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Transcript
Workshop: Using Visualization in Teaching Introductory E&M AAPT National Summer Meeting, Edmonton, Alberta, Canada. Organizers: John Belcher, Peter Dourmashkin, Carolann Koleci, Sahana Murthy P17- 1 MIT Class: Feeling Magnetic Fields Magnetic Forces on Charges Magnetic Dipoles Experiment: Dipoles in B Fields P17- 2 The Biot-Savart Law Current element of length ds carrying current I produces a magnetic field: 0 I d s rˆ dB 2 4 r Moving charges are currents too… o q v x rˆ B 2 4 r P17- 3 Ampere’s Law: B d s 0 I enc . B Long Circular Symmetry I B (Infinite) Current Sheet X X X X X X X X X X X X X X B X X Solenoid = 2 Current Sheets X X X X X X X X X X X X Torus P17- 4 Review: Right Hand Rules 1. 2. 3. 4. Torque: Thumb = torque, fingers show rotation Create: Thumb = I, Fingers (curl) = B Feel: Thumb = I, Fingers = B, Palm = F Moment: Fingers (curl) = I, Thumb = Moment P17- 5 Demonstration: TV in Field P17- 6 How a CRT Works: It could… P17- 7 How a CRT Works: More Typical P17- 8 How a CRT Works P17- 9 Moving Charges Feel Magnetic Force FB q v B Magnetic force perpendicular both to: Velocity v of charge and magnetic field B P17- 10 Reminder: B Field Units Since FB q v B newton N N B Units 1 1 coulomb meter/seco nd C m s A m This is called 1 Tesla (T) 4 1 T = 10 Gauss (G) P17- 11 Putting it Together: Lorentz Force Charges Feel… FE qE FB q v B Electric Fields Magnetic Fields F q E vB This is the final word on the force on a charge P17- 12 Application: Velocity Selector What happens here? P17- 13 Velocity Selector Particle moves in a straight line when Fnet E q(E v B) 0 v B P17- 14 PRS Question: Hall Effect P17- 15 PRS: Hall Effect A conducting slab has current to the right. A B field is applied out of the page. Due to magnetic forces on the charge carriers, the bottom of the slab is at a higher electric potential than the top of the slab. B I V > V(Top) On the basis of this experiment, the sign of the charge carriers carrying the current in the slab is: 0% 1. Positive 0% 2. Negative 0% 3. Cannot be determined 4. I don’t know 0% 0 P17- 16 PRS Answer: Hall Effect Answer: 1. Here the charge carriers are positive B I V > V(Top) Look at the force on the carriers. If positive, they are flowing to the right, and F will be down. If negative they are flowing to the left and F will be down (don’t forget the sign of q!) So either way the force is down. But we know that the result is a higher potential at the bottom – positive charges are moving down. So the carriers are positive P17- 17 What Kind of Motion in Uniform B Field? P17- 18 Cyclotron Motion (1) r : radius of the circle mv 2 mv qvB r r qB (2) T : period of the motion 2 r 2 m T v qB (3) : cyclotron frequency v qB 2 f r m P17- 19 Collections of Charges: Current Carrying Wires P17- 20 Demonstration: Jumping Wire P17- 21 Magnetic Force on Current-Carrying Wire Current is moving charges, and we know that moving charges feel a force in a magnetic field P17- 22 Magnetic Force on Current-Carrying Wire FB qv B m charge B s charge mB s FB I L B P17- 23 PRS Question: Parallel Current Carrying Wires P17- 24 0 PRS: Parallel Wires Consider two parallel current carrying wires. With the currents running in the same direction, the wires are 0% 0% 0% 0% 0% 1. 2. 3. 4. 5. I I 1 2 attracted (likes attract?) repelled (likes repel?) pushed another direction not pushed – no net force I don’t know P17- 25 PRS Answer: Parallel Wires Answer: 1. The wires are attracted I1 creates a field into the page at I2. That makes a force on I2 to the left. X I I 1 2 I2 creates a field out of the page at I1. That makes a force on I1 to the right. P17- 26 Demonstration: Parallel & Anti-Parallel Currents P17- 27 Summary Magnetic Force FB qv B dFB Id s B FB I L B P17- 28 Can we understand why? Whether they attract or repel can be seen in the shape of the created B field (Animation) (Animation) P17- 29 Field Pressures and Tensions: A Way To Understand the qVxB Magnetic Force P17- 30 Tension and Pressures Transmitted by E and B E & B Fields: • Transmit tension along field direction (Field lines want to pull straight) • Exert pressure perpendicular to field (Field lines repel) P17- 31 Example of E Pressure/Tension (Animation) Positive charge in uniform (downward) E field Electric force on the charge is combination of 1. Pressure pushing down from top 2. Tension pulling down towards bottom P17- 32 Example of B Pressure/Tension (Animation) Positive charge moving out of page in uniform (downwards) B field. Magnetic force combines: 1. Pressure pushing from left 2. Tension pulling to right P17- 33 PRS Question: Field Strength P17- 34 PRS: Field Strength A B C 0% 0% 0% 0% Where is the pictured field the strongest? 1. A 2. B 3. C 4. I don’t know 0 P17- 35 PRS Answer: Field Strength A B C Answer: 3. The field is the strongest at C Line density is proportional to field strength P17- 36 Example of B Pressure/Tension (Animation) Both cases: repelling “pressure” arises from HIGH field strength HIGH energy density P17- 37 Loops of Current P17- 38 Group Problem: Current Loop Place rectangular current loop in uniform B field 1) What is the net force on this loop? 2) What is the net torque on this loop? 3) Describe the motion the loop makes ĵ k̂ î P17- 39 Torque on Rectangular Loop ˆ τ IABj A A nˆ ab nˆ : area vector nˆ kˆ , B=B ˆi k̂ ĵ x î τ IA B Familiar? No net force but there is a torque P17- 40 Magnetic Dipole Moment Define Magnetic Dipole Moment: μ IAnˆ IA Then: τ μB Analogous to τ pE t tends to align with B P17- 41 Animation: Another Way To Look At Torque External field connects to field of magnet and “pulls” the dipole into alignment P17- 42 Demonstration: Galvanometer P17- 43 Magnetic Dipole Moment μ IAnˆ IA P17- 44 PRS Question: Force on Magnetic Dipole P17- 45 PRS: Dipole in Field From rest, the coil above will: 0% 0% 0% 0% 0% 0% 0% 0% 1. 2. 3. 4. 5. 6. 7. 8. rotate clockwise, not move rotate counterclockwise, not move move to the right, not rotate move to the left, not rotate move in another direction, without rotating both move and rotate neither rotate nor move I don’t know :00P17- 46 PRS Answer: Dipole in Field Answer: 1. Coil will rotate clockwise (not move) No net force so no center of mass motion. BUT Magnetic dipoles rotate to align with external field (think compass) P17- 47 Dipoles don’t move??? This dipole rotates but doesn’t feel a net force But dipoles CAN feel force due to B. What’s up? P17- 48 Something New Dipoles in Non-Uniform Fields: Force P17- 49 Force on Magnetic Dipole? We Want to Know: What is the force on this dipole? P17- 50 PRS Question: Force on Magnetic Dipole P17- 51 PRS: Dipole in Field The current carrying coil above will feel a net force 0% 0% 0% 0% 1. 2. 3. 4. upwards downwards of zero I don’t know 0 P17- 52 PRS Answer: Dipole in Field Answer: 2. Feels downward force The I ds x B forces shown produce a net downward force P17- 53 Can just sum I ds x B forces Is there another way? P17- 54 Energy of Magnetic Dipole U Dipole -μ B This equation gives you a general way to think about what dipoles will do in B fields P17- 55 Magnetic Dipole Moments μ IAnˆ IA Generate: Feel: 1) Torque to align with external field 2) Forces as for bar magnets U Dipole -μ B P17- 56 Force on Magnetic Dipole Alternate Thought #1 Where does the dipole want to be? P17- 57 Think Using Energy U Dipole -μ B Where does dipole go to reduce its energy? Aligned dipoles seek high fields! Force here is down P17- 58 Force on Magnetic Dipole Alternate Thought #2 What makes the field pictured? P17- 59 Force on Magnetic Dipole N S N S Bar magnet below dipole, with N pole on top It is aligned with the dipole pictured, they attract! P17- 60 PRS Questions: Force on Dipole P17- 61 PRS: Dipole in Field 0 The current carrying coil above will feel a net force 0% 0% 0% 0% 1. 2. 3. 4. upwards downwards of zero I don’t know P17- 62 PRS Answer: Dipole in Field S N S N Answer: 2. The coil feels a force down Many ways to know this: I ds x B forces Energy (aligned seeks high B) Equivalent bar magnets P17- 63 PRS: Free Dipoles If a number of dipoles are randomly scattered through space, after a while they 0% 0% 0% 0% 1. 2. 3. 4. Attract (move together) Repel (move apart) Basically stay put I don’t know 0 P17- 64 PRS Answer: Free Dipoles Answer: 1. Free Dipoles Attract • • Torque on dipole aligns it with the local field Dipole then moves toward stronger field — closer to another dipole Shockwave P17- 65 Some Fun: Magnetic Levitation P17- 66 Put a Frog in a 16 T Magnet… For details: http://www.hfml.sci.kun.nl/levitate.html P17- 67 How does that work? First a BRIEF intro to magnetic materials P17- 68 Para/Ferromagnetism Applied external field B0 tends to align the atomic magnetic moments (unpaired electrons) P17- 69 Diamagnetism Everything is slightly diamagnetic. Why? More later. If no unpaired electrons then this effect typically dominates. P17- 70 Back to Levitation P17- 71 Levitating a Diamagnet N S S N 1) Create a strong field (with a field gradient!) 2) Looks sort of like dipole field 3) Toss in a frog (diamagnet) 4) Looks like a bar magnet pointing opposite the field 5) Seeks lower field (force up) which balances gravity Most importantly, in a certain region it is stable: Restoring force always towards the center P17- 72 Using B to Levitate For details: http://www.hfml.sci.kun.nl/levitate.html P17- 73 Using B to Levitate For details: http://www.hfml.sci.kun.nl/levitate.html P17- 74 Using B to Levitate For details: http://www.hfml.sci.kun.nl/levitate.html P17- 75 Using B to Levitate For details: http://www.hfml.sci.kun.nl/levitate.html P17- 76 Demonstration: Levitating Magnet over Superconductor P17- 77 Perfect Diamagnetism: “Magnetic Mirrors” N S S N P17- 78 Perfect Diamagnetism: “Magnetic Mirrors” N S S N No matter what the angle, it floats -- STABILITY P17- 79 Using B to Levitate For details: http://www.hfml.sci. kun.nl/levitate.html P17- 80 Levitate Magnet with your Fingers? P17- 81 Well… and a lifting magnet N S Why need diamagnetic stabilization? 1) Magnet seeks STRONG field, wants to snap up to lifter 2) Downward oscillation will move it to region where field gradient is too weak to lift it Diamagnetic sheets above, below prevent these effects, since they repel the floating magnet P17- 82 Experiment 4: Magnetic Forces on Dipoles This is a little tricky. We will lead you through with lots of PRS questions P17- 83 First: Set up current supply • Open circuit (disconnect a lead) • Turn current knob full CCW (off) • Increase voltage to ~12 V § This will act as a protection: V<12 V • Reconnect leads in Helmholtz mode • Increase current to ~1 A P17- 84 Field Profiles VERY UNIFORM! Single Coil Helmholtz Anti-Helmholtz ZERO FIELD! P17- 85 PRS Prediction: Dipole in Helmholtz P17- 86 :00 PRS: Dipole in Helmholtz A randomly aligned dipole at the center of a Helmholtz coil will feel: 0% 1. a force but not a torque 0% 2. a torque but not a force 0% 3. both a torque and a force 4. neither force nor torque 0% P17- 87 PRS Answer: Dipole in Helmholtz Answer: 2. a torque but not a force The Helmholtz coil makes a UNIFORM FIELD Dipole feels only torque (need gradient for F) P17- 88 Next: Dipole in Helmholtz (Q1-2) • • • • • Set in Helmholtz Mode (~1 A) Turn off current Put dipole in center (0 on scale) Randomly align using bar magnet Turn on current What happens? P17- 89 PRS Prediction: Reverse Helmholtz P17- 90 0 PRS: Reverse Helmholtz Using aligned dipole, flip the field. Ideally the dipole will feel: 0% 0% 0% 0% 1. 2. 3. 4. a force but not a torque a torque but not a force both a torque and a force neither force nor torque P17- 91 PRS Answer: Reverse Helmholtz Answer: 4. IDEALLY neither force nor torque The dipole is exactly anti-aligned, so the torque is 0. Still uniform field means still no force. P17- 92 Next: Reverse Helmholtz (Q3) Starting from end of previous (aligned dipole at center) • Turn off current • VERY CAREFULLY (don’t bump!) Reverse leads at power supply • Turn on current What happens? P17- 93 PRS Predictions: Moving in Helmholtz P17- 94 PRS: Moving in Helmholtz 0 When moving through the above field profile, a dipole will: 0% 1. Never rotate 0% 2. Rotate once 3. Rotate twice 0% P17- 95 PRS Answer: Moving in Helmholtz Answer: 1. The dipole will never rotate The dipole is always aligned with the field so it will never rotate P17- 96 PRS: Moving in Helmholtz 0 When pulling the dipole through the above field profile, the spring stretch direction will: 0% 0% 0% 0% 1. 2. 3. 4. Always be the same Change once Change twice Change three times P17- 97 PRS Answer: Moving in Helmholtz Answer: 2. The direction will change once The dipole always wants to be at the peak field, so when below it the force is up, when above it the force is down. P17- 98 Next: Moving in Helmholtz (Q4-5) • • • • Keep in Helmholtz Mode (~1 A) Lower dipole to bottom Randomly align (is it possible?) Slowly & smoothly raise to well above What happens (torque? force?) P17- 99 PRS Prediction: Dipole in Anti-Helmholtz P17-100 PRS: Anti-Helmholtz :00 A randomly aligned dipole at the center of an AntiHelmholtz coil will feel: 0% 0% 0% 0% 1. 2. 3. 4. a force but not a torque a torque but not a force both a torque and a force neither force nor torque P17-101 PRS Answer: Anti-Helmholtz Answer: 1. A force but not a torque No field no torque Field gradient force P17-102 Next: Dipole in Anti-Helmholtz (Q6-7) • • • • • Set in Anti-Helmholtz Mode (~2 A) Turn off current Put dipole in center (0 on scale) Randomly align using bar magnet Turn on current What happens? P17-103 PRS Predictions: Moving in Anti-Helmholtz P17-104 PRS: Moving in Anti-Helmholtz 0 When moving through the above field profile, a dipole will: 0% 1. Never rotate 0% 2. Rotate once (at sign change) 3. Rotate twice (at slope changes) 0% P17-105 PRS Answer: Moving in Anti-HH Answer: 2. Dipole rotates once at sign change The dipole always wants to align with the field so when it crosses through zero it will rotate P17-106 PRS: Moving in Helmholtz :00 When pulling the dipole through the above field profile, the spring stretch direction will: 0% 0% 0% 0% 1. 2. 3. 4. Always be the same Change once Change twice Change three times P17-107 PRS Answer: Moving in Anti-HH F F F F Answer: 4. Force direction changes 3 times The dipole always wants to seek the strongest field, so the force reverses 3 times P17-108 Next: Moving in Anti-Helmholtz (Q8-9) • • • • Keep in Anti-Helmholtz Mode (~2 A) Lower dipole to bottom Randomly align (is it possible?) Slowly & smoothly raise to well above What happens (torque? force?) P17-109 Moving in Anti-Helmholtz (Q8-9) Force 0, then flips Where does it want to be? F F Bottom F Field reverses, so does dipole Top F NOTE: Field Up/Down Motion Up/Down Force 0, then flips P17-110 PRS Questions: Force from Single Coil Fields from Coils P17-111 PRS: Single Coil A 0 B C A field-aligned dipole located as pictured feels forces: 0% 0% 0% 0% 0% 1. 2. 3. 4. 5. FA > F B > F C FA > F B ~ F C FB > F A ~ F C FA ~ F B ~ F C No force, only a torque P17-112 PRS Answer: Single Coil A B C Answer: 3. FB > FA ~ FC The force goes like the slope of the field. It is ~ 0 at A & C, non-zero at B. P17-113 PRS: Current Carrying Coils :00 The above coils have 0% 1. parallel currents that attract 0% 2. parallel currents that repel 3. opposite currents that attract 0% 4. opposite currents that repel 0% P17-114 PRS Answer: I Carrying Coils Answer: 4. Opposite currents that repel Look at the field lines at the edge between the coils. They are jammed in, want to push out. Also, must be in opposite directions P17-115 Force on Dipole from Dipole: Anti-Parallel Alignment P17-116 Force on Dipole from Dipole: Parallel Alignment P17-117 Applications P17-118 Speakers P17-119 Speakers P17-120 DC Motor P17-121