W12D2 Exam 3 Review Summary of Presentations and Concept Questions 1 W06D2 Magnetic Forces and Sources of Magnetic Fields W06D2 Magnetic Force on Current Carrying Wire Sources of Magnetic Fields Biot Savart Law Reading Course Notes Sections 8 3 9 1 9 2 2 Outline Magnetic Force on Current Carrying Wire Sources of Magnetic Fields Biot Savart Law 3 1 Review Lorentz Force Law Force on charged particles in electric and magnetic fields Felec qE Fmag q v B Electric Force Magnetic Force F q E v B 4 Magnetic Force on Current Carrying Wire dFmag dqv B d s dq dqv dq d s Id s dt dt dFmag Id s B Fmag I d s B wire 5 Magnetic Force on Current Carrying Wire If the wire is a uniform magnetic field then Fmag I d s B wire If the wire is also straight then Fmag I L B 6 2 Magnetic Field of Moving Charge Moving charge with velocity v produces magnetic field P r o q v x r B 4 r 2 r unit vector directed from charged object to P 0 4 10 7 T m A 1 permeability of free space 7 The Biot Savart Law Current element of length d s carrying current I produces a magnetic field at the point P I d s r dB 0 4 r 2 B r 0 4 Id s r r r r 3 wire 8 The Right Hand Rule 2 dir d s z dir B P z r 9 3 The Biot Savart Law Infinite Wire Magnetic Field of an Infinite Wire Carrying Current I from Biot Savart I B 0 k 2 y See W06D3 Problem Solving http web mit edu 8 02t www materials ProblemSolving solution05 pdf More generally I B 0 2 r 10 Magnetic Field Generated by a Current Loop http web mit edu viz EM visualizations magnetostatics calculatingMagneticFields RingMagInt RingMagIntegration htm 11 Example Coil of Radius R In the circular part of the coil d s r d s r ds I I r ds I Biot Savart I d s r 0 I ds dB 0 4 r 2 4 r 2 I R d 0 I d 0 4 R 2 4 R I 2 I B 0 d 0 4 R 0 2R 12 4 W07D1 Magnetic Dipoles Force and Torque on a Dipole Experiment 2 W07D1 Magnetic Dipoles Torque and Force on a Dipole Experiment 2 Magnetic Dipole in a Helmholtz Coil http web mit edu 8 02t www materials Experiments expMagForcesDipoleHelmholtz pdf Reading Course Notes Sections 8 4 8 6 4 8 10 4 8 13 9 5 9 9 13 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 14 Magnetism Bar Magnet Like poles repel opposite poles attract 15 5 Conservation of Magnetic Flux qin E S d A 0 B d A 0 S 16 Magnetic Dipole Moment IA n IA http web mit edu viz EM visualizations magnetostatics calculatingMagneticFields RingMagField RingMagField htm Magnetic Force on Current Loop in Uniform Field If the current loop is a uniform magnetic field then Fmag I d s B 0 loop becasue I d s 0 loop 18 6 Torque on Current Loop Place rectangular current loop in uniform B field Magnetic moment points out of the page j k i B B B j IAB j torque tries to align the magnetic moment vector in the direction of the magnetic field 19 Dipoles don t move This dipole rotates but doesn t feel a net force in a uniform magnetic field But dipoles can feel magnetic force 20 Force on Magnetic Dipole in NonUniform Field N S N S What makes the field pictured Bar magnet below dipole with N pole on top It is aligned with the dipole pictured they attract 21 7 Force on Magnetic Dipole U Dipole B F B F B Fz 0 0 z z N S N S Bz z 22 Magnetic Field Profiles for Experiment 2 W09D1 Sources of Magnetic Fields Ampere s Law Today s Reading Assignment Course Notes Sections 9 3 9 4 9 7 9 10 2 24 8 3rd Maxwell Equation Ampere s Law B d s J 0 n da closed path open surface Open surface is bounded by closed path 25 Current Enclosed J Current density I enc J n dA open surfacce S Current enclosed is the flux of the current density through an open surface S bounded by the closed path Because the unit normal to an open surface is not uniquely defined this expression is unique up to a plus or minus sign 26 Ampere s Law The Idea In order to have a non zero line integral of magnetic field around a closed path there must be current punching through any area with path as boundary 27 9 Sign Conventions Right Hand Rule B d s 0 closed path J n da open surface Integration direction clockwise for line integral requires that unit normal points into page for open surface integral Current positive into page negative out of page 28 Sign Conventions Right Hand Rule B d s 0 J n da closed path open surface Integration direction counterclockwise for line integral requires that unit normal points out of page for open surface integral Current positive out of page negative into page 29 Applying Ampere s Law 1 Identify regions in which to calculate B field 2 Choose Amperian closed path such that by symmetry B is zero or constant magnitude on the closed path B times length B ds or zero I enc 0 J n da 4 Calculate current enclosed 3 Calculate oriented closed path open surface 5 Apply Ampere s Law to solve for B check signs closed path B d s 0 J n da open surface 30 10 Example Infinite Wire Region 1 Outside wire r R Cylindrical symmetry Amperian Circle B field counterclockwise B d s B ds B 2 r 0 I enc 0 I I B 0 2 r 31 Magnetic Field of Solenoid Horiz comp cancel loosely wound tightly wound For ideal solenoid B is uniform inside zero outside 32 Magnetic Field of Ideal Solenoid Using Ampere s law Think B d s along sides 2 and 4 B 0 along side 3 B d s B d s B d s B d s B d s 1 Bl I enc nlI 2 3 0 4 0 0 n of turns per unit length B d s Bl 0 nlI n N L turns unit length B 0 nlI 0 nI l 33 11 Ampere s Law Infinite Current Sheet B I B Amperian Loops B is Constant Parallel OR Perpendicular OR Zero I Penetrates 34 Surface Current Density A very thin sheet of current of width w carrying a current I in the positive z direction has a surface current density K Kk K I w For sheet of thickness d width w and current I I Jdw Kw J K d 35 Biot Savart …
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