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

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

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

Point in Cartesian Coordinate System 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

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

Position Vector in Cartesian Coordinates Unit Vectors Components 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

Example of Position Vector in Cartesian Coordinates 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

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

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

Dr. Milica Markovic, EEE 161 Applied Electromagnetics Addition of Vectors Head to Tail Rule Parallelogram Rule 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

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

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

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

Distance Vector Magnitude and Unit Vector 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

Vector Multiplication Scalar or Dot Product Vector or Cross Product Scalar Triple Product Vector Triple Product 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

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

Scalar Product in Cartesian Coordinate System 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

Dr. Milica Markovic, EEE 161 Applied Electromagnetics 5-min Practice 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

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

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

Vector Product in Cartesian Coordinate System 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

Properties of Cross Product 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

Direction of Vector Product 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

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

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

Differential Length - Cart Coord 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

Differential Surface – Cart Coord 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

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

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

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

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

Differential Length –Cyl Coord 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

Differential Surface – Cyl Coord 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

Differential Volume Cyl Coord 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

Magnitude Transformation Relations Cyl Coord – Cart Coord 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

Unit Vectors Transformation Relations Cyl-Cart 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

Cylindrical-Cartesian Coordinates 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

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

Differential Length – Spherical Coord. 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

Differential Surface – Spher. Coord 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

Differential Volume- Spher Coord 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

Distance Between Two Points 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

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

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

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

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

Dr. Milica Markovic, EEE 161 Applied Electromagnetics A Del Operator Del operator is used to define Gradient Divergence Laplacian Curl. 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

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

Gradient of a Scalar Field 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

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

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

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

Dr. Milica Markovic, EEE 161 Applied Electromagnetics Flux of a vector Weak Strong Number of vector lines “flowing” through a surface 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

Dr. Milica Markovic, EEE 161 Applied Electromagnetics Divergence of a Vector 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

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

Curl of a Vector = Rotation (Curling) of Field Direction perpendicular to vector field. 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

Finding the direction of curl with paddle 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

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

Dr. Milica Markovic, EEE 161 Applied Electromagnetics Laplacian of a Scalar Divergence of Gradient Scalar field is harmonic if: (Laplace’s Equation) 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

Solenoidal or Divergenceless Field Field has no source or sink. 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

Irrotational or Potential Field 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

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

Classification of Vector Fields 9/17/2018 Dr. Milica Markovic, EEE 161 Applied Electromagnetics

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