What Produces Magnetic Field? Properties of Magnetic versus Electric Fields

1. Physics 121 - Electricity and Magnetism Lecture 09 - Charges & Currents in Magnetic FieldsY&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

2. “Permanent Magnets”: North- and South- seeking poles • Natural magnets known in antiquity • Compass - Earth’s 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 DIPOLES ALIGN IN B FIELD Magnetic Field Electrostatic & gravitational forces act through vector fields Now: Magnetic force (vector but different): • Force law first: Effect of a given B field on charges & currents • Next Lecture: How to create B field • 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  Maxwell’s equations, electromagnetic waves (optional)

3. 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. 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.

4. Cut up a bar magnet small, complete magnets Test monopole and magnetic field ? Single magnetic poles have never been found Magnetic poles always occur as dipole pairs.. Test charge and electric field Single electric poles exist There is no magnetic monopole... ...dipoles are the basic units Electrostatic Gauss Law Magnetic Gauss Law 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 Differences between magnetic & electrostatic field

5. “LORENTZ FORCE” Units: 1 “GAUSS” = 10-4 Tesla Magnetic Force on a Charged Particle DefineB by the magnetic force FB it exerts on a charged particle moving with a velocity v Typical Magnitudes: Earth’s field: 10-4 T. Bar Magnet: 10-2 T. Electromagnet: 10-1 T. See Lecture 01 for cross product definitions and examples • FB is proportional to speed v as well as charge q and field B. • FB is geometrically complex – depends on cross product • F = 0 if v is parallel to B. • F is otherwise normal to plane of both v and B. • FB reverses sign for opposite sign of charge • Strength also depends on qv [current x length]. • Electric force can do work on a charged particle… • BUT magnetic force cannot do work on moving particles since FB.v = 0.

6. A simple geometry y • charge +q • B along –z direction • v along +x direction x z CONVENTION More simple examples Head Tail • charge -q • B into page • Fm is still down • |Fm| = qv0B0 • charge +q • Fm is down • |Fm| = qv0B0 • charge +q or -q • v parallel to B • |Fm| = qv0B0sin(0) • charge +q • Fm is out of • the page • |Fm| = qv0B0 x x x x x x x x x x x x Magnetic force geometry examples:

7. force for +e a) Find the magnetic force on an electron: Negative sign means force is opposite to result of using the RH rule b) Acceleration of electron: Direction is the same as that of the force: Fov = 0 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

8. 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 below is NOT possible? • 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.

9. 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 shows 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.

10. Magnetic field lines – Similarities to E • The tangent to a magnetic field line at any point gives the direction of B at that point; • The spacing of the lines represents the magnitude of B– the magnetic field is stronger where the lines are closer together, and conversely. • Differences from E field lines • B field direction is not that of the force on charges • Field lines have no end or beginning – closed curves But magnetic dipoles will align with field Magnetic Units and Field Line Examples • SI unit of magnetic field: Tesla (T) • 1T = 1 N/[Cm/s] = 1 N/[Am] = 104 gauss

11. y v CW rotation Fm B v Fm x Fm v B B z Set If v not normal to B particle spirals around B Radius of the path Period • 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. • The rotation direction for a positive and negative particles is opposite. Cyclotron angular frequency Charged Particles Circle at Constant Speed in a Uniform Magnetic Field: Cyclotron Frequency • B uniform in z direction, 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

12. gap magnetic field Cyclotron particle accelerator Early nuclear physics research, Biomedical applications • Inject charged particles in center • Charged “Dees” reverse polarity as • particles cross gap to accelerate them. • Particles spiral out in magnetic • field as they gain KE and are detected. • Frequency of polarity reversal can be • constant: it does not depend on speed • or radius of path - it can be constant!

13. Earth’s field shields us from the Solar Wind and produces the Aurora Earth’s magnetic field deflects charged solar wind particles (cyclotron effect) protecting the Earth and making life possible (magnetosphere). Some solar wind particles spiral around the Earth’s magnetic field lines producing the Aurora at high latitudes. They can also be trapped.

14. 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? B A C D E

15. Add the electrostatic and magnetic forces Does F = ma change if you observe this while moving at constant velocity v’? Velocity Selector: Crossed E and B fields CONVENTION FBD of q OUT IN • B out of paper • E up and normal to B • + charge • v normal to both E & B E Fe B v OPPOSED FORCES Equilibrium when... Fm • Independent of charge q • Charges move through fields un-deflected • Use to select particles with a particular velocity Charged particle in bothE and B fields

16. Measuring e/m ratio for the electron (J. J. Thompson, 1897) CRT screen L y e- electron gun + - - + - + + - - + Use crossed B and E fields to measure vx B points into the page, perpendicular to both E and Vx FM points along –y, opposite to FE (negative charge) flight time t = L / vx ( vx is constant) AdjustB until beam deflection = 0 (FE cancels FM) to find vx and time t First: Apply only E field in –y direction y = deflection of beam from center of screen (position for E = 0, B = 0)

17. ionized atoms or molecules accelerated through potential V some types use velocity selector here Molecular mass Mass spectrum of a peptide showing the isotopic distribution. Relative abundances of molecules plotted as functions of the ratio of mass m to charge z. Mass spectrometer - Another cyclotron effect device: Separates particles with different charge/mass ratios ionized isotopes or molecules are separated, forming a “spectrum

18. Force due to crossed E and B fields 9-4: The figure shows four choices 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 velocity produces the greatest magnitude of the net force? E A v v D B B v v C

19. Free electrons (negative) flowing: • Drift velocity vd is opposite to current (along wire) • Lorentz force on an electron = - evd x B (normal to wire) • Motor effect: wire is pushed or pulled by the charges CONVENTION OUT IN dq is the charge moving in wire segment whose length dx = -vddt Recall: dq = - i dt, Note: dx is in direction of current (+ charges) Integrate along the whole length L of the wire (assume B is constant ) iL Fm uniform B Force on a straight wire carrying current in a B field

20. F is in F = 0 F = 0 B A current-carrying loop experiences A torque (but zero net force) i F is out Motor effect on a wire: which direction is the pull?

21. Example: A3.0 A. current flows along +x in a 1.0 m wire parallel to the x-axis. Find the magnitude and direction of the magnetic force on the wire, assuming a uniform magnetic field given by: y Only the component of B perpendicular to the current-length contributes to the force Example: Same as above, but now the field has a z-component x z Whatever the direction of the field B, there can never be a force component along the current.

22. Apply • to each side of the loop x z z into paper y • 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 down into paper along –y • rotation axis x • Lower sketch: • |F2|= |F4| = i.b.Bcos(q): Forces cancel, • same line of action  zero torque Torque on a current loop in a magnetic field • Upper sketch: y-axis toward viewer • Loop can rotate about y-axis • B field is along z axis • |F1|= |F3| = iaB: Forces cancel • but net torque is not zero!

23. q x z (B) z y -x Maximum torque: maximum torque zero torque Sinusoidal variation: Stable when parallels Restoring torque  oscillations Field PRODUCED by a current loop is a magnetic dipole field Torque on a current-carrying loop: Motor Effect

24. 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 • ELECTRIC MOTOR: • Reverse current I when torque • t changes sign (q = 0 ) • Use mechanical “commutator” • for AC or DC • Multiple turns of wire increase torque • N turns assumed in flat, planar coil • Real motors use multiple coils (smooth torque)

25. q 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 q Electron (intrinsic) m ~ 9.3×10-24 J/T ELECTRIC DIPOLE MAGNETIC DIPOLE MOMENT TORQUE POTENTIAL ENERGY Current loops are basic magnetic dipoles Represent loop as a vector • Magnetic dipole moment mmeasures • strength of response to external B field • strength of loop as source of a dipole field

26. The DC Motor

27. A B C B D E Torque and potential energy of a dipole 9-5: In which configuration is the torque on the dipole at it’s maximum value? 9-6: In which configuration is the potential energy of the dipole at it’s smallest value?