Lecture 14. Magnetic Forces on Currents. Outline: Hall Effect. Magnetic Force on a Wire Segment. Torque on a Current-Carrying Loop. Lecture 13: Magnetic Forces on Moving Charges - we considered individual charges; - and ignored electrostatic interactions among the charges in the electron beam. 1
Hall Effect Current-carrying conductors in external magnetic field: two systems of charges - mobile current carriers and immobile ion lattice - and only the current carriers are affected by B. Edwin Hall (1855-1938) In the steady state, the magnetic force on moving charges is compensated by the electrostatic force due to uncompensated surface charge: n the density of mobile carriers W the width of the conductor t the thickness of the conductor -Hall voltage The sign of V H depends on the sign of the charge of current carriers, the magnitude 1/n. 5
Hall Effect (cont d) Example: typical two-dimensional (n t) charge density in Si field-effect transistor (in the on state) is10 16 1/m 2.Let srun a0.1a currentandplacethetransistorin a 0.1T magnetic field: 0.1 0.1 1.6 10 1 10 #6! " Hall effect measurements allow determination of -charge carrier density and -mobility in conductors and semiconductors. 6
Magnetic Force on a Current-Carrying Conductor The electrons drift along the wire because the net (el.+mag.) force on them in the direction normal to the wire is 0. However, positively charge ions are at rest, they feel only an uncompensated electric force: $ -force per one ion $ -force per unit length of the conductor The force on a wire segment of length %&: %& 7
Jumping Wire Experiment Example: a straight horizontal length of wire has a mass of!/&=10 g/m; it carries a current of 1A.What are the magnitude and direction of the minimum magnetic field needed to suspend the wire in the Earth s gravitational field?!( &! & &(!/& ( 0.01*(/! 10!/+" 1 0.1 8
Appendix I: Ampere s Motor 9
Net force 0, torque 40,2 Dipoles in Uniform Fields 2/2 2/2 axis of rotation 2 Electric Magnetic +q -q,- /. / -01+ / 10
Torque on a Current Loop Consider > ( in the loop s plane): " 5 < = 0, 6 2 8 6 2 " 6 2 5 9: 86 5 9: 56 9: 2 " Magnetic dipole moment: ; 56 (A the loop s area) Direction of ;: the right-hand rule. / In general:,;, ; +C/ (compare with,-) If there are N turns, the total magneticdipolemomentis ;?, 0 if / 0 @,180 @ 11
Potential Energy of a Current Loop in Magnetic Field,; ; +C /. / ; 01+ / +q -q unstable equilibrium /,(25%. / stable equilibrium 0 E/2 E / %. / %/ / F 12
Magnetic Dipole vs. Electric Dipole vs. Current-carrying loop Similarity: Bar magnet Electric dipole - a magnet s magnetic field is very similar to a dipole s electric field at points far from the dipoles; - both repel/attract each other; - bothalignalongthefieldlines. Difference: - unlike electric dipoles, magnetic poles cannot be separated; - magnets have no effect on stationary charges. 14
Magnetic Dipoles (cont d) Current-carrying loop Bar magnet Compass needle The Earth's North Magnetic Pole is actually a magneticsouthpole and the Earth's South Magnetic Pole is a magnetic north pole. 15
Next time: Lecture 15: Magnetic Fields of Currents. 28.1-28.5 16
HI Pressure on Solenoid Walls (see L15) G MNL 0 HIOHJ ; @ ; @ P P linear current density (per unit length) Pressure on the solenoid walls due to the interaction of currents with B ext produced by all other elements of the solenoid: %& JKL Total force per unit area 1! " (pressure) on a current-carrying sheet: -force per one wire of length dl Q JKL P JKL P ; @P 2 1 2; @ HI " Thus, the pressure equals to the energy density in the solenoid. Problem of ultra-strong mag. fields = problem of mechanical stability of solenoids TheLANLpulsedmagnet:delivers~100Tforabout 15 milliseconds. The magnet consists of an outer coilset(notshown),andasmallercoilinsertedinto thehighfieldregionoftheoutercoilset. The LANL 80 Tesla pulsed test magnet before and after 10 pulses. The magnetic pressure acting on thewallsofthecoilis 4 10 4 bar. http://www.gizmag.com/100-tesla-pulsed-magnet/21946/ 17
Appendix II: Magnetic Dipole in Non-Uniform Magnetic Field http://www.ru.nl/hfml/research/levitation/diamagnetic/ Paramagnetic response: the induced ;is parallel to Diamagnetic response: the induced ; is antiparallel to ; % % % % Top view Levitation of a diamagnetic object Andre Geim Nobel 2010 IgNobel 2000 In contrast, the induced electric dipoles are oriented along the electric field, they are always attracted to the region of a stronger field. 18