Physics 121 - Electricity and Magnetism Lecture 09 - Charges & Currents in Magnetic FieldsY&F Chapter 27, Sec. 1 - 8 • What Produces Magnetic Field? • Comparing 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 • Motor Effect: Force on a Current-Carrying Wire • Torque on a Current Loop • The Magnetic Dipole Moment • Summary
“Permanent Magnets”: North- and South- seeking poles • Natural magnets interact via B field • Compass – Earth has a magnetic field • Dipoles align to an external magnetic field • Can make magnets - ferromagnetic materials • cooled in B field (Fe, Ni, Co) • Materials like plastic, copper, wood,… only • slightly affected (para- and dia- magnetism) ATTRACTS REPELS UNIFORM S N Field Lines • Force law: A given B field exerts force on moving charges or currents • Next week: how to create B field using… • Currents in loops of wire or moving charges • Intrinsic spins of e-,p+ elementary currents magnetic dipoles • Spins can align permanently to form natural magnets • Not covered: B & E fields Maxwell’s equations, electromagnetic waves Magnetic Field Electrostatic & Gravitational forces act through vector fields Now: Magnetic force on moving charge is due to
Magnetic Field • New force acts at a distance through new vector magnetic field • Source: moving electric charge (i.e., current) • North pole (N) and south pole (S) • Opposite poles attract • Like poles repel. • Field lines show the direction and magnitude of B. Similarities between electric and magnetic fields Electric Field • Force acts at a distance through vector electric field • Source: stationary or moving electric charge. • Positive (+) & negative (-) charge. • Opposite charges attract • Like charges repel. • Field lines show the direction and magnitude of E.
Cut up a bar magnet small, complete magnets There is no “magnetic charge”p such that No single N or S magnetic poles have ever been seen. Magnets are always dipoles. Complicated force on charge in B field. Test charge feels electric field Single electric poles exist Force parallel to E-field There is no magnetic monopole... ...dipoles are the basic units Electrostatic Gauss Law Magnetic Gauss Law Magnetic field exerts force on moving charges (current) only Magnetic flux through each and every Gaussian surface = 0 Magnetic field lines are closed curves - no beginning or end Differences between magnetic & electrostatic field
Units: 1 “GAUSS” = 10-4 Tesla Magnetic force on a moving, charged particle Point charge particle with velocity v in magnetic field B Typical Magnitudes: Earth’s field: 10-4 T. Bar Magnet: 10-2 T. Electromagnet: 10-1 T. See Lecture 01 for cross product • FB is proportional to speed v, charge q, and field B. • FB depends on complex geometry using cross product • FB = 0 if v is parallel to B • FB is normal to plane of both v and B (use right hand rule) • FB reverses sign if charge sign reverses. • B field created by moving charge 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.
y • charge +q • B along –z direction • v along +x direction x q z CONVENTION More simple examples Head Tail • charge -q • B into page • Fm is still down • |Fm| = qv0B0 +q • charge +q • Fm is down • |Fm| = qv0B0 -q • charge +q or -q • v parallel to B • |Fm| = qv0B0sin(0) • charge +q • Fm is out of • the page • |Fm| = qv0B0 +q x x x x x x x x x x x x Examples with simple geometries:
force for +e b) Find the electron’s acceleration (mechanics still applies): Direction is the same as that of the force: Fov = 0 A numerical example • Single electron moving in plane of sketch • v = 107 m/s along +x • B = 10-3 T. out of page along +y • z axis is down a) Find the magnetic force on the electron: Negative sign means force is opposite to result of using the RH rule
When is magnetic force = zero? 9-1: Suppose that a particle in a magnetic field has zero magnetic force on it. Which situation below could not be the case? • 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.
y y y y y C A B B v v x x x x x B z z z z z B v E D v B B v 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 does the magnetic force point along the +x axis ? Hint: Use Right Hand Rule.
Demonstration Source: Pearson Study Area - VTD Chapter 28 Magnet and Electron Beam https://mediaplayer.pearsoncmg.com/assets/secs-vtd38_bendelectronbeam Discussion: • You will need the right hand rule. • How did the demo create the electron beam? • In what way does the magnetic force act on the beam? • Why is the beam visible with or without the magnet?
Magnetic field lines – Similarities to E • 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. Strong field where the lines are close together. • Differences from E field lines • B field direction is not direction of force on charges • B field lines have no beginning or end – closed curves But magnetic dipoles will align with B field Magnetic Units and Field Line Examples • SI unit of magnetic field: Tesla (T) • 1T = 1 N/[Cm/s] = 1 N/[Am] = 104 gauss
y v + charge CW rotation FM B v FM x FM v B B z Set Radius of the path If v has a component alongB the particle spirals around B Period • tc and ωc 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. • Positive and negative particles rotate in opposite directions. Cyclotron angular frequency Charged Particles Circle at Constant Speed in a Uniform Magnetic Field: Centripetal Force • Uniform B along z. v in x-y plane tangent to path. • FM is normal to both v&B ... so Power = F.v = 0. • Magnetic force cannot change particle’s speed or KE. • Force direction does change (rotates) • A charged particle moving in a plane perpendicular to a B field circles in the plane with constant speed v • FM is a centripetal force (Uniform Circular Motion)
gap magnetic field - + Cyclotron particle accelerator Early nuclear physics research, Biomedical applications • Inject charged particles in center at S • Charged plates (“Dees”) reverse polarity • of E field in gap with frequency wc. • This accelerates particles crossing gap. • Particles spiral out along field lines as • they gain KE until they enter the beam. • Frequency of Dee polarity reversal can • be constant! It does not depend on • speed or radius of path. So particles • can gain energy every time they cross • the gap..
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.
Circulating Charged Particle 9-3: The 5 figures show circular paths of two particles having the same speed in a uniform magnetic field B, which points into the page. One particle is a proton; the other is an electron (much less massive). Which figure is physically reasonable? B A C D E
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 v v D B B v v C
dq = idt is the charge moving in wire segment whose length dx = -vddt. dx is along current direction. Integrate along the length L of the wire (assume B is uniform ) iL Fm uniform B Force on a straight wire carrying current in a B field • Current flows opposite to electron’s drift velocity vd • Lorentz force on an electron = - evd x B (normal to wire) • Electrons transmit force to the wire structure • Motor effect: wire is pushed rightward by the charges
Demonstration Source: Pearson Study Area - VTD Chapter 27 Current-Carrying Wire in Magnetic Field https://mediaplayer.pearsoncmg.com/assets/secs-vtd39_jumplongwire Discussion: • What expression tells us the force magnitude and • direction on the wire? • The field exerts force directly on conduction electrons. How does it get transmitted to the lattice of ions in the wire?
F is into slide • What is the effect on a current loop in a magnetic field? Answer: • zero net force …but • a torque about an axis • in space F = 0 F = 0 B i F is out of slide Motor effect: direction of the pull on a wire?
x z Maximum torque: Restoring Torque: maximum torque zero torque Rotational oscillation about Current loop in B field behaves like electric dipole in E field – MAGNETIC DIPOLE Torque on a current loop in magnetic field
q z (B) q y -x Torque on a loop in external magnetic field is normal to both m and B. Current loops are basic magnetic dipoles Represents a current loop as a vector • Magnetic dipole moment mmeasures: • strength of response to external B field • strength as source of a dipole field
ELECTRIC DIPOLE MAGNETIC DIPOLE SAMPLE DIPOLE VALUES [J / T] Small bar magnet m ~ 5 J/T MOMENT Earth m ~ 8.0×1022 J/T TORQUE Proton (intrinsic) m ~ 1.4×10-26 J/T Electron (intrinsic) m ~ 9.3×10-24 J/T POTENTIAL ENERGY Comparison: Magnetic & Electric Dipole Moments • Magnetic dipole moments are basic • building blocks • …..but…. • Electric dipoles are composed of • smaller components (charges)
ELECTRIC MOTOR: • Reverse current I when torque • t changes sign (q = 0 ) • Use mechanical “commutator” • for AC or DC • MOVING COIL GALVANOMETER • Basis of most 19th & 20th century • analog instruments: voltmeter, • ohmmeter, ammeter, speedometer, • 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) Devices using the Motor Effect
A B C B D E 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 magnitude? 9-6: In which configuration (see below) does the potential energy of the dipole have it’s minimum value?
Add the electrostatic and magnetic forces Velocity Selector: Crossed E and B fields CONVENTION OUT IN • B out of paper • E up and normal to B • q is + charge • v normal to both E & B Free Body Diagram for q E Fe B v Fm Can have equilibrium if... OPPOSED FORCES • Independent of charge q • Charges with speed v are un-deflected • Can select particles with a particular velocity Charged particle in bothE and B fields
Measuring e/m ratio for the electron (J. J. Thompson, 1897) 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) Need vx to find flight time t: adjustB until beam deflection y = 0 flight time t = L / vx ( vx is constant)
Ionized molecules or isotopes are accelerated • varying masses & q/m • same potential V & KE Ionized particle paths have separate radii, forming “mass spectrum” some types use velocity selector here 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. Mass spectrometer - Another cyclotron effect device: Separates molecules with different charge/mass ratios
Upper sketch: y-axis toward viewer • Lower sketch: y-axis toward right • Loop can rotate about y-axis • B field is along z axis • Apply • to each of 4 sides of the loop x z • |F1|= |F3| = iaB: • Forces cancel but net torque is not zero! z into paper y • |F2|= |F4| = i.b.Bcos(q): Forces cancel, • Same line of action zero torque x • Moment arms for F1 & F3 equal b.sin(q)/2 • Force F1 produces CW torque equal to • t1 = i.a.B.b.sin(q)/2 • Same for F3 • Torque vector is along –y (rotation axis) Torque on a rectangular current loop in a magnetic field
Maximum Torque CCW Zero Torque Cross Commutator Gap Maximum Torque CCW (Reversed) The DC Motor