EEE 161 Applied Electromagnetics Dr Milica Markovic 11252020
- Slides: 64
EEE 161 Applied Electromagnetics Dr. Milica Markovic 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 1
Chapter 1 VECTORS 11/25/2020 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 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 3
Point in Cartesian Coordinate System 11/25/2020 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! 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 5
Position Vector in Cartesian Coordinates Unit Vectors Components 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 6
Example of Position Vector in Cartesian Coordinates 11/25/2020 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 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 9
Addition of Vectors • Head to Tail Rule • Parallelogram Rule 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 10
Negative Vector Negative Sign Changes Direction! 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 11
Subtraction of Vectors • First we change direction of vector B • Then we add A and –B up! 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 12
Distance Vector Can be represented by two position vectors , . Coordinates of points B and E 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 13
Distance Vector Magnitude and Unit Vector 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 14
Vector Multiplication • • Scalar or Dot Product Vector or Cross Product Scalar Triple Product Vector Triple Product 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 15
Scalar Product • Theta is the smaller angle between two vectors • Projection of vector B in the direction of vector A (the green line) 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 16
Scalar Product in Cartesian Coordinate System 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 17
5 -min Practice 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 18
Vector Product 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 19
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Vector Product in Cartesian Coordinate System 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 21
Properties of Cross Product 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 22
Direction of Vector Product 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 23
Chapters 2 and 3 COORDINATE SYSTEMS AND VECTOR CALCULUS 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 24
Cartesian Coordinates 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 25
Differential Length - Cart Coord 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 26
Differential Surface – Cart Coord 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 27
Differential Volume – Cart Coord Volume is base times height 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 28
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! 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 29
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Differential Length –Cyl Coord 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 32
Differential Surface – Cyl Coord 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 33
Differential Volume Cyl Coord 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 34
Magnitude Transformation Relations Cyl Coord – Cart Coord 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 35
Unit Vectors Transformation Relations Cyl-Cart 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 36
Cylindrical-Cartesian Coordinates 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 37
Position Vector in Spherical Coordinates Three coordinates r, θ and Φ. Position vector in Cylindrical Coordinates is only in the R direction! 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 38
Differential Length – Spherical Coord. 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 39
Differential Surface – Spher. Coord 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 40
Differential Volume- Spher Coord 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 41
Distance Between Two Points 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 42
Line Integral 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 43
Surface Integral 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 44
Volume Integral 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 45
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. 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 46
A Del Operator Del operator is used to define 1. Gradient 2. Divergence 3. Laplacian 4. Curl. 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 47
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Gradient of a Scalar Field 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 49
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11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 51
Directional Derivative 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 52
Flux of a vector Weak Strong Number of vector lines “flowing” through a surface 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 53
Divergence of a Vector 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 54
Divergence Theorem Flux through a closed surface 11/25/2020 Volume integral thorough of divergence over a volume ~ this is usually easier to find. Dr. Milica Markovic, EEE 161 Applied Electromagnetics 55
Curl of a Vector = Rotation (Curling) of Field Direction perpendicular to vector field. 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 56
Finding the direction of curl with paddle 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 57
Stoke’s Theorem Circulation of vector A 11/25/2020 Surface integral of the curl of A over the surface bounded by S Dr. Milica Markovic, EEE 161 Applied Electromagnetics 58
Laplacian of a Scalar Divergence of Gradient Scalar field is harmonic if: (Laplace’s Equation) 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 59
Solenoidal or Divergenceless Field has no source or sink. 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 60
Irrotational or Potential Field 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 61
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Classification of Vector Fields 11/25/2020 Dr. Milica Markovic, EEE 161 Applied Electromagnetics 63
1. 2. 3. 4. 5. 6. 7. 11/25/2020 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 64
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