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

2. Week 13, Day 1
MIT Class: Sources of Magnetic Fields Creating Fields: Biot-Savart Experiment: Magnetic Fields Ampere’s Law 2
Class 30 2

3. Week 05, Day 1
Magnetic Fields
Class 12

4. Gravitational – Electric Fields
Week 05, Day 1
Gravitational – Electric Fields
Mass m Charge q (±)
Create:
Feel:
Also saw… Dipole p
Creates:
Feels:
Class 12

5. Magnetism – Bar Magnet Like poles repel, opposite poles attract
Week 05, Day 1
Magnetism – Bar Magnet
Like poles repel, opposite poles attract
Class 12

6. Demonstration: Magnetic Field Lines from Bar Magnet
Week 05, Day 1
Demonstration: Magnetic Field Lines from Bar Magnet
Class 12

7. Week 05, Day 1
Demonstration: Compass (bar magnet) in Magnetic Field Lines from Bar Magnet
Class 12

8. Magnetic Field of Bar Magnet
Week 05, Day 1
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
Class 12

9. Bar Magnets Are Dipoles!
Week 05, Day 1
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
Class 12

10. Magnetic Monopoles? Electric Dipole Magnetic Dipole -q q When cut:
Week 05, Day 1
Magnetic Monopoles?
Electric Dipole
Magnetic 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
Class 12

11. PRS: B Field inside a Magnet
Week 05, Day 1
PRS: B Field inside a Magnet
Class 12

12. PRS: Magnetic Field Lines
Week 07, Day 1
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
Class 15

13. PRS Answer: Magnetic Field Lines
Week 07, Day 1
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
Class 15

14. Magnetic Field of the Earth
Week 06, Day 1
Magnetic Field of the Earth
Also a magnetic dipole!
North magnetic pole located in southern hemisphere 14
Class 14 14

15. Fields: Grav., Electric, Magnetic
Week 05, Day 1
Fields: Grav., Electric, Magnetic
Mass m Charge q (±)
No Magnetic Monopoles!
Create:
Feel:
Dipole p Dipole m
Create:
Feel:
Class 12

16. What is B? B is the magnetic field It has units of Tesla (T)
Week 05, Day 1
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
Class 12

17. How Big is a Tesla? Earth’s Field Brain (at scalp) Refrigerator Magnet
Week 05, Day 1
How Big is a Tesla?
Earth’s Field
Brain (at scalp)
Refrigerator Magnet
Inside MRI
Good NMR Magnet
Biggest in Lab
Biggest in Pulsars
5 x 10-5 T = 0.5 Gauss
~1 fT
3 T
18 T
150 T (pulsed)
Class 12

18. Week 05, Day 1
How do we create fields?
Class 12

19. What creates fields? Magnets – more about this later
Week 05, Day 1
What creates fields?
Magnets – more about this later
The Earth How’s that work?
Class 12

20. Magnetic Field of the Earth
Week 05, Day 1
Magnetic Field of the Earth
Also a magnetic dipole!
North magnetic pole located in southern hemisphere
(for now)
Class 12

21. What creates fields? Magnets – more about this later
Week 05, Day 1
What creates fields?
Magnets – more about this later
The Earth How’s that work?
Moving charges!
Class 12

22. Electric Field Of Point Charge
Week 05, Day 1
Electric Field Of Point Charge
An electric charge produces an electric field:
: unit vector directed from q to P
Class 12

23. Magnetic Field Of Moving Charge
Week 05, Day 1
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
Class 12

24. Week 05, Day 1
Recall: Cross Product
Class 12

25. Notation Demonstration
Week 05, Day 1
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”
Class 12

26. Cross Product: Magnitude
Week 05, Day 1
Cross Product: Magnitude
Computing magnitude of cross product A x B:
area of parallelogram
Class 12

27. Cross Product: Direction
Week 05, Day 1
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
Class 12

28. Cross Product: Signs Cross Product is Cyclic (left column)
Week 05, Day 1
Cross Product: Signs
Cross Product is Cyclic (left column)
Reversing A & B changes sign (right column)
Class 12

29. PRS Questions: Right Hand Rule
Week 05, Day 1
PRS Questions: Right Hand Rule
Class 12

30. PRS: Cross Product B A up down left right into page out of page
Week 07, Day 1 15
PRS: Cross Product
What is the direction of A x B given the following two vectors? A B up
down
left
right
into page
out of page
Cross product is zero (so no direction)
Class 15

31. PRS Answer: Cross Product
Week 07, Day 1
PRS Answer: Cross Product
Answer: 5. A x B points into the page A B
Using your right hand, thumb along A, fingers along B, palm into page
Class 15

32. PRS: Cross Product B A up down left right into page out of page
Week 07, Day 1
PRS: Cross Product
What is the direction of A x B given the following two vectors? A B up
down
left
right
into page
out of page
Cross product is zero (so no direction) 15
Class 15

33. PRS Answer: Cross Product
Week 07, Day 1
PRS Answer: Cross Product
Answer: 6. A x B points out of the page A B
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
Class 15

34. Continuous charge distributions: Currents & Biot-Savart
Week 05, Day 1
Moving
Continuous charge distributions: Currents & Biot-Savart ^
Class 12

35. From Charges to Currents?
Week 05, Day 1
From Charges to Currents?
Class 12

36. Week 05, Day 1
The Biot-Savart Law
Current element of length ds carrying current I produces a magnetic field:
(Shockwave)
Class 12

37. Week 05, Day 1
The Right-Hand Rule #2
Class 12

38. Animation: Field Generated by a Moving Charge
Week 05, Day 1
Animation: Field Generated by a Moving Charge
Class 12

39. Demonstration: Field Generated by Wire
Week 05, Day 1
Demonstration: Field Generated by Wire
Class 12

40. Example : Coil of Radius R
Week 05, Day 1
Example : Coil of Radius R
Consider a coil with radius R and current I
Find the magnetic field B at the center (P)
Class 12

41. Example : Coil of Radius R
Week 05, Day 1
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
Class 12

42. Example : Coil of Radius R
Week 05, Day 1
Example : Coil of Radius R
In the circular part of the coil…
Biot-Savart:
Class 12

43. Example : Coil of Radius R
Week 05, Day 1
Example : Coil of Radius R
Consider a coil with radius R and current I
Class 12

44. Example : Coil of Radius R
Week 05, Day 1
Example : Coil of Radius R
Notes:
This is an EASY Biot-Savart problem:
No vectors involved
This is what I would expect on exam
Class 12

45. PRS Questions: B fields Generated by Currents
Week 05, Day 1
PRS Questions: B fields Generated by Currents
Class 12

46. PRS: Biot-Savart 15 The magnetic field at P points towards the
Week 07, Day 1
PRS: Biot-Savart 15
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)
Class 15

47. PRS Answer: Biot-Savart
Week 07, Day 1
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.
Class 15

48. PRS: Bent Wire The magnetic field at P is equal to the field of: 15
Week 07, Day 1
PRS: Bent Wire 15
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
Class 15

49. PRS Answer: Bent Wire Answer: 2. Semicircle + infinite wire
Week 07, Day 1
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
Class 15

50. Group Problem: B Field from Coil of Radius R
Week 05, Day 1
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 P I
What is B at point P?
Class 12

51. Group Problem: B Field from Coil of Radius R
Week 05, Day 1
Group Problem: B Field from Coil of Radius R
Consider a coil with radius R and carrying a current I
WARNING:
This is much harder than the previous problem. Why??
What is B at point P?
Class 12

52. Experiment: Magnetic Fields: Bar Magnets & Wire Coils
Week 06, Day 1
Experiment: Magnetic Fields: Bar Magnets & Wire Coils 52
Class 14 52

53. PRS Question: Part I: B Field from Bar Magnet
Week 06, Day 1
PRS Question: Part I: B Field from Bar Magnet 53
Class 14 53

54. PRS: Bar Magnet B Field Up Down Right Left Up & right Up & left
Week 07, Day 2
Thinking of your map of the B field lines from part 1, assume that your magnet and compass were on the table in the orientation shown.
The red end of the compass points: Up
Down
Right
Left
Up & right
Up & left
Down & right
Down & left N S ? 54
Class 16 54

55. PRS Answer: Bar Magnet B Field
Week 07, Day 2
PRS Answer: Bar Magnet B Field
Answer: 7. Down & right N S
If you only had to consider the bar magnet (for example, if you were very close to it) the compass would point to the right. But the Earth’s magnetic field (pointing toward geographic North) pulls the field down. 55
Class 16 55

56. Visualization: Bar Magnet & Earth’s Magnetic Field
Week 06, Day 1
Visualization: Bar Magnet & Earth’s Magnetic Field 56
Class 14 56

57. PRS Question: Part 3: B Field from Helmholtz
Week 06, Day 1
PRS Question: Part 3: B Field from Helmholtz 57
Class 14 57

58. PRS: Helmholtz
Week 07, Day 2
Identify the three field profiles that you measured as Single (Sgl), Helmholtz (Hh) or Anti-Helmholtz (A-H): A B C
The curves, A, B & C are respectively:
Sgl, Hh, A-H
Hh, A-H, Sgl
A-h, Sgl, Hh
Sgl, A-H, Hh
A-H, Hh, Sgl
Hh, Sgl, A-H 58
Class 16 58

59. PRS Answer: Helmholtz Answer: 6. Helmholtz, Single, Anti-Helmholtz
Week 07, Day 2
PRS Answer: Helmholtz
Answer: 6. Helmholtz, Single, Anti-Helmholtz
Note that the Helmholtz mode creates a very uniform field near the center while the field from the Anti-Helmholtz is zero at the center. The single coil peaks at the coil’s center. 59
Class 16 59

60. Last Time: Creating Magnetic Fields: Biot-Savart
Week 06, Day 1
Last Time: Creating Magnetic Fields: Biot-Savart 60
Class 14 60

61. Week 06, Day 1
The Biot-Savart Law
Current element of length ds carrying current I produces a magnetic field:
Moving charges are currents too… 61
Class 14 61

62. Today: 3rd Maxwell Equation: Ampere’s Law
Week 06, Day 1
Today: 3rd Maxwell Equation: Ampere’s Law
Analog (in use) to Gauss’s Law 62
Class 14 62

63. Week 06, Day 1
Gauss’s Law – The Idea
The total “flux” of field lines penetrating any of these surfaces is the same and depends only on the amount of charge inside 63
Class 14 63

64. Week 06, Day 1
Ampere’s Law: The Idea
In order to have a B field around a loop, there must be current punching through the loop 64
Class 14 64

65. Ampere’s Law: The Equation
Week 06, Day 1
Ampere’s Law: The Equation
The line integral is around any closed contour bounding an open surface S.
Ienc is current through S: 65
Class 14 65

66. PRS Questions: Ampere’s Law
Week 06, Day 1
PRS Questions: Ampere’s Law 66
Class 14 66

67. PRS: Ampere’s Law a positive number a negative number zero
Week 07, Day 2
PRS: Ampere’s Law
Integrating B around the loop shown gives us:
a positive number
a negative number
zero
:15 67
Class 16 67

68. PRS Answer: Ampere’s Law
Week 07, Day 2
PRS Answer: Ampere’s Law
Answer: 3. Total penetrating current is zero, so 68
Class 16 68

69. PRS: Ampere’s Law a positive number a negative number zero
Week 07, Day 2
PRS: Ampere’s Law
Integrating B around the loop shown gives us:
a positive number
a negative number
zero 15 69
Class 16 69

70. PRS Answer: Ampere’s Law
Week 07, Day 2
PRS Answer: Ampere’s Law
Answer: 2.
Net penetrating current is out of the page, so field is counter-clockwise (opposite path) 70
Class 16 70

71. infinite current sheet
Week 06, Day 1
Biot-Savart vs. Ampere
Biot-Savart Law
general
current source
ex: finite wire
wire loop
Ampere’s law
symmetric
ex: infinite wire
infinite current sheet 71
Class 14 71

72. Week 06, Day 1
Applying Ampere’s Law
Identify regions in which to calculate B field Get B direction by right hand rule
Choose Amperian Loops S: Symmetry B is 0 or constant on the loop!
Calculate
Calculate current enclosed by loop S
Apply Ampere’s Law to solve for B 72
Class 14 72

73. Always True, Occasionally Useful
Week 06, Day 1
Always True, Occasionally Useful
Like Gauss’s Law,
Ampere’s Law is always true
However, it is only useful for calculation in certain specific situations, involving highly symmetric currents.
Here are examples… 73
Class 14 73

74. Example: Infinite Wire
Week 06, Day 1
Example: Infinite Wire I
A cylindrical conductor has radius R and a uniform current density with total current I
Find B everywhere
Two regions:
(1) outside wire (r ≥ R)
(2) inside wire (r < R) 74
Class 14 74

75. Ampere’s Law Example: Infinite Wire
Week 06, Day 1
Ampere’s Law Example: Infinite Wire I B I
Amperian Loop:
B is Constant & Parallel
I Penetrates 75
Class 14 75

76. Example: Infinite Wire
Week 06, Day 1
Example: Infinite Wire
Region 1: Outside wire (r ≥ R)
Cylindrical symmetry 
Amperian Circle
B-field counterclockwise 76
Class 14 76

77. Example: Infinite Wire
Week 06, Day 1
Example: Infinite Wire
Region 2: Inside wire (r < R)
Could also say: 77
Class 14 77

78. Example: Infinite Wire
Week 06, Day 1
Example: Infinite Wire 78
Class 14 78

79. Group Problem: Non-Uniform Cylindrical Wire
Week 06, Day 1
Group Problem: Non-Uniform Cylindrical Wire I
A cylindrical conductor has radius R and a non-uniform current density with total current:
Find B everywhere 79
Class 14 79

80. Applying Ampere’s Law In Choosing Amperian Loop:
Week 06, Day 1
Applying Ampere’s Law
In Choosing Amperian Loop:
Study & Follow Symmetry
Determine Field Directions First
Think About Where Field is Zero
Loop Must
Be Parallel to (Constant) Desired Field
Be Perpendicular to Unknown Fields
Or Be Located in Zero Field 80
Class 14 80

81. Week 06, Day 1
Other Geometries 81
Class 14 81

82. Week 06, Day 1
Two Loops 82
Class 14 82

83. Two Loops Moved Closer Together
Week 06, Day 1
Two Loops Moved Closer Together 83
Class 14 83

84. Week 06, Day 1
Multiple Wire Loops 84
Class 14 84

85. Multiple Wire Loops – Solenoid
Week 06, Day 1
Multiple Wire Loops – Solenoid 85
Class 14 85

86. Demonstration: Long Solenoid
Week 06, Day 1
Demonstration: Long Solenoid 86
Class 14 86

87. Magnetic Field of Solenoid
Week 06, Day 1
Magnetic Field of Solenoid
Horiz.
comp.
cancel
loosely wound
tightly wound
For ideal solenoid, B is uniform inside & zero outside 87
Class 14 87

88. Magnetic Field of Ideal Solenoid
Week 06, Day 1
Magnetic Field of Ideal Solenoid
Using Ampere’s law: Think! 88
Class 14 88

89. Group Problem: Current Sheet
Week 06, Day 1
Group Problem: Current Sheet y
A sheet of current (infinite in the y & z directions, of thickness 2d in the x direction) carries a uniform current density:
Find B for x > 0 89
Class 14 89

90. Ampere’s Law: Infinite Current Sheet
Week 06, Day 1
Ampere’s Law: Infinite Current Sheet B I B
Amperian Loops:
B is Constant & Parallel OR Perpendicular OR Zero
I Penetrates 90
Class 14 90

91. Solenoid is Two Current Sheets
Week 06, Day 1
Solenoid is Two Current Sheets
Field outside current sheet should be half of solenoid, with the substitution:
This is current per unit length (equivalent of l, but we don’t have a symbol for it) 91
Class 14 91

92. (Infinite) Current Sheet
Week 06, Day 1
Ampere’s Law: I B
Long
Circular
Symmetry
(Infinite) Current Sheet X B X =
2 Current Sheets
Solenoid
Torus 92
Class 14 92

93. Week 06, Day 1
Brief Review Thus Far… 93
Class 14 93

94. Maxwell’s Equations (So Far)
Week 06, Day 1
Maxwell’s Equations (So Far) 94
Class 14 94