1. Chapter 23

2. Potential Energy

3. Gravitation-Electrical, what’s the difference?so

4. Potential i.e. Voltage
Potential is actually potential energy per unit charge
V=U/q1
Volts (V)= Joule/Coulomb or J/C
Recall E-field units were N/C
Now we have a new unit for electric field: V/m
So now we see that the electric field is expression of how the voltage is changing over space

5. The coolest thing about voltage
It is a SCALAR!
But it can tell us about the E-field and Forces!

6. How to calculate V Point charge (I put the minus sign in q)
Collection of point charges
Distribution of charge

7. Potential Difference (i.e. the Voltmeter)
Potential difference between point A and point B

8. Equipotential Surfaces
kq/r1 q
kq/r2
E-field Lines
Electric field lines are always perpendicular to equipotential surfaces

9. Equipotentials and Conductors
Recall that all the charge of an isolated conductor resides on the surface
All excess charge placed on an isolated conductor will distribute itself so that the surface will come to same potential because:
If there is more or less charge between two points on the surface, there will be current flow until equilibrium re-established
Furthermore, the electric field lines must be perpendicular to the surface at every point
Because if there were components tangential to the surface, there would be

10. E from V

11. So

12. For radial E-fields only

13. Going back to “Work” Recall: But the work done is independent of path
Just like in gravity

14. Back to where we started
What if:
Well, from point a to point a seems facetious, what about 0 to 2p?
Such an integral is called a closed path integral and is expressed

15. Curls again Recall Curl(v) Curl=0 Curl=+, out of page
(Right Hand Rule)
Curl=0

16. Stoke Theorem

17. So

18. Funky Shapes and the Method of Recursion
Needless to say, Gauss’s Law works well for 3 shapes:
Sphere
Cylinder
Big, Flat Surfaces
But what if you have a funny shape?

19. Some Definitions
Recall that
What is this quantity?

20. Laplacian

21. Laplace’s Equation This equation applies to many physical systems
Gravitation
Heat flow EM
Quantum Mechanics

22. Relax! A numerical method to solve Laplace’s equation Requires
Method of Relaxation
Method of Recursion
Finite Element Method
Big Brothers
Discrete Ordinate (reactor physics)
Finite Difference Time Domain (RF engineering)
Requires
Known boundary conditions
The ability to break geometry into small regular shaped volume elements
Computer with sufficient memory (PVM)

23. Averaging Based on nearest neighbor averaging 5 5 5 ?
?=( )/4 =1.25
This is the estimate of the voltage in this square

24. 2-d and 3-d Loop thru multiple times
You save the new values in a temporary array
The new values replace the old values and you loop again
For 3-d objects, there are 6 surfaces

25. More Realism By a simple average, you are assuming perfect conduction
For different materials, you can slow up the transfer of voltage, heat, etc by using
Heat: different thermal conductivity values
Electricity: different electrical conductivity values
Fluids: different viscosities

26. Electric Field Lines
1.0
3.5
2.0
0.0 4 5