MIT Class: Sources of Magnetic Fields Creating Fields: Biot-Savart Experiment: Magnetic Fields Ampere’s Law

1. Workshop: Using Visualization in Teaching Introductory E&MAAPT National Summer Meeting, Edmonton, Alberta, Canada.Organizers: John Belcher, Peter Dourmashkin, Carolann Koleci, Sahana Murthy

2. MIT Class: Sources of Magnetic FieldsCreating Fields: Biot-SavartExperiment: Magnetic Fields Ampere’s Law 2

3. Magnetic Fields

4. Gravitational – Electric Fields Mass m Charge q (±) Create: Feel: Also saw… Dipole p Creates: Feels:

5. Magnetism – Bar Magnet Like poles repel, opposite poles attract

6. Demonstration:Magnetic Field Linesfrom Bar Magnet

7. Demonstration:Compass (bar magnet) in Magnetic Field Linesfrom Bar Magnet

8. Magnetic Field of Bar Magnet (1) A magnet has two poles, North (N) and South (S) (2) Magnetic field lines leave from N, end at S

9. Bar Magnets Are Dipoles! • Create Dipole Field • Rotate to orient with Field Is there magnetic “mass” or magnetic “charge?” NO! Magnetic monopoles do not exist in isolation

10. Magnetic Monopoles? Magnetic Dipole Electric Dipole -q q When cut: 2 monopoles (charges) When cut: 2 dipoles Magnetic monopoles do not exist in isolation Another Maxwell’s Equation! (2 of 4) Gauss’s Law Magnetic Gauss’s Law

11. PRS:B Field inside a Magnet

12. PRS: Magnetic Field Lines The picture shows the field lines outside a permanent magnet The field lines inside the magnet point: • Up • Down • Left to right • Right to left • The field inside is zero • I don’t know 15

13. PRS Answer: Magnetic Field Lines Answer: 1. They point up inside the magnet Magnetic field lines are continuous. E field lines begin and end on charges. There are no magnetic charges (monopoles) so B field lines never begin or end

14. Magnetic Field of the Earth Also a magnetic dipole! North magnetic pole located in southern hemisphere 14

15. Fields: Grav., Electric, Magnetic Mass m Charge q (±) No Magnetic Monopoles! Create: Feel: Dipole p Dipole m Create: Feel:

16. What is B? B is the magnetic field It has units of Tesla (T) This class & next: creating B fields Next two classes: feeling B fields

17. How Big is a Tesla? 5 x 10-5 T = 0.5 Gauss ~1 fT 3 T 18 T 150 T (pulsed) • Earth’s Field • Brain (at scalp) • Refrigerator Magnet • Inside MRI • Good NMR Magnet • Biggest in Lab • Biggest in Pulsars

18. How do we create fields?

19. What creates fields? Magnets – more about this later The Earth How’s that work?

20. Magnetic Field of the Earth Also a magnetic dipole! North magnetic pole located in southern hemisphere (for now)

21. What creates fields? Magnets – more about this later The Earth How’s that work? Moving charges!

22. Electric Field Of Point Charge An electric charge produces an electric field: : unit vector directed from q to P

23. Magnetic Field Of Moving Charge Moving charge with velocity v produces magnetic field: P unit vector directed from q to P permeability of free space

24. Recall:Cross Product

25. Notation Demonstration X X X X X X X X X X X X X X X X OUT of page “Arrow Head” INTO page “Arrow Tail”

26. Cross Product: Magnitude Computing magnitude of cross product A x B: area of parallelogram

27. Cross Product: Direction Right Hand Rule #1: For this method, keep your hand flat! 1) Put Thumb (of right hand) along A 2) Rotate hand so fingers point along B 3) Palm will point along C

28. Cross Product: Signs Cross Product is Cyclic (left column) Reversing A & B changes sign (right column)

29. PRS Questions:Right Hand Rule

30. B A PRS: Cross Product What is the direction of A x B given the following two vectors? • up • down • left • right • into page • out of page • Cross product is zero (so no direction) 15

31. B A PRS Answer: Cross Product Answer: 5. A x B points into the page Using your right hand, thumb along A, fingers along B, palm into page

32. B A PRS: Cross Product What is the direction of A x B given the following two vectors? • up • down • left • right • into page • out of page • Cross product is zero (so no direction) 15

33. B A PRS Answer: Cross Product Answer: 6. A x B points out of the page Using your right hand, thumb along A, fingers along B, palm out of page Also note from before, one vector flipped so result does too

34. Moving Continuous charge distributions:Currents & Biot-Savart ^

35. From Charges to Currents?

36. The Biot-Savart Law Current element of length ds carrying current I produces a magnetic field: (Shockwave)

37. The Right-Hand Rule #2

38. Animation:Field Generated by a Moving Charge

39. Demonstration:Field Generated by Wire

40. Example : Coil of Radius R Consider a coil with radius R and current I Find the magnetic field B at the center (P)

41. Example : Coil of Radius R Consider a coil with radius R and current I • 1) Think about it: • Legs contribute nothing • I parallel to r • Ring makes field into page • 2) Choose a ds • 3) Pick your coordinates • 4) Write Biot-Savart

42. Example : Coil of Radius R In the circular part of the coil… Biot-Savart:

43. Example : Coil of Radius R Consider a coil with radius R and current I

44. Example : Coil of Radius R • Notes: • This is an EASY Biot-Savart problem: • No vectors involved • This is what I would expect on exam

45. PRS Questions:B fields Generated by Currents

46. 15 PRS: Biot-Savart The magnetic field at P points towards the • +x direction • +y direction • +z direction • -x direction • -y direction • -z direction • Field is zero (so no direction)

47. PRS Answer: Biot-Savart Answer: 3. B(P) is in the +z direction (out of page) The vertical line segment contributes nothing to the field at P (it is parallel to the displacement). The horizontal segment makes a field out of the page.

48. PRS: Bent Wire The magnetic field at P is equal to the field of: • a semicircle • a semicircle plus the field of a long straight wire • a semicircle minus the field of a long straight wire • none of the above 15

49. PRS Answer: Bent Wire Answer: 2. Semicircle + infinite wire All of the wire makes B into the page. The two straight parts, if put together, would make an infinite wire. The semicircle is added to this to get the complete field

50. P I Group Problem:B Field from Coil of Radius R Consider a coil made of semi-circles of radii R and 2R and carrying a current I What is B at point P?