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UW-Madison PHYSICS 208 - Magnetism

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This Lecture:From previous lecture:•RC circuits •Membrane electrical currents •Introduction to Magnetic fieldMagnetism•Sources of magnetic field: moving charges•Biot & Savart Law•current loop and magnetic dipole•Lorentz force and motion of charge in a B-fieldMTE 2: Ch 2103 5:30-7pm on Oct 31 Contact me and Prof. Rzchowskiafter this lecture for Alternate Exams (also by email asap)This week again my office hrs on Wed 11:30am-1pm2Contents of MTE2!Electric Potential and Field !Capacitors and capacitance !Current and resistance, Ohm’s law!DC Circuits and Kirchoff’s laws !RC circuits!Lorentz force and motion of charge in a magnetic field!Biot and Savart!Ampere’s law!Read chapters 28 (no 28.2)-29-30-31-32Magnetic Fields!A vector quantity (B)!compass needle traces B field lines and points towards N!B-field lines start in N and go to S !they do not start or stop (no magnetic monopoles)Iron filings show pattern of B-field linesElectric vs Magnetic Field Lines!Similarities!Density gives strength!Arrow gives direction!Leave +, North!Enter -, South!Differences!Start/Stop on electric charge!No Magnetic Charge ! lines are continuous!!SI unit of magnetic field: tesla (T)!CGS unit: gauss (G): 1 T = 104 G (Earth surface 0.5 G)Refrigerator magnet 5 x 10-3 TSources of magnetic field!Magnetic Fields are created by moving electric charge!!1819 Hans Christian Oersted!Discovered the relationship between electricity and magnetism!An electric current in a wire deflected a nearby compass needle (2 kind of magnetism due to permanent magnets and currents or the same?)!1820’s!Faraday and Henry!Further connections between electricity and magnetism!A changing magnetic field creates an electric field!Maxwell!A changing electric field produces a magnetic field7Surprise! Electric current produces magnetic field!Current (flow of electric charges ) in wire produces magnetic field.!The magnetic field aligns compass needleCurrentMagnetic field8Magnetic field from a currentIron filings align with magnetic field linesMagnetic field loops around the current. (Right hand rule)9Current loop as magnetic dipoleA current loop is an electromagnet: it behaves like a permanent magnet. Two current loops with current in the same direction attract like 2 magnets (since the opposite poles are closer)10Biot-Sarvart Law!Each element of current produces a contribution to the magnetic field.r"IdsB out of pageFor a single charge in motion! B =µ04"qv # urr2For a wire with current! dB =µ04"Ids # urr2permeability of free space! µ0= 4"#10$7Tm / A = 1.257 #10$6Tm / Aur! qv " IdsField from a circular loopAt the center of the loop:! dB =µ04"Idl sin#R2#= 90o$ % $ $ dB =µ04"IdlR2! B =µ04"IR2dl#! B =µ04"IR22"R =µ0I2R12Field on the axis of a circular loop! dB =µ04"Ids(x2+ R2)orthogonal! dBx=µ04"Ids(x2+ R2)cos#=µ04"Ids(x2+ R2)R(x2+ R2)1/ 2! Bx=µ04"I2"R2(x2+ R2)3 / 2For |x|>>R! Ex=14"#02 px3vertical components sum to give 0B is along x also when x <0!! Bx=µ04"I2"R2x3=µ04"2µx3magnetic dipole moment magnitude:µ = A ISimilar to electric field on the axis of a dipoledirection: # to loop plane according to RH ruleLorentz force FB = q v x B!Magnitude of force is proportional to!Charge of particle, q!Speed of particle, v!Strength of magnet field, B!sin("), "=angle between B and v!Tesla = N s /(C m) = kg /(Cs)!If there is also an electric field: total force:The magnetic force acts only a moving charged particle 14Magnetic force on charged particle ! FB= q vB sin"Direction of force is perpendicular to both B and v ! F = qv " Bvector ‘cross product’Right Hand Rule!FB is zero when v and B are parallel or antiparallel!FB is a maximum when v and B are perpendicular15Effect of uniform magnetic field!Effect of uniform B-field on charged particle!If charged particle is not moving - no effect!If particle is moving: force perpendicularto both field and velocity!the charge sign must be accounted for16Magnetic Force and Work!The magnetic force NEVER does work on the particle.Magnetic force perpendicular to the particle displacement => the magnetic force does not do work!B field can alter direction of velocity, but not the speed or the kinetic energyFx x x x x x x x x x x x x x x x x xvBqBqF = 0$ $ $ $ $ $ $ $$ $ $ $ $ $ $ $ vBqvFQuick QuizThe three charges below have equal charge and speed, but are traveling in different directions in a uniform magnetic field. Which particle experiences the greatest magnetic force? 1 2 3 Same B123F = q v B sin(")Quick QuizThe three charges below have equal charge and speed, but are traveling in different directions in a uniform magnetic field. The force on particle 3 is in the same direction as the force on particle 1. 1) True 2) False B123Velocity points in two different directions.RHR determines force direction - different!F = q v B sin(")!Force on each charge!Force on length ds of wire19Magnetic Force on a Current! FB= I L " B! qv " B! qv " Ids! dF = Ids " B20‘Homopolar motor’A homopolar motor has a magnetic field along the axis of rotation and an electric current produced by the battery that moves radially through the magnet.The resulting Lorentz force in the tangential direction produces a torque in the magnet, which is free to rotate with the attached screw. The name homopolar refers to the absence of polarity change.BIdsF! FB= I L " B21Trajectory in Constant B FieldFFvx x x x x x x x x x x x x x x x x xvBqx x x x x x x x x x x x x x x x x x• Suppose a charge q of mass m enters B field with velocity v as shown below. (v#B) What will be the path q follows? !Force is always # to velocity and B. !Path is a circle. F/m is the centripetal acceleration of the particle in circular orbit. 22Radius of Circular Orbit•Lorentz force when v # B:• Centripetal acceleration:• Newton's 2nd Law: ! F = ma!!Radius of orbit related to properties of the particle m/q,to its velocity and to B-field23Your experiment in labe-1) Turn on electron ‘gun’%V‘gun’2) Turn on magnetic field B (Helmotz coils)R3) Measure electron charge/mass as a function of !V, R and BB! 12mv2= q"V! v2= 2qm"V! R2=m2v2q2B2= 2qm"Vm2q2B2! qm=2"VR2B224An example: B-field due to a long straight wire with current!All current elements produce B out of pagexa! r = x2+ a2"Add them all up!dsur!! r =asin"! x =atan"# dx = $ad"sin2"! dB =µ04"Ids # urr2$ dB =µ04"Isin%r2dx ==µ04"Isin("&')r2dx =µ04"Isin'r2dx ==


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UW-Madison PHYSICS 208 - Magnetism

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