1. Fields and Waves I Lecture 8 K. A. Connor Y. Maréchal
Intro to Fields and Math of Fields
K. A. Connor
Electrical, Computer, and Systems Engineering Department
Rensselaer Polytechnic Institute, Troy, NY
Y. Maréchal
Power Engineering Department
Institut National Polytechnique de Grenoble, France
Welcome to Fields and Waves I
Before I start, can those of you with pagers and cell phones please turn them off? Thanks.
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2. J. Darryl Michael – GE Global Research Center, Niskayuna, NY
These Slides Were Prepared by Prof. Kenneth A. Connor Using Original Materials Written Mostly by the Following:
Kenneth A. Connor – ECSE Department, Rensselaer Polytechnic Institute, Troy, NY
J. Darryl Michael – GE Global Research Center, Niskayuna, NY
Thomas P. Crowley – National Institute of Standards and Technology, Boulder, CO
Sheppard J. Salon – ECSE Department, Rensselaer Polytechnic Institute, Troy, NY
Lale Ergene – ITU Informatics Institute, Istanbul, Turkey
Jeffrey Braunstein – Chung-Ang University, Seoul, Korea
Materials from other sources are referenced where they are used. Most figures from Ulaby’s textbook.
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3. Electrostatics Electric Field of a Point Charge 9 November 2018
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4. The math we have seen so far:
Overview
The math we have seen so far:
Phasor notation
Partial differential equations
The wave equation
For fields, we need to review vector calculus
Vector notation
Coordinate systems
Line, area and volume integrals
Gradient, divergence and curl
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5. 3 systems of coordinates
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3 systems of coordinates
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6. Vector representation
3 PRIMARY COORDINATE SYSTEMS:
Choice is based on symmetry of problem
RECTANGULAR
CYLINDRICAL
SPHERICAL
Examples:
Sheets - RECTANGULAR
Wires/Cables - CYLINDRICAL
Spheres - SPHERICAL
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7. Cartesian coordinates basics
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
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8. Cartesian coordinates Dot product
Definition
Meaning of dot product
Dot Product
(scalar)
Magnitude of vector
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9. Cartesian coordinates Cross product
Definition
Meaning of the cross product
Cross Product
(VECTOR)
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10. Cylindrical coordinates Basics
Ulaby
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11. Cylindrical coordinates Unit vectorsf 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
In cylindrical coordinates, both
and
are functions of f
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12. Cylindrical coordinates Dot product
UNIT VECTORS:
Cylindrical representation uses: r ,f , z
Dot Product
(SCALAR) r f z P x y
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13. Spherical coordinates Basics
Ulaby
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14. Spherical coordinates Unit vectorsz P q r
The Unit Vectors imply :
Points in the direction of increasing r f y x
Points in the direction of increasing q
Points in the direction of increasing j
In spherical coordinates,
and
are functions of f
and q
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15. Spherical coordinates Dot product
UNIT VECTORS:
Spherical representation uses: r ,q , f
Dot Product
(SCALAR) z P q r x f y
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16. Vector representation 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
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17. Vector representation : Examples
In spherical coordinate system
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18. Differential calculus
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Differential calculus
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19. Differential calculus : introduction
Example x y 2 6 3 7
integration over 2 “delta” distances dx dy
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20. Differential lengths 1. Rectangular Coordinates:
Unit is in “meters”
When you move a small amount in x-direction, the distance is dx
In a similar fashion, you generate dy and dz
Generate:
( dx, dy, dz )
2. Cylindrical Coordinates:
Distance = r df x y df r
Differential distances:
( dr, rdf, dz )
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21. Differential lengths P 3. Spherical Coordinates:x z y q x y df
r sinq
Distance = r sinq df
Differential distances:
( dr, rdq, r sinq df )
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22. Representation of differential lengths dl in the 3 coordinate systems
rectangular
cylindrical
spherical
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23. Differential surfaces and volumes
Example of surface differentials or
Representation of differential surface element:
Vector is NORMAL to surface
Differential volume ( a scalar)
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24. Differential volumes : Cartesian coordinates
Ulaby
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25. Differential volumes: cylindrical coordinates
Ulaby
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26. Differential volumes: cylindrical coordinates
Ulaby
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27. Differential volumes : spherical coordinates
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28. Differential volumes : spherical coordinates
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Ulaby

29. Integrals calculations
How to perform integrals (surface or volumes ) ?
What is the right system of coordinates ?
What is kept constant?
What does the limits stand for ?
What are the differentials?
See that on the following examples …
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30. Find the following area
Area calculus
Find the following area
Cylindrical area
r=5, 30°<f<60°, 0<z<3cm
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31. Find the following surface area
Area calculus
Find the following surface area
Spherical area
30<q<60, 0<f<2p, r=3cm
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32. Volume calculus : examples
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33. Volume calculus : examples
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34. Volume calculus : examples
Volume in cylindrical coordinates
Calculate the volume by integration of
1<r<3, 0<f<p/3, -2<z<2
The electric charge density inside a sphere is given by 4cos2(q) . Find the total charge Q contained in a sphere of radius 2cm.
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35. Volume calculus : examples
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36. The curl , gradient and divergence operators
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The curl , gradient and divergence operators
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37. Curl operator : Basics NOTATION: Implies a CLOSED LOOP Integral
measures circulation or Curl of B
Example of a uniform field B in the x direction
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38. Main property (Stokes’s theorem)
The Curl operator
The curl operator
Main property (Stokes’s theorem)
NOTATION:
Result of this operation is a VECTOR
This is NOT a CROSS-PRODUCT
To calculate CURL, use formulas in the textbook
Surface integral on right is surface enclosed by line on the left
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39. Curl operator : physical example
Plot of magnetic field direction and modulus
Current flowing in an infinite wire
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40. Rotation or Curl operator Examples
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42. Examples
Calculate over the two shaded surfaces and compare to the previous result.
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43. Gradient operator The gradient operator Main property
GRADIENT measures CHANGE in a SCALAR FIELD
the result is a VECTOR pointing in the direction of increase
For a Cartesian system:
You will find that
ALWAYS IF
, then
and
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44. Gradient : physical example
Current flow simulation
V=10
V=0
Current modulus in color shades
And arrows
Potential in red equi lines
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45. Gradient examples
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46. Surface integrals dx Note that all 3 coordinates are involved dzy dx
Note that all 3 coordinates are involved dz
, measures flux of
, through a surface
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47. Surface integrals and flux
Example - FLUID FLOW
For,
, there is flow through
But,
, there is no flow
Hence,
, measures FLUX
Example: Let y=2, x=0 to 3 and z = -1 to 1
, then,
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48. Divergence operator Measures Flux through any surface
“Global” quantities
Measures Flux through closed surfaces
is related to
, is a “local” measure of flux property
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49. The divergence operator
Notation:
NOT a DOT product but has similar features
Result is a SCALAR, composed of derivatives
in Cartesian coordinates
Divergence Theorem:
Volume integral on right is volume enclosed by surface on the left
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50. Divergence operator : physical example
2 chips on a PCB
Temperature in color shade, Iso values of temperature in red lines
The integral heat flux through the surface of a chip = amount of heat included in its volume
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51. Examples
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52. Operators summary Curl Gradient Divergence
Measures the circulation of a vector field
Gradient
Measures the change in a scalar field
Divergence
Measures the flux of a vector field trough a surface
Result is a VECTOR
Result is a VECTOR
Result is a SCALAR
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53. Visual Electromagnetics for Mathcad
On Main Course Info Page
Download the Mathcad Explorer
Download Visual Electromagnetics
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