Week #3

1. Week #3 Gauss’ Law September 7, 2005

2. What’s up Doc?? • At this moment I do not have quiz grades unless I get them at the last minute. • There was a short WebAssign due and there is a rather long set of problems also due next Monday. • First EXAM is on Friday the 16th. • May be pushed out a week because I will not be here on the 23rd so it would be a good test date. • Material will be the same. • Will decide this week.

3. Gauss’s Law http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss 1777-1855

4. Recall what we done so far? • We were given Coulomb’s Law • We defined the electric field. • Calculated the Electric Field given a distribution of charges using Coulomb’s Law. (Units: N / C)

5. A Question: • Given the magnitude and direction of the Electric Field at a point, can we determine the charge distribution that created the field? • Is it Unique? • Question … given the Electric Field at a number of points, can we determine the charge distribution that caused it? • How many points must we know??

6. Another QUESTION: Solid Surface Given the electric field at EVERY point on a closed surface, can we determine the charges that caused it??

7. The Answer … The answer to the question is buried in something called GAUSS’ LAW or Gauss’s Law or The Law Of Gauss or The Chapter that Everybody hates!

8. The “Area Vector” • Consider a small area. • It’s orientation can be described by a vector NORMAL to the surface. • We usually define the unit normal vector n. • If the area is FLAT, the area vector is given by An, where A is the area. • A is usually a differential area of a small part of a general surface that is small enough to be considered flat.

9. Area Vector Area

10. q The normal component of a vector The normal vector to a closed surface is DEFINED as positive if it points OUT of the surface. Remember this definition!

11. E DA ENORMAL ANOTHER DEFINITION:Element of Flux through a surface DF=|ENORMAL| x |DA| (a scalar)

12. “Element” of Flux of a vector E leaving a surface n is a unit OUTWARD pointing vector.

13. q This flux was LEAVING the closed surface

14. Definition of TOTAL FLUX through a surface

15. Total Flux • F=Flux leaving the surface • It is a SCALAR • You can only do this integration explicitly for “well behaved” surfaces and fields.

16. VisualizingFlux n is the OUTWARD pointing unit normal.

17. Definition: A Gaussian Surface Any closed surface that is near some distribution of charge

18. Remember Component of E perpendicular to surface. This is the flux passing through the surface and n is the OUTWARD pointing unit normal vector! n E q q A

19. Flux is -EL2 ExampleCube in a UNIFORM Electric Field Flux is EL2 E area L Note sign E is parallel to four of the surfaces of the cube so the flux is zero across these because E is perpendicular to A and the dot product is zero. Total Flux leaving the cube is zero

20. Simple Example r q

21. Gauss’ Law Flux is total EXITING the Surface. n is the OUTWARD pointing unit normal. q is the total charge ENCLOSED by the Gaussian Surface.

22. Simple ExampleUNIFORM FIELD LIKE BEFORE E No Enclosed Charge A A E E

23. Q L Line of Charge

24. Line of Charge From SYMMETRY E is Radial and Outward

25. What is a Cylindrical Surface?? Ponder

26. Drunk Looking at A Cylinder from its END Circular Rectangular

27. Infinite Sheet of Charge +s h E cylinder We got this same result from that ugly integration!

28. Materials • Conductors • Electrons are free to move. • In equilibrium, all charges are a rest. • If they are at rest, they aren’t moving! • If they aren’t moving, there is no net force on them. • If there is no net force on them, the electric field must be zero. • THE ELECTRIC FIELD INSIDE A CONDUCTOR IS ZERO!

29. More on Conductors • Charge cannot reside in the volume of a conductor because it would repel other charges in the volume which would move and constitute a current. This is not allowed. • Charge can’t “fall out” of a conductor.

30. Isolated Conductor Electric Field is ZERO in the interior of a conductor. Gauss’ law on surface shown Also says that the enclosed Charge must be ZERO. Again, all charge on a Conductor must reside on The SURFACE.

31. Charged Conductors Charge Must reside on the SURFACE - - E=0 - - E - s Very SMALL Gaussian Surface

32. Charged Isolated Conductor • The ELECTRIC FIELD is normal to the surface outside of the conductor. • The field is given by: • Inside of the isolated conductor, the Electric field is ZERO. • If the electric field had a component parallel to the surface, there would be a current flow!

33. Isolated (Charged) Conductor with a HOLE in it. Because E=0 everywhere inside the surface. So Q (total) =0 inside the hole Including the surface.

34. A Spherical Conducting Shell withA Charge Inside. A Thinker!

35. Insulators • In an insulator all of the charge is bound. • None of the charge can move. • We can therefore have charge anywhere in the volume and it can’t “flow” anywhere so it stays there. • You can therefore have a charge density inside an insulator. • You can also have an ELECTRIC FIELD in an insulator as well.

36. E O r Example – A Spatial Distribution of charge. Uniform charge density = r = charge per unit volume (Vectors) A Solid SPHERE

37. Outside The Charge R r O E Old Coulomb Law!

38. Graph E r R

39. s s ++++++++ ++++++++ A A E Charged Metal Plate E E is the same in magnitude EVERYWHERE. The direction is different on each side.

40. Apply Gauss’ Law s s ++++++++ ++++++++ A A E Same result!

41. Negatively ChargedISOLATED Metal Plate -s -s • - E is in opposite direction but Same absolute value as before

42. Bring the two plates together +s1 -s1 +s1 -s1 A B e e As the plates come together, all charge on B is attracted To the inside surface while the negative charge pushes the Electrons in A to the outside surface. This leaves each inner surface charged and the outer surface Uncharged. The charge density is DOUBLED.

43. Result is ….. +2s1 -2s1 A +s -s E=0 E=0 E B e e

44. VERY POWERFULL IDEA • Superposition • The field obtained at a point is equal to the superposition of the fields caused by each of the charged objects creating the field INDEPENDENTLY. Remember It!!

45. Problem #1Trick Question Consider a cube with each edge = 55cm. There is a 1.8 mC charge In the center of the cube. Calculate the total flux exiting the cube. NOTE: INDEPENDENT OF THE SHAPE OF THE SURFACE! Easy, yes??

46. Problem #2(15 from text) Note: the problem is poorly stated in the text. Consider an isolated conductor with an initial charge of 10 mC on the Exterior. A charge of +3mC is then added to the center of a cavity. Inside the conductor. (a) What is the charge on the inside surface of the cavity? (b) What is the final charge on the exterior of the cavity? +10 mC initial +3 mC added

47. a m,q both given as is a +s Another Problem from the book Charged Sheet Gaussian Surface

48. -2 a m,q both given as is a a T qE +s mg Free body diagram

49. -3 (all given)

50. a A VERY Hard One! A uniformly charged VOLUME With volume charge density=r What is E at the end of the vector? VECTOR!!! (We already did this)