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

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

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**Workshop: Using Visualization in Teaching Introductory**E&MAAPT National Summer Meeting, Edmonton, Alberta, Canada.Organizers: John Belcher, Peter Dourmashkin, Carolann Koleci, Sahana Murthy**MIT Class: Sources of Magnetic FieldsCreating Fields:**Biot-SavartExperiment: Magnetic Fields Ampere’s Law 2**Gravitational – Electric Fields**Mass m Charge q (±) Create: Feel: Also saw… Dipole p Creates: Feels:**Magnetism – Bar Magnet**Like poles repel, opposite poles attract**Demonstration:Compass (bar magnet) in Magnetic Field**Linesfrom Bar Magnet**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**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**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**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**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**Magnetic Field of the Earth**Also a magnetic dipole! North magnetic pole located in southern hemisphere 14**Fields: Grav., Electric, Magnetic**Mass m Charge q (±) No Magnetic Monopoles! Create: Feel: Dipole p Dipole m Create: Feel:**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**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**What creates fields?**Magnets – more about this later The Earth How’s that work?**Magnetic Field of the Earth**Also a magnetic dipole! North magnetic pole located in southern hemisphere (for now)**What creates fields?**Magnets – more about this later The Earth How’s that work? Moving charges!**Electric Field Of Point Charge**An electric charge produces an electric field: : unit vector directed from q to P**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**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”**Cross Product: Magnitude**Computing magnitude of cross product A x B: area of parallelogram**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**Cross Product: Signs**Cross Product is Cyclic (left column) Reversing A & B changes sign (right column)**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**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**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**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**Moving**Continuous charge distributions:Currents & Biot-Savart ^**The Biot-Savart Law**Current element of length ds carrying current I produces a magnetic field: (Shockwave)**Example : Coil of Radius R**Consider a coil with radius R and current I Find the magnetic field B at the center (P)**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**Example : Coil of Radius R**In the circular part of the coil… Biot-Savart:**Example : Coil of Radius R**Consider a coil with radius R and current I**Example : Coil of Radius R**• Notes: • This is an EASY Biot-Savart problem: • No vectors involved • This is what I would expect on exam**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)**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.**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**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**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?