Fields and Waves I

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Fields and Waves I

• Lecture 9
• Introduction to Electrostatics and Gauss Law
• K. A. Connor
• Electrical, Computer, and Systems Engineering
  Department
• Rensselaer Polytechnic Institute, Troy, NY

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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. Those listed as Ulaby are figures
from Ulabys textbook.
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Overview

• Review of Math for Fields
• Vector notation
• Coordinate systems
• Line, area and volume integrals
• Gradient, divergence and curl
• Electrostatics
• Sources Charges and Charge Distributions
• Coulombs Law
• Gauss Law

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Maxwells Equations
Electrostatics

• Integral Form

• Differential Form

0
0
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Electrostatics
Note the Math Operations

• Integral Form

• Differential Form

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Electrostatics What is its value?

• Capacitance
• I-V Characteristics of Devices
• Useful when the system size is small compared to

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Capacitance

• Charge-Voltage Method
• Energy Method

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Electrostatics

• Basic Configurations

Ulaby
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ELECTRICAL CHARGES
SOURCE of Electrostatic E-Field is CHARGE
Examples of various charge distributions
1. Point charge - Q (units of Coulomb)
- model individual particle (eg. Electron) or a
well-localized group of charge particles
2. Volume Charge Density - r or rv (units of
Coulomb/m3)
- large of particles - ignore discrete nature
to smooth out distribution
Eg. Doped Region of Semiconductor, e-beam in a
cathode ray tube
( Beam has finite radius )
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ELECTRICAL CHARGES
Other examples of Charge Distribution..
3. Surface Charge Density - r or rs (units of
Coulomb/m2)
Eg. Very thin charge layer on conductor surface
4. Line Charge Density - r or rl (units of
Coulomb/m)
- not as physically realizable Eg. Model for a
wire, electron beam from far
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ELECTRICAL CHARGES
Maxwells equation
More generally,
Derived from
or
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Example 1

• Charge density
• Find the charge in the spherical volume
  containing a charge distribution
  .
• Evaluate the result analytically. Then find a
  numerical result when meters and
  .
• b. A surface charge on a disk increases linearly
  from in the center to
  at the outer edge where r 2 meters. Find
  the total charge on the disk.

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Example 1
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Fields and Waves I

• Force and fields

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COULOMBS LAW
(force), between point charges
Q1
Unit vector in r-direction
Force on Charge 2 by Charge 1
R
Q2
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COULOMBS LAW - E Field
,of Q1 is
Unit vector pointing away from Q1
Then,
- we work with E-Field because Maxwells
equations written in those terms
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E-FIELDS
, is a VECTOR Field
How do we represent it?
- Field points in the direction that a q test
charge would move
Represent using Arrows Direction and Length
Proportional to Magnitude or strength of E-Field
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Figure from Ulaby
Ulaby
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E-FIELDS
Computation of E-fields from multiple charges
Example DIPOLE - 2 separated opposite polarity
point charges
Apply superposition of Fields
Planes of symmetry
Horizontal axis Ex cancels, Ey adds
Q vector
Vertical axis only Ey component
-Q
Resulting vector
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E-FIELDS - Some examples
http//shinliang.blogspot.com/2009/04/21-coulombs-
law.html
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E-FIELDS - Some examples
Note, in the upper right figure, that four times
as many field lines leave the 4 positive charge
as leave the 1 charge. All of the field lines
end at infinity, as they do with a single
positive charge.
http//people.seas.harvard.edu/jones/cscie129/nu_
lectures/lecture6/field_vis/e_vis.html
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Lines of Symmetry
E-FIELDS - Dipole
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E-FIELDS
How would the DIPOLE field lines change if the
charges were the same polarity?
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E-FIELDS
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Example 2

• Sketch the electric field lines for the electric
  quadrupole shown. Sketch the planes for which
  you expect the field to be symmetric. After
  completing your sketch, verify your result with
  the applet at http//hibp.ecse.rpi.edu/crowley/ja
  va/Efield/App.htm . Dipole results can be seen
  with the applet or with the Mathcad worksheet for
  Sect. 3.6.2

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Example 2
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Example 2 with Finite Element Computation
Plot of the electric field (direction and
magnitude)
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Fields and Waves I

• Calculating the electric field thanks to the
  Gauss Law

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MAXWELLS FIRST EQUATION GAUSS LAW
Enclosed Charge
Differential Form
Integral Form
- dv integral over volume enclosed by ds
integral
For vacuum and air - think of D and E as being
the same
D vs E depends on materials
constant
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Example 3

• Gauss Law
• Show that the electric field of a point charge
  satisfies Gauss Law by integrating
  over the surface of a sphere of radius a.

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Example 3
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GAUSS LAW - strategy
Use Gauss Law to find D and E in symmetric
problems
Get D or E out of integral
Always look at symmetry of the problem - and take
advantage of this
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GAUSS LAW - use of symmetry
Example A sheet of charge
- charges are infinite in extent on say x,y plane
, is sum due to all charges
Arbitrary Point P
, points in

• all other components cancel

• only a function of z (not x or y)

Surface of infinite extent of charge
Can write down
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Example 4

• Setting up the Problem
• For the three charge distributions, find the
  direction of the electric field, the surface on
  which the field is a constant and the flux is
  nonzero, and sketch the surfaces.

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Example 4
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GAUSS LAW
,is constant. For example a planar sheet of
charge, where z is constant
To use GAUSS LAW, we need to find a surface that
encloses the volume
GAUSSIAN SURFACE - takes advantage of symmetry
- when r is only a f(r)
- when r is only a f(z)
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GAUSS LAW
Use Gaussian surface to pull this out of
integral
Integral now becomes
Usually an easy integral for surfaces under
consideration
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GAUSS LAW
Z a
Z -a
a slab of charge
By symmetry
From symmetry
If r0 gt 0, then
Z0
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GAUSS LAW
First get
in region z lt a and create a surface at
arbitrary z
Use Gaussian surface with top at z z and the
bottom at -z
Note Gaussian Surface is NOT a material boundary
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GAUSS LAW
0, since
Evaluate LHS
These two integrals are equal
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GAUSS LAW
Key Step Take E out of the Integral
Computation of enclosed charge
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GAUSS LAW
(drop the prime)
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GAUSS LAW
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GAUSS LAW
As before,
Computation of enclosed charge
Note that the z-integration is from -a to a
there is NO CHARGE outside zgta
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GAUSS LAW
Once again,
For the region outside zgta
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GAUSS LAW
-a
z
a
Note E-field is continuous
Plot of E-field as a function of z for planar
example
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Example 5

• Full Gauss Law Solution
• A charge distribution with cylindrical symmetry
  is shown. The inner cylinder has a uniform
  charge density . The outer shell has
  a surface charge density such
  that the total charge on the outer shell is the
  negative of the total charge in the inner
  cylinder. Ignore end effects.

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Example 5

1. Find the electric field for all r.
2. Check your answer by evaluating the divergence
   and curl of the electric field.
3. What is the closed line integral of the electric
   field around the contour shown?
4. Express the surface charge density in terms of
   the volume charge density.

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Example 5
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Example 5
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Example 5
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Example 5
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Ulaby
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Using Gauss Law to find E

• Recognize the coordinate system.
• Using symmetry, determine which components of the
  field exist.
• Identify a Gaussian surface for which the sides
  are either parallel to or perpendicular to the
  field components. This surface is arbitrary in
  size.
• Determine the total charge within that surface.
  The charges can be distributed on lines, surfaces
  or in volumes.

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Using Gauss Law to find E

• Evaluate the electric flux passing through the
  Gaussian surface.
• If the field is parallel to the surface
• If the field is perpendicular to the surface,
• where the subscript refers to the direction of
  the surface.
• Note that a high level of symmetry is necessary
  to make these simplifications.

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Using Gauss Law to find E

• Now equate the two sides of Gauss Law to find E
• Remember that the Gaussian surface is arbitrary
  in position so the surface area is a function.
  For example for a spherical surface

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Using Gauss Law to find E

• There is a short write-up on this topic in the
  Supplementary Materials
• http//hibp.ecse.rpi.edu/7Econnor/education/Fiel
  ds/gauss_law.pdf