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

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

Week 05, Day 1 Magnetic Fields Class 12

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Week 05, Day 1 Recall: Cross Product Class 12

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Week 06, Day 1 Other Geometries 81 Class 14 81

Week 06, Day 1 Two Loops 82 Class 14 82

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

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

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

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

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

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

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

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

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

(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

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

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