

Differential Length - Cart Coord 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 28

Differential Surface – Cart Coord 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 29

Differential Volume – Cart Coord Volume is base times height 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 30

Position Vector in Cylindrical Coordinates Three coordinates r, θ and z. Θ= 60 deg Position vector in Cylindrical Coordinates has only r and z directions! 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 31

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Differential Length –Cyl Coord 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 34

Differential Surface – Cyl Coord 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 35

Differential Volume Cyl Coord 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 36

Magnitude Transformation Relations Cyl Coord – Cart Coord 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 37

Unit Vectors Transformation Relations Cyl-Cart 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 38

Cylindrical-Cartesian Coordinates 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 39

Position Vector in Spherical Coordinates Three coordinates r, θ and Φ. Position vector in Cylindrical Coordinates is only in the R direction! 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 40

Differential Length – Spherical Coord. 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 41

Differential Surface – Spher. Coord 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 42

Differential Volume- Spher Coord 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 43

Distance Between Two Points 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 44

Line Integral 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 45

Surface Integral 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 46

Volume Integral 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 47

Representation of Vector Fields are usually represented by arrows. 1. 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. 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 48

A Del Operator Del operator is used to define 1. Gradient 2. Divergence 3. Laplacian 4. Curl. 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 49

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Gradient of a Scalar Field 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 51

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Directional Derivative 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 54

Flux of a vector Weak Strong Number of vector lines “flowing” through a surface 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 55

Divergence of a Vector 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 56

Divergence Theorem Flux through a closed surface 10/31/2021 Volume integral thorough of divergence over a volume ~ this is usually easier to find. Dr. Milica Markovic, EEE 161 Applied Electromagnetics 57

Curl of a Vector = Rotation (Curling) of Field Direction perpendicular to vector field. 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 58

Finding the direction of curl with paddle 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 59

Stoke’s Theorem Circulation of vector A 10/31/2021 Surface integral of the curl of A over the surface bounded by S Dr. Milica Markovic, EEE 161 Applied Electromagnetics 60

Laplacian of a Scalar Divergence of Gradient Scalar field is harmonic if: (Laplace’s Equation) 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 61

Solenoidal or Divergenceless Field has no source or sink. 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 62

Irrotational or Potential Field 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 63

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Classification of Vector Fields 10/31/2021 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 65

1. 2. 3. 4. 5. 6. 7. 10/31/2021 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 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 66