Lesson 7 Gauss’s Law and Electric Fields

1. Lesson 7Gauss’s Law and Electric Fields

2. Today, we will: • learn the definition of a Gaussian surface • learn how to count the net number of field lines passing into a Gaussian surface • learn Gauss’s Law of Electricity • learn about volume, surface, and linear charge density • learn Gauss’s Law of Magnetism • show by Gauss’s law and symmetry that the electric field inside a hollow sphere is zero Class 18

3. Section 1Visualizing Gauss’s Law

4. Gaussian Surface A Gaussian surface is any closed surface surface that encloses a volume Gaussian surfaces include: balloons boxes tin cans Gaussian surfaces do not include: sheets of paper loops

5. Counting Field Lines To count field lines passing through Gaussian surfaces: Count +1 for every line that passes out of the surface. Count ─1 for every line that comes into the surface. +1 ─1

6. We have a +2 charge and a ─2 charge. Electric Field Lines

7. What is the net number of field lines passing through the Gaussian surface? Electric Field Lines

8. What is the net number of field lines passing through the Gaussian surface? Electric Field Lines +8

9. What is the net number of field lines passing through the Gaussian surface? Electric Field Lines

10. What is the net number of field lines passing through the Gaussian surface? Electric Field Lines +8

11. What is the net number of field lines passing through the Gaussian surface? Electric Field Lines

12. What is the net number of field lines passing through the Gaussian surface? Electric Field Lines ─8

13. What is the net number of field lines passing through the Gaussian surface? Electric Field Lines

14. What is the net number of field lines passing through the Gaussian surface? Electric Field Lines ─8

15. What is the net number of field lines passing through the Gaussian surface? Electric Field Lines

16. What is the net number of field lines passing through the Gaussian surface? Electric Field Lines 0

17. What is the net number of field lines passing through the Gaussian surface? Electric Field Lines

18. What is the net number of field lines passing through the Gaussian surface? Electric Field Lines 0

19. From the field lines coming out of this box, what can you tell about what’s inside? Electric Field Lines

20. The net charge inside must be +1 (if we draw 4 lines per unit of charge). Electric Field Lines

21. The net number of electric field lines passing through a Gaussian surface is proportional to the charge enclosed within the Gaussian surface. Gauss’s Law of Electricity

22. Section 2Charge Density

23. Charge Volume Charge Area Charge Length Volume: ρ = Surface: σ = Linear: λ = Charge Density

24. In general, charge density can vary with position. In this case, we can more carefully define density in terms of the charge in a very small volume at each point in space. The density then looks like a derivative: Charge Density You need to understand what we mean by this equation, but we won’t usually need to think of density as a derivative.

25. Section 3Gauss’s Law of Magnetism

26. If magnetic field lines came out from point sources like electric field lines, then we would have a law that said: The net number of magnetic field lines passing through a Gaussian surface is proportional to the magnetic charge inside. Gauss’s Law and Magnetic Field Lines N

27. But we have never found a magnetic monopole. - The thread model suggests that there is no reason we should expect to find a magnetic monopole as the magnetic field as we know it is only the result of moving electrical charges. - The field line model suggests that there’s no reason we shouldn’t find a magnetic monopole as the electric and magnetic fields are both equally fundamental. Gauss’s Law and Magnetic Field Lines

28. What characteristic would a magnetic monopole field have? Gauss’s Law and Magnetic Field Lines

29. What characteristic would a magnetic monopole field have? Gauss’s Law and Magnetic Field Lines

30. All known magnetic fields have field lines that form closed loops. So what can we conclude about the number of lines passing through a Gaussian surface? Gauss’s Law and Magnetic Field Lines

31. The net number of magnetic field lines passing through any Gaussian surface is zero. Gauss’s Law of Magnetism

32. Section 4Gauss’s Law and Spherical Symmetry

33. The charge density, ρ, can vary with r only. Below, we assume that the charge density is greatest near the center of a sphere. Spherically Symmetric Charge Distribution

34. Outside the distribution, the field lines will go radially outward and will be uniformly distributed. Spherically Symmetric Charge Distribution

35. The field is the same as if all the charge were located at the center of the sphere! Spherically Symmetric Charge Distribution

36. Now consider a hollow sphere of inside radius r with a spherically symmetric charge distribution. Inside a Hollow Sphere

37. There will be electric field lines outside the sphere and within the charged region. The field lines will point radially outward because of symmetry. But what about inside? Inside a Hollow Sphere

38. Draw a Gaussian surface inside the sphere. What is the net number of electric field lines that pass through the Gaussian surface? Inside a Hollow Sphere

39. The total number of electric field lines from the hollow sphere that pass through the Gaussian surface inside the sphere is zero because there is no charge inside. Inside a Hollow Sphere

40. 1. We could have some lines come in and go out again… How can we get zero net field lines? … but this violates symmetry!

41. 2. We could have some radial lines come in and other radial lines go out… How can we get zero net field lines? … but this violates symmetry, too!

42. How can we get zero net field lines? 3. Or we could just have no electric field at all inside the hollow sphere.

43. How can we get zero net field lines? 3. Or we could just have no electric field at all inside the hollow sphere. This is the only way it can be done!

44. The Electric Field inside a Hollow Sphere Conclusion: the static electric field inside a hollow charged sphere with a spherically symmetric charge distribution must be zero.

45. Today, we will: • learn how to use Gauss’s law and symmetry to find the electric field inside a spherical charge distribution • show that all the static charge on a conductor must reside on its outside surface • learn why cars are safe in lightning but cows aren’t Class 19

46. Electric field lines do not start or end outside charge distributions, but that can start or end inside charge distributions. Spherically Symmetric Charge Distribution

47. What is the electric field inside a spherically symmetric charge distribution? Spherically Symmetric Charge Distribution

48. Inside the distribution, it is difficult to draw field lines, as some field lines die out as we move inward. – We need to draw many, many field lines to keep the distribution uniform as we move inward. Spherically Symmetric Charge Distribution

49. But we do know that if we drew enough lines, the distribution would be radial and uniform in every direction, even inside the sphere. Spherically Symmetric Charge Distribution

50. Let’s draw a spherical Gaussian surface at radius r. Spherically Symmetric Charge Distribution r