1. Notes 11 ECE 3318 Applied Electricity and Magnetism Gauss’s Law II
Spring 2019
Prof. David R. Jackson
Dept. of ECE
Notes 11
Gauss’s Law II
Notes prepared by the EM Group University of Houston

2. Infinite uniform line charge
Example
Infinite uniform line charge
Find the electric field vector

3. Example (cont.)
Assume

4. Example (cont.)
Side view

5. Example (cont.)
Hence,
We then have

6. Example (cont.)
Summary

7. Note About Cylindrical Coordinates
Note: In cylindrical coordinates, the LHS is always:
Assumption:
Cylindrical Gaussian surface
Assumption

8. Example This example illustrates when Gauss’s Law is not useful.
E has more than one component !
E is not a function of only  !
Note:
Although Gauss’s law is still valid, it is not useful in helping us to solve the problem. We must use Coulomb’s law.
Finite uniform line charge

9. Infinite cylinder of non-uniform volume charge density
Example
Infinite cylinder of non-uniform volume charge density
Find the electric field vector everywhere
Note:
This problem would be very difficult to solve using Coulomb’s law!

10. Example (cont.)
(a)  < a so

11. Example (cont.) so
Side view

12. Example (cont.)
Hence so

13. Example (cont.)
(b)  > a so

14. See if you can prove this to yourself!
Example (cont.)
Hence, we have
Note:
Outside the cylinder, the electric field is the same as that coming from an equivalent line charge located on the z axis at the center.
See if you can prove this to yourself!

15. Example (cont.)
Summary or

16. Find the electric field vector everywhere
Example
Infinite sheet of uniform surface charge density
Find the electric field vector everywhere

17. Example (cont.)
Assume
Consider first z > 0

18. Example (cont.) so
Assume
We then have

19. Example (cont.) For the charge enclosed we have
Hence, from Gauss’s law we have so
We then also have:
Therefore

20. Example (cont.)
Summary

21. Example From superposition: (a) x > h (b) 0 < x < h
(c) x < 0
(b) < x < h

22. Example (cont.)
Choose:
(a) x > h
(b) < x < h
(c) x < 0

23. Ideal parallel-plate capacitor
Example (cont.)
Ideal parallel-plate capacitor
Metal plates
Note:
The metal plates support the charge, but they themselves do not produce an electric field.
0 < x < h

24. Infinite slab of uniform volume charge density
Example
Infinite slab of uniform volume charge density
Find the electric field vector everywhere

25. Example (cont.)
Assume
(since Ex (x) is a continuous function)

26. (as was done for the sheet of charge).
Example (cont.)
(a) x > d / 2
Alternative choice:
Another choice of Gaussian surface would be a symmetrical surface, symmetrical about x = 0
(as was done for the sheet of charge).

27. Example (cont.)
Note: If we define
then
(sheet formula)

28. Example (cont.)
(b) 0 < x < d / 2 r

29. Example (cont.) Summary Note:
In the second formula we had to introduce a minus sign, while in the third one we did not.