1. Fields and Waves I Lecture 9 K. A. Connor
Introduction to Electrostatics and Gauss’ Law
K. A. Connor
Electrical, Computer, and Systems Engineering Department
Rensselaer Polytechnic Institute, Troy, NY
Welcome to Fields and Waves I
Before I start, can those of you with pagers and cell phones please turn them off? Thanks.

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. Those listed as Ulaby are figures from Ulaby’s textbook.
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3. Review of Math for Fields
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
Coulomb’s Law
Gauss’ Law
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4. Maxwell’s Equations Integral Form Differential Form Electrostatics
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5. Electrostatics Integral Form Differential Form
Note the Math Operations
Integral Form
Differential Form
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6. Electrostatics – What is its value?
Capacitance
I-V Characteristics of Devices
Useful when the system size is small compared to
&
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7. Capacitance Charge-Voltage Method Energy Method 18 September 2018
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8. Electrostatics Basic Configurations Ulaby 18 September 2018
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9. SOURCE of Electrostatic E-Field is CHARGE
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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10. Other examples of Charge Distribution……..
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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11. ELECTRICAL CHARGES Maxwell’s equation: More generally, Derived from:
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12. 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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13. Example 1
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14. Fields and Waves I
Force and fields

15. (force), between point charges
COULOMB’S LAW
(force), between point charges Q1
Unit vector in r-direction
Force on Charge 2 by Charge 1 R Q2
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16. COULOMB’S LAW - E Field ,of Q1 is Then,
Unit vector pointing away from Q1
Then,
- we work with E-Field because Maxwell’s equations written in those terms
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17. Proportional to Magnitude or strength of E-Field
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
Point charge
Proportional to Magnitude or strength of E-Field
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18. Figure from Ulaby
Ulaby
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19. Computation of E-fields from multiple charges:
Example: DIPOLE - 2 separated opposite polarity point charges +Q
Apply superposition of Fields x y
Planes of symmetry:
Horizontal axis: Ex cancels, Ey adds -Q
+Q vector
Vertical axis: only Ey component -Q
Resulting vector
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20. E-FIELDS - Some examples
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21. 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.
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22. E-FIELDS - Dipole Lines of Symmetry 18 September 2018
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23. E-FIELDS
How would the DIPOLE field lines change if the charges were the same polarity?
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24. E-FIELDS
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25. 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 . Dipole results can be seen with the applet or with the Mathcad worksheet for Sect
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26. Example 2
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27. Example 2 with Finite Element Computation
Plot of the electric field (direction and magnitude)
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28. Calculating the electric field thanks to the Gauss’ Law
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Calculating the electric field thanks to the Gauss’ Law

29. MAXWELL’S 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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30. 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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31. Example 3
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32. Use Gauss’ Law to find D and E in symmetric problems
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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33. 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) x y z
Surface of infinite extent of charge
Can write down:
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34. 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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35. Example 4
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36. For example a planar sheet of charge, where z is constant
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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37. Use Gaussian surface to “pull” this out of integral
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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38. Example of using GAUSS’ law to find
Z = a
-a < z < a
Z > a ; z< -a
Z = -a
“a slab of charge”
By symmetry:
From symmetry
If r0 > 0, then
Z=0
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39. in region |z| < a and create a surface at arbitrary z
GAUSS’ LAW
First get
in region |z| < 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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40. GAUSS’ LAW =0, since Evaluate LHS: These two integrals are equal
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41. Key Step: Take E out of the Integral
GAUSS’ LAW
Key Step: Take E out of the Integral
Computation of enclosed charge
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42. GAUSS’ LAW
(drop the prime)
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43. GAUSS’ LAW Need to find , for |z| > a Z = z’ r0 Z = -z’
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44. Computation of enclosed charge
GAUSS’ LAW
As before,
Computation of enclosed charge
Note that the z-integration is from -a to a ;
there is NO CHARGE outside |z|>a
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45. For the region outside |z|>a
GAUSS’ LAW
Once again,
For the region outside |z|>a
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46. Note: E-field is continuous
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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47. 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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48. Find the electric field for all r.
Example 5
Find the electric field for all r.
Check your answer by evaluating the divergence and curl of the electric field.
What is the closed line integral of the electric field around the contour shown?
Express the surface charge density in terms of the volume charge density.
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49. Example 5
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50. Example 5
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51. Example 5
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52. Example 5
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53. Ulaby
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54. 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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55. 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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56. 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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57. Using Gauss’ Law to find E
There is a short write-up on this topic in the Supplementary Materials:
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