1. Physics 014
Gauss’ Law

2. F P

3. Topics
Flux
Gaussian Surfaces

4. Flux
Consider an airstream of uniform velocity at a small square loop of area A.

5. Flux

6. Flux
Define Φ as the volume flow rate at which air flows through the loop

7. Flux

8. Flux

9. Flux For flux involving electrostatics, we must include vectors
Define an area vector as being a vector with magnitude equal to an area but with direction perpendicular to the plane of the area (pointing outward)

10. Flux

11. Flux of an Electric Field
Consider the flux through a surface divided into small squares
Take the electric field as constant over an individual square

12. Flux of Electric Field

13. Flux of Electric Field
We can then think of the flux of the electric field for the surface as

14. Flux of Electric Field In the limit as the areas become differentials
Circle denotes integral over a closed surface!

15. Flux of Electric Field
Example: A nonuniform electric field given by the following, pierces a cubical surface. What is the flux through the right face?

16. Flux of Electric Field

17. Flux of Electric Field

18. ???
The figure shows a cube of face area A immersed in a uniform field E that has the positive direction of the z axis. In terms of E and A, what is the flux through (a) the front face (xy plane) (b) the rear face (c) the top face, (d) the whole cube?

19. Gauss’ Law

20. Gauss’ Law

21. Flux of Electric Field
Example: For a cylindrical surface of radius R in electric field , what is the flux of the field through this closed surface?

22. Flux of Electric Field

23. Flux of Electric Field

24. Gauss’ Law
Gauss’ Law relates net flux through a closed surface to the net charge qenc enclosed by that surface

25. S1: flux out, +q
S2: flux in, -q
S3: no (net) flux, (net) charge
S4: no (net) flux, (net) charge

26. ???
The figure shows three situations in which a Gaussian cube sits in an electric field. The arrows and values indicate directions of field lines and magnitudes. In which situation does the cube enclose (a) a net positive charge, (b) a net negative charge, (c) a zero net charge?

27. Gauss’ Law
Example: The figure shows five charged lumps of plastic and a neutral coin. The Gaussian surface S is indicated. What is the net flux through S if q1=q4=+3.1nC, q2=q5=-5.9 nC, and q3=-3.1 nC?

28. Gauss’ Law

29. Gauss’ Law

30. Gauss’ Law and Coulomb’s Law

31. Gauss’ Law and Coulomb’s Law
E and A are in same direction. So by
symmetry, use only magnitudes
E has same magnitude everywhere on Gaussian
surface, so take out of integral

32. ???
There is a net flux, Φi , through a Gaussian sphere of radius r enclosing an isolated charged particle. Suppose the enclosing Gaussian surface is changed to (a) a larger Gaussian sphere, (b) a Gaussian cube with edge length equal to r, and (c) a Gaussian cube with edge length equal to 2r. In each case is the net flux through the new Gaussian surface greater than, less than, or equal to Φi ?