1. W02D2 Gauss’s Law
Class 02

2. Announcements
Math Review Tuesday Week Three Tues from 9-11 pm in Vector Calculus
PS 2 due Week Three Tuesday Tues at 9 pm in boxes outside or
W02D3 Reading Assignment Course Notes: Chapter Course Notes: Sections 3.6, 3.7, 3.10
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3. Outline Electric Flux Gauss’s Law
Calculating Electric Fields using Gauss’s Law
Class 04

4. The first Maxwell Equation!
Gauss’s Law
The first Maxwell Equation!
A very useful computational technique to find the electric field when the source has ‘enough symmetry’.
Carl Friedrich is so money!
Class 04

5. Gauss’s Law – The Idea
The total “flux” of field lines penetrating any of these closed surfaces is the same and depends only on the amount of charge inside
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6. Gauss’s Law – The Equation
Electric flux (the surface integral of E over closed surface S) is proportional to charge enclosed by the volume enclosed by S
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7. Electric Flux
Case I: E is a uniform vector field perpendicular to planar surface S of area A
Our Goal: Always reduce problem to finding a surface where we can take E out of integral and get simply E*Area
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8. Electric Flux
Case II: E is uniform vector field directed at angle to planar surface S of area A
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9. Concept Question: Flux
The electric flux through the planar surface below (positive unit normal to left) is: +q -q
positive.
negative.
zero.
Not well defined.
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10. Concept Question Answer: Flux
Answer: 2. The flux is negative. +q -q
The field lines go from left to right, opposite the assigned normal direction. Hence the flux is negative.

11. Open and Closed Surfaces
A rectangle is an open surface — it does NOT contain a volume
A sphere is a closed surface — it DOES contain a volume
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12. Area Element: Closed Surface
Case III: not uniform, surface curved
For closed surface, is normal to surface and points outward ( from inside to outside)
if points out
if points in
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13. Group Problem Electric Flux: Sphere
Consider a point-like charged object with charge Q located at the origin. What is the electric flux on a spherical surface (Gaussian surface) of radius r ?
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14. Arbitrary Gaussian Surfaces
explain
True for all surfaces such as S1, S2 or S3
Why? As area gets bigger E gets smaller
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15. Gauss’s Law
Note: Integral must be over closed surface
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16. Concept Question: Flux thru Sphere
The total flux through the below spherical surface is +q
positive (net outward flux).
negative (net inward flux).
zero.
Not well defined.
Class 04

17. Concept Question Answer: Flux thru Sphere
Answer: 3. The total flux is zero +q
We know this from Gauss’s Law:
No enclosed charge  no net flux.
Flux in on left cancelled by flux out on right

18. Concept Question: Gauss’s Law
The grass seeds figure shows the electric field of three charges with charges +1, +1, and -1, The Gaussian surface in the figure is a sphere containing two of the charges. The electric flux through the spherical Gaussian surface is
Positive
Negative
Zero
Impossible to determine without more information.

19. Concept Question Answer: Gauss’s Law
Answer 3: Zero. The field lines around the two charged objects inside the Gaussian surface are the field lines associated with a dipole, so the charge enclosed in the Gaussian surface is zero. Therefore the electric flux on the surface is zero.
Note that the electric field E is clearly NOT zero on the surface of the sphere. It is only the INTEGRAL over the spherical surface of E dotted into dA that is zero.

20. Choosing Gaussian Surface
True for all closed surfaces
Useful (to calculate electric field ) for some closed surfaces for some problems with lots of symmetry.
Desired E: Perpendicular to surface and uniform on surface.
Flux is EA or -EA.
Other E: Parallel to surface. Flux is zero
Class 04

21. Symmetry & Gaussian Surfaces
Desired E: perpendicular to surface and constant on surface. So Gauss’s Law useful to calculate electric field from highly symmetric sources
Source Symmetry
Gaussian Surface
Spherical
Concentric Sphere
Cylindrical
Coaxial Cylinder
Planar
Gaussian “Pillbox”
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22. Virtual Experiment Gauss’s Law Applet
Bring up the Gauss’s Law Applet and answer the experiment survey questions
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23. Applying Gauss’s Law
Based on the source, identify regions in which to calculate electric field.
Choose Gaussian surface S: Symmetry
Calculate
Calculate qenc, charge enclosed by surface S
Apply Gauss’s Law to calculate electric field:
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24. Examples: Spherical Symmetry Cylindrical Symmetry Planar Symmetry
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25. Group Problem Gauss: Spherical Symmetry
+Q uniformly distributed throughout non-conducting solid sphere of radius a. Find everywhere.
Let students try problem, summarize result at end. Decide how much time you let students work on this while observing what they are doing.
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26. Concept Question: Spherical Shell
We just saw that in a solid sphere of charge the electric field grows linearly with distance. Inside the charged spherical shell at right (r<a) what does the electric field do? a Q
Zero
Uniform but Non-Zero
Still grows linearly
Some other functional form (use Gauss’ Law)
Can’t determine with Gauss Law
Class 04

27. Concept Question Answer: Flux thru Sphere
Answer: 1. Zero Q a
Spherical symmetry
 Use Gauss’ Law with spherical surface.
Any surface inside shell contains no charge
 No flux
E = 0!

28. Demonstration Field Inside Spherical Shell (Grass Seeds):
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29. Worked Example: Planar Symmetry
Consider an infinite thin slab with uniform positive charge density Find a vector expression for the direction and magnitude of the electric field outside the slab. Make sure you show your Gaussian closed surface.
This could either be a group problem or a worked example depending on the time left.
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30. Gauss: Planar Symmetry
Symmetry is Planar
Use Gaussian Pillbox
Note: A is arbitrary (its size and shape) and should divide out
Do this one as a worked example
Gaussian
Pillbox
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31. Gauss: Planar Symmetry
Total charge enclosed:
NOTE: No flux through side of cylinder, only endcaps +
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32. Concept Question: Superposition
Three infinite sheets of charge are shown above. The sheet in the middle is negatively charged with charge per unit area , and the other two sheets are positively charged with charge per unit area Which set of arrows (and zeros) best describes the electric field?

33. Concept Question Answer: Superposition
Answer The fields of each of the plates are shown in the different regions along with their sum.

34. Group Problem: Cylindrical Symmetry
Week 03, Day 2
Group Problem: Cylindrical Symmetry
An infinitely long rod has a uniform positive linear charge density Find the direction and magnitude of the electric field outside the rod. Clearly show your choice of Gaussian closed surface. 34
Class 07 34

35. Electric Field for Charged Infinite Plane
Dipole: E falls off like 1/r3
Spherical charge: E falls off like 1/r2
Line of charge: E falls off like 1/r
(infinite)
Plane of charge: E uniform on
(infinite) either side of plane