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Physics 121 - Electricity and Magnetism Lecture 09 - Charges & Currents in Magnetic Fields Y&F Chapter 27, Sec. 1 - 8 • • • • • • • • • • • • What Produces Magnetic Field? Properties of Magnetic versus Electric Fields Force on a Charge Moving through Magnetic Field Magnetic Field Lines A Charged Particle Circulating in a Magnetic Field – Cyclotron Frequency The Cyclotron, the Mass Spectrometer, the Earth’s Field Crossed Electric and Magnetic Fields The e/m Ratio for Electrons Magnetic Force on a Current-Carrying Wire Torque on a Current Loop: the Motor Effect The Magnetic Dipole Moment Summary Copyright R. Janow Fall 2016 Magnetic Field E and g Electrostatic & Gravitational forces act through vector fields Now: Magnetic force on moving charge acts through B magnetic field Force law first: Effect of a given B field on charges & currents in wires How to create B field (details next week): • Currents in loops of wire • Intrinsic spins of e-,p+ elementary currents magnetic dipole moment • Spins can align permanently to form natural magnets B & E fields Maxwell’s equations, electromagnetic waves (not covered) “Permanent Magnets”: North- and South- seeking poles • • • • Natural magnets Compass – Earth has a magnetic field Ferromagnetic materials magnetize when cooled in B field (Fe, Ni, Co) - Spins align into“domains” (magnets) Other materials (plastic, copper, wood,…) slightly or not affected (para- and dia- magnetism) ATTRACTS REPELS UNIFORM S N Copyright R. Janow Fall 2016 DIPOLES ALIGN IN B FIELD Electric Field versus Magnetic Field • • • • • • • Electric force acts at a distance through electric field. Vector field, E. Source: electric charge. • Positive (+) & negative (-) charge. Opposite charges attract Like charges repel. Field lines show the direction and magnitude of E. • • • • • • Magnetic force acts at a distance through magnetic field. Vector field, B Source: moving electric charge (current, even in permanent magnets). North pole (N) and south pole (S) Opposite poles attract Like poles repel. Field lines show the direction and magnitude of B. Copyright R. Janow Fall 2016 Differences between magnetic & electrostatic field Test charge and electric field F E qE Single electric poles exist Test monopole p and magnetic field ? FB pB Single magnetic poles have never been Found. Magnetic poles always occur as dipole pairs.. Cut up a bar magnet small, complete magnets There is no magnetic monopole... ...dipoles are the basic units Electrostatic Gauss Law S q E dA enc 0 Magnetic Gauss Law S B dA 0 Magnetic flux through each and every Gaussian surface = 0 Magnetic field exerts force on moving charges (current) only Magnetic field lines are closed curves - no beginning or end Copyright R. Janow Fall 2016 Magnetic Force on a Charged Particle Particle moves with a velocity v through a magnetic field B | FB | qvBsin() FB qv B Typical Magnitudes: Earth’s field: 10-4 T. Bar Magnet: 10-2 T. Electromagnet: 10-1 T. “LORENTZ FORCE” Units: 1 Tesla Newtons Coulombm / s 1 “GAUSS” = 10-4 Tesla See Lecture 01 for cross product definitions and examples • FB is proportional to speed v, charge q, and field B. • FB depends on complex geometry using cross product o FB = 0 if v is parallel to B. o FB is normal to plane of both v and B. o FB reverses sign for opposite sign of charge o Strength of B field also depends on qv [current x length]. • Electrostatic force can do work on a charged particle…BUT… … magnetic force cannot do work on moving particles since FB.v = 0. Copyright R. Janow Fall 2016 Magnetic force geometry examples: A simple geometry B B0k̂ v v0 î y • charge +q • B along –z direction • v along +x direction Fm î k̂ ĵ B Fm qv B x Fm q v0 B0 ĵ CONVENTION v z Head Tail More simple examples Fm F • charge +q • Fm is down • |Fm| = qv0B0 v0 B v0 B • charge +q • Fm is out of the page • |Fm| = qv0B0 x x x x x x x v0 x Fm x x B x v0 x î ĵ • • • • k̂ charge -q B into page Fm is still down |Fm| = qv0B0 • charge +q or -q • v parallel to B • |Fm| = qv0B0sin(0) Fm 0 Copyright R. Janow Fall 2016 A numerical example • Electron beam moving in plane of sketch • v = 107 m/s along +x • B = 10-3 T. out of page along +y • z axis is down B Fm e v force for +e a) Find the magnetic force on an electron : Fm - e v B 1.6x1019 x107 x103 x sin(90o ) k̂ Fm 1.6x1015 k̂ Negative sign means force is opposite to result of using the RH rule b) Acceleration of electron : Fm 1.6 x 1015 N 15 a k̂ 1.76 x 10 k̂ 31 me 9 x 10 Kg 2 [m/s ] Direction is the same as that of the force: Fov = 0 Copyright R. Janow Fall 2016 When is magnetic force = zero? 9-1: A particle in a magnetic field is found to have zero magnetic force on it. Which of the situations could not be the case? A. B. C. D. E. The particle is neutral. The particle is stationary. The motion of the particle is parallel to the magnetic field. The motion of the particle is opposite to the magnetic field. All of them are possible. FB q vB sin( ) Copyright R. Janow Fall 2016 Direction of Magnetic Force 9-2: The figures show five situations in which a positively charged particle with velocity v travels through a uniform magnetic field B. For which situation is the direction of the magnetic force along the +x axis ? Hint: Use Right Hand Rule. A B y y v C y B B B x x v z z z D E y v y v B v z B x x z Copyright R. Janow Fall 2016 x Magnetic Units and Field Line Examples • SI unit of magnetic field: Tesla (T) – • Magnetic field lines – Similarities to E – – • 1T = 1 N/[Cm/s] = 1 N/[Am] = 104 gauss The tangent to a magnetic field line at any point gives the direction of B at that point (not direction of force). The spacing of the lines represents the magnitude of B – the magnetic field is stronger where the lines are closer together, and conversely. At surface of neutron star 108 T Near big electromagnet 1.5 T Inside sunspot 10-1 T Near small bar magnet 10-2 T At Earth’s surface 10-4 T In interstellar space 10-10 T Differences from E field lines – – B field direction is not that of the force on charges B field lines have no end or beginning – closed curves But magnetic dipoles will align with B field Copyright R. Janow Fall 2016 Charged Particles Circle at Constant Speed in a Uniform Magnetic Field: Centripetal Force Choose B uniform along z, v in x-y plane tangent to path FB is normal to both v & B ... so Power = F.v = 0. Magnetic force cannot change particle’s speed or KE FB is a centripetal force, motion is UCM A charged particle moving in a plane perpendicular to a B field circles in the plane with constant speed v mv 2 Set FB qvB r mv Radius of r the path qB Period 2r 2m c v qB Cyclotron 2 qB angular c c m frequency FB qv B y v CW rotation Fm v B Fm x Fm B v B z If v not normal to B particle spirals around B v causes circling v para is cons tan t and ω do not depend on velocity. Fast particles move in large circles and slow ones in small circles All particles with the same charge-to-mass ratio have the same period. Copyright R. Janow Fall 2016 Positive and negative particles rotate in opposite directions. Cyclotron particle accelerator Early nuclear physics research, Biomedical applications magnetic field gap • Inject charged particles in center at S • Charged plates (“Dees”) reverse polarity of E field in gap with frequency c. This accelerates particles crossing gap. • Particles spiral out in magnetic field as they gain KE and are detected. +- mv r qB • Frequency of Dee polarity reversal can be constant! It does not depend on speed or radius of path. c 2 qB c m Copyright R. Janow Fall 2016 Earth’s field shields us from the Solar Wind and produces the Aurora Earth’s magnetic field deflects charged solar wind particles (via Lorentz force) . Protects Earth . Makes life possible (magnetosphere). Solar wind particles spiral around the Earth’s magnetic field lines producing the Aurora at high latitudes. They can also be trapped. Copyright R. Janow Fall 2016 Circulating Charged Particle 9-3: The figures show circular paths of two particles having the same speed in a uniform magnetic field B, which is directed into the page. One particle is a proton; the other is an electron (much less massive). Which figure is physically reasonable? r B A D C E Copyright R. Janow Fall 2016 mv qB Charged particle in both E and B fields Add the electrostatic Ftot ma Ftot qE qv B also and magnetic forces Does F = ma change if you observe this while moving at constant velocity v’? Velocity Selector: Crossed E and B fields B E OUT • B out of paper • E up and normal to B • q is + charge • v normal to both E & B v Fe qE (up) Fm qv B (down) Equilibrium when... Ftot 0 CONVENTION qE qvB v E /B • Independent of charge q • Charges with speed v are un-deflected Copyright R. Janow • Can select particles with a particular velocity IN FBD of q E Fe B v Fm OPPOSED FORCES Fall 2016 Measuring e/m ratio for the electron (J. J. Thompson, 1897) + + + E + CRT screen + L y e- electron gun - For E = 0, B = 0 beam hits center of screen Add E field in –y direction… y = deflection of beam from center Add a crossed B field (into page, E remains) FM points along –y, opposite to FE (negative charge) - FE,y qE ma y - - - e ay m E q-e y 21 ayt2 ay 2y/t2 FM ev B FE eE Adjust B until beam deflection y = 0 (FE cancels FM) to find vx and time t when y 0 BL t E vx E / B 2y E2 a y 2 2y 2 2 t BL flight time t = L / vx ( vx is constant) e E 2y 2 2 1.76 x 1011 C/kg m BL Copyright R. Janow Fall 2016 Mass spectrometer - Another cyclotron effect device: Separates molecules with different charge/mass ratios v r some types use velocity selector here 2qV m mv qB Ionized molecules or isotopes are accelerated • varying masses & q/m • same potential V Ionized particle paths have separate radii, forming “mass spectrum” 1 2mV r B q q 2V 2 2 m Br Molecular mass Mass spectrum of a peptide shows the isotopic distribution. Relative abundances are plotted as functions of the ratio of mass m to charge z. Copyright R. Janow Fall 2016 Force due to crossed E and B fields 9-4: The figure shows four possible directions for the velocity vector v of a positively charged particle moving through a uniform electric field E (into the page) and a uniform magnetic field B (pointing to the right). The speed is E/B. Which direction of the velocity produces zero net force? E A D v v v v B B C Fnet qE qV B Copyright R. Janow Fall 2016 Force on a straight wire carrying current in a B field • Current is flowing opposite to electron’s drift velocity vd • Lorentz force on electron = - evd x B (normal to wire) • Motor effect: wire is pushed right by the charges vd is opposite to current direction dq is the charge moving in wire dFm vd B dq segment whose length dx = -v dt d Recall: dq = - i dt, Note: dx is in direction of current (+ charges) dFm i (vddt) B i dx B Integrate along the whole length L of the wire (assume B is constant ) L length of wire, parallel to current Fm i L B iL Fm uniform B Copyright R. Janow Fall 2016 Motor effect on a wire: which direction is the pull? Fm i L B F iL replaces qv Fm 0 for iL parallelto B F F is into slide F=0 F=0 A current-carrying loop experiences A torque (but zero net force) i B F is out of slide Copyright R. Janow Fall 2016 Torque on a current loop in a magnetic field • Upper sketch: y-axis toward viewer • Lower sketch: y-axis toward right • Loop can rotate about y-axis • B field is along z axis x • Apply Fm i L B z to each side of the loop • |F1|= |F3| = iaB: • Forces cancel but net torque is not zero! • Lower sketch: • |F2|= |F4| = i.b.Bcos(): Forces cancel, same line of action zero torque Net force on loop 0 z into paper x y • Moment arms for F1 & F3 equal b.sin()/2 • Force F1 produces CW torque equal to 1 = i.a.B.b.sin()/2 • Same for F3 • Torque vector is along –y (rotation axis) Net torque iabBsin() Copyright R. Janow Fall 2016 Motor Effect: Torque on a current loop in B field x z z (B) y -x Torque iabBsin() i A n̂ B A area of loop Maximum torque: max iAB n̂ B Restoring Torque: () max sin() Oscillates about B B Current loop in B field is like electric dipole maximum torque Field PRODUCED BY a current loop is a magnetic dipole field Copyright R. Janow Fall 2016 zero torque Current loops are basic magnetic dipoles Represent current Magnetic dipole moment m N i A n̂ loop as a vector B Newton - m Joule Dimensions m ampere - m Tesla Tesla N number of turns in the loop 2 m B A area of loop B m m Bsin() Magnetic dipole moment m measures • strength of response to external B field • strength as source of a dipole field ELECTRIC DIPOLE MAGNETIC DIPOLE MOMENT p qd m NiA TORQUE e p E m m B POTENTIAL ENERGY Ue p E SAMPLE VALUES [J / T] Small bar magnet m ~ 5 J/T Earth m ~ 8.0×1022 J/T Proton (intrinsic) m ~ 1.4×10-26 J/T Electron (intrinsic) m ~ 9.3×10-24 J/T UM d mBsin()d Um m BCopyright R. Janow B Fall 2016 ELECTRIC MOTOR: MOVING COIL GALVANOMETER • Basis of most 19th & 20th century • Reverse current I when torque changes sign ( = 0 ) analog instruments: voltmeter, ohmmeter, ammeter, speedometer, • Use mechanical “commutator” for AC or DC gas gauge, …. • Coil spring calibrated to balance torque at proper mark on scale • Multiple turns of wire increase torque • N turns assumed in flat, planar coil • Real motors use multiple coils (smooth torque) NiA n̂ B A area of loop ab or NiACopyright Bsin(R. )Janow Fall 2016 The DC Motor m B Maximum Torque CCW Zero Torque Cross Commutator Gap Maximum Torque CCW (Reversed) Copyright R. Janow Fall 2016 Torque and potential energy of a magnetic dipole 9-5: In which configuration (see below) does the torque on the dipole have it’s maximum value? m m B 9-6: In which configuration (see below) does the potential energy of the dipole have it’s smallest value? Um m B A B C B D E Copyright R. Janow Fall 2016