Intro to Fields and Math of Fields

1
Fields and Waves I

• Lecture 8
• 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

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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 Ulabys
textbook.
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Electrostatics
Electric Field of a Point Charge
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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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Fields and Waves I

• 3 systems of coordinates

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Vector representation
3 PRIMARY COORDINATE SYSTEMS

• RECTANGULAR

• CYLINDRICAL

• SPHERICAL

Examples
Sheets - RECTANGULAR
Wires/Cables - CYLINDRICAL
Spheres - SPHERICAL
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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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Cartesian coordinates Dot product

• Definition
• Meaning of dot product

Dot Product (scalar)
Magnitude of vector
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Cartesian coordinates Cross product

• Definition
• Meaning of the cross product

Cross Product
(VECTOR)
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Cylindrical coordinates Basics
Ulaby
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Cylindrical coordinates Unit vectors
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
f
In cylindrical coordinates, both
and
are functions of
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Cylindrical coordinates Dot product
UNIT VECTORS
Cylindrical representation uses r ,f , z
Dot Product
(SCALAR)
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Spherical coordinates Basics
Ulaby
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Spherical coordinates Unit vectors
z
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
f
In spherical coordinates,
and
are functions of
and
q
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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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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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Vector representation Examples

• In spherical coordinate system

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Fields and Waves I

• Differential calculus

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Differential calculus introduction

• Example

• integration over 2 delta distances

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Differential lengths
1. Rectangular Coordinates
When you move a small amount in x-direction, the
distance is dx
In a similar fashion, you generate dy and dz
( dx, dy, dz )
Generate
2. Cylindrical Coordinates
Differential distances
( dr, rdf, dz )
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Differential lengths
3. Spherical Coordinates
Distance r sinq df
Differential distances
( dr, rdq, r sinq df )
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Differential lengths

• Representation of differential lengths dl in the
  3 coordinate systems

rectangular
cylindrical
spherical
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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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Differential volumes Cartesian coordinates
Ulaby
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Differential volumes cylindrical coordinates
Ulaby
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Differential volumes cylindrical coordinates
Ulaby
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Differential volumes spherical coordinates
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Differential volumes spherical coordinates
Ulaby
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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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Area calculus

• Find the following area
• Cylindrical area
• r5, 30ltflt60, 0ltzlt3cm

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Area calculus

• Find the following surface area
• Spherical area
• 30ltqlt60, 0ltflt2p, r3cm

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Volume calculus examples
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Volume calculus examples
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Volume calculus examples

• Volume in cylindrical coordinates
• Calculate the volume by integration of
• 1ltrlt3, 0ltfltp/3, -2ltzlt2
• 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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Volume calculus examples
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Fields and Waves I

• The curl , gradient and divergence operators

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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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The Curl operator

• The curl operator
• Main property (Stokess 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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Curl operator physical example
Plot of magnetic field direction and modulus
Current flowing in an infinite wire
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Rotation or Curl operator Examples
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Examples
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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
ALWAYS
You will find that
IF
, then
and
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Gradient physical example

• Current flow simulation

V10
V0
Current modulus in color shades And
arrows Potential in red equi lines
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Gradient examples
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Surface integrals
Note that all 3 coordinates are involved
, measures flux of
, through a surface
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Surface integrals and flux
Example - FLUID FLOW
, there is flow through
For,
But,
, there is no flow
Hence,
, measures FLUX
Example Let y2, x0 to 3 and z -1 to 1
, then,
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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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Divergence operator

• The divergence operator

NOT a DOT product but has similar features
Notation
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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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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Examples
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Operators summary

• Curl
• 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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