Lesson 8 Ampère’s Law and Differential Operators

1. Lesson 8Ampère’s Law and Differential Operators

2. Section 1Visualizing Ampère’s Law

3. Amperian Loop An Amperian loop is any closed loop Amperian loops include: a circle a square a rubber band Amperian loops do not include: a balloon a piece of string (with two ends)

4. Take a wire with current coming out of the screen. The Field Contour of a Wire

5. The field contour is made of half-planes centered on the wire. The Field Contour of a Wire

6. We draw arrows on each plane pointing in the direction of the magnetic field. The Field Contour of a Wire

7. We draw an Amperian loop around the wire. Amperian Loops

8. We wish to count the “net number” of field lines pierced by the Amperian loop. Amperian Loops

9. First, we put an arrow on the loop in an overall counterclockwise direction. Amperian Loops

10. To count the net number of surfaces pierced by Amperian loop, we add +1 when the loop is “in the direction’ of the plane and −1 when it is “opposite the direction” of the plane. Counting Surfaces Pierced +1 +1 +1 +1 +1 −1 +1 +1 −1 +1 +1 +1 +1 −1 +1 +1 +1 +1 −1 +1 +1 +1 +1 +1

11. Note there are “+1” appears 20 times and “-1” appears 4 times. Counting Surfaces Pierced +1 +1 +1 +1 +1 −1 +1 +1 −1 +1 +1 +1 +1 −1 +1 +1 +1 +1 −1 +1 +1 +1 +1 +1

12. The net number of surfaces pierced by the Amperian loop us therefore +16. Counting Surfaces Pierced +1 +1 +1 +1 +1 −1 +1 +1 −1 +1 +1 +1 +1 −1 +1 +1 +1 +1 −1 +1 +1 +1 +1 +1

13. What is the net number of surfaces pierced by each of these Amperian loops? Other Amperian Loops

14. What is the net number of surfaces pierced by each of these Amperian loops? Other Amperian Loops

15. What is the net number of surfaces pierced by each of these Amperian loops? Other Amperian Loops

16. The net numbers of surfaces pierced by each of these loops is 16. Other Amperian Loops

17. What is the net number of surfaces pierced by this Amperian loop? Other Amperian Loops

18. This time the net number of surfaces pierced by the loop is 0. Why? Other Amperian Loops

19. This is the same loop we saw earlier, but now only 8 surfaces are pierced, since there are only 8 surfaces extending outward from the wire. Other Wires.

20. There are 8 surfaces coming from the wire because the current through the wire is half as much as it was before. Other Wires.

21. The net number of perpendicular surfaces pierced by an Amperian loop is proportional to the current passing through the loop. Ampère’s Law

22. Always traverse the Amperian loop in a (generally) counterclockwise direction. • If the number of surfaces pierced N>0, the current comes out of the screen. • If N<0, the current goes into the screen. Sign Convention

23. Section 2Cylindrically Symmetric Current Density

24. There are three kinds of charge density (ρ,σ,λ) • There is one kind of current density (current/unit area) Current Density

25. The current passing through a small gate of area ΔA is Current Density small gate

26. The total current passing through the wire is the sum of the current passing through all small gates. Current Density small gates

27. The current density, j, can vary with r only. Below, we assume that the current density is greatest near the axis of the wire. Cylindrically Symmetric Current Distribution

28. Cylindrically Symmetric Current Distribution Outside the distribution, the field contour is composed of surfaces that are half planes, uniformly spaced.

29. Cylindrically Symmetric Current Distribution Inside the distribution, it is difficult to draw perpendicular surfaces, as some surfaces die out as we move inward. – We need to draw many, many surfaces to keep them equally spaced as we move inward.

30. Cylindrically Symmetric Current Distribution But we do know that if we draw enough surfaces, the distribution of the surfaces will be uniform, even inside the wire.

31. Cylindrically Symmetric Current Distribution Let’s draw a circular Amperian loop at radius r. r

32. Cylindrically Symmetric Current Distribution Now we split the wire into two parts – the part outside the Amperian loop and the part inside the Amperian loop. r r

33. Cylindrically Symmetric Current Distribution The total electric field at r will be the sum of the electric fields from the two parts of the wire. r r

34. The total number of perpendicular surfaces pierced by the Amperian loop is zero because there is no current passing through it. Inside a Hollow Wire r

35. 1. We could have all the surfaces pierced twice, one in the positive sense and one in the negative… How can we get zero net surfaces? … but this violates symmetry!

36. 2. We could have some surfaces oriented one way and some the other… How can we get zero net surfaces? … but this violates symmetry, too!

37. How can we get zero net surfaces? 3. Or we could just have no surfaces at all inside the hollow wire. This is the only way it can be done!

38. The Magnetic Field inside a Hollow Wire If the current distribution has cylindrical symmetry, the magnetic field inside a hollow wire must be zero.

39. Since the magnetic field inside a hollow wire is zero, the total magnetic field at a distance r from the center of a solid wire is the field of the “core,” the part of the wire within the Amperian loop. Cylindrically Symmetric Current Distribution r r

40. r Cylindrically Symmetric Current Distribution Outside the core, the magnetic field is the same as that of a thin wire that has the same current as the total current passing through the Amperian loop.

41. r Cylindrically Symmetric Current Distribution Inside a cylindrically symmetric current distribution, the magnetic field is:

42. Section 3Uniform Current Density

43. A wire of radius R with a uniform current distribution has a total charge i passing through it. What is the magnetic field at r < R ? Example: Uniform Current Distribution

44. r A wire of radius R with a uniform current distribution has a total charge i passing through it. What is the magnetic field at r < R ? Example: Uniform Current Distribution

45. r A wire of radius R with a uniform current distribution has a total charge i passing through it. What is the magnetic field at r < R ? Example: Uniform Current Distribution The current density is uniform, so:

46. r A wire of radius R with a uniform current distribution has a total charge i passing through it. What is the magnetic field at r < R ? Example: Uniform Current Distribution Therefore:

47. Example: Uniform Current Distribution

48. Example: Uniform Current Distribution

49. Example: Uniform Current Distribution

50. Example: Uniform Current Distribution