1. Flux Capacitor (Operational)
Physics 2113
Jonathan Dowling
Flux Capacitor (Operational)
Physics 2113 Lecture 10
Gauss’ Law III
Carl Friedrich Gauss
1777 – 1855

2. What? The Flux!
Changing Area: dA E
Constant Area: A

3. Gauss’ Law: General Case
Consider any ARBITRARY CLOSED surface S -- NOTE: this “Gaussian Surface” does NOT have to be a “real” physical object!
The TOTAL ELECTRIC FLUX through S is proportional to the TOTAL CHARGE ENCLOSED!
The results of a complicated integral is a very simple formula: it avoids long calculations! S E +q
(One of Maxwell’s 4 equations!)

4. Examples What is Flux Through Surfaces: S1 = S2 = S3 = S4 = +q/ε0

5. What is total charge inside?

6. Gauss’ Law: Insulating Plate
Infinite INSULATING plane with uniform charge density s
E is NORMAL (perpendicular) to plane
Construct Gaussian box as shown
Surface Charge;
σ = q/A
Units: [C/m2]
For an insulator, E=σ/2ε0, and for a conductor, E=σ/ε0.

7. Recall Disk of Charged Sheet From Last Week!
Add the Vectors!
Horrible Integral!
Trig Substitution!
So Hard We Didn’t Do It!
If the Disk Has Large Radius (R>> z) …
Blah, Blah, Blah…
With Gauss’s Law We Got Same Answer With Two Lines of Algebra!

8. Insulating and Conducting PlanesQ
Insulating Plate: Charge Distributed Homogeneously.
Conducting Plate: Charge Distributed on the Outer Surfaces.
Electric Field Inside a Conductor is ZERO!
Q/2

9. Two Insulating Sheets

10. The field from the plates cancels out so ignore them and use Coulombs inverse square law for the central charge only.

11. E does not pass through a conductor
Two Conducting Sheets
E does not pass through a conductor
Formula for E different by Factor of 2
7.68
4.86
4.86
7.68
12.54

12. Gauss’ Law: Spherical Symmetry
Consider a POINT charge q & pretend that you don’t know Coulomb’s Law
Use Gauss’ Law to compute the electric field at a distance r from the charge
Use symmetry:
place spherical surface of radius R centered around the charge q
E has same magnitude anywhere on surface
E normal to surface r q

14. Electric Fields With Spherical Symmetry: Shell Theorem
A spherical shell has a charge of +10C and a point charge of –15C at the center.
What is the electric field produced OUTSIDE the shell?
If the shell is conducting?
Field Inside a Conductor is ZERO! E r
E=k(15C)/r2
E=k(5C)/r2
E=0
And if the shell is insulating?
Charged Shells
Behave Like a Point Charge of Total Charge “Q” at the Center
Once Outside the Last Shell!
Conducting

15. Electric Fields With Insulating Sphere

16. Summary
Electric Flux: a Surface Integral (Vector Calculus!); Useful Visualization: Electric Flux Lines Like Wind Through a Window.
Gauss’ Law Provides a Very Direct Way to Compute the Electric Flux.