Lecture 14. Magnetic Forces on Currents. Hall Effect. Magnetic Force on a Wire Segment. Torque on a Current-Carrying Loop.

Transcription

1 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

2 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 ( ) 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

3 Hall Effect (cont d) Example: typical two-dimensional (n t) charge density in Si field-effect transistor (in the on state) is /m 2.Let srun a0.1a currentandplacethetransistorin a 0.1T magnetic field: #6! " Hall effect measurements allow determination of -charge carrier density and -mobility in conductors and semiconductors. 6

4 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

5 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!/+"

6 Appendix I: Ampere s Motor 9

7 Net force 0, torque 40,2 Dipoles in Uniform Fields 2/2 2/2 axis of rotation 2 Electric Magnetic +q -q,- /. / -01+ / 10

8 Torque on a Current Loop Consider > ( in the loop s plane): " 5 < = 0, " : : 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 11

9 Potential Energy of a Current Loop in Magnetic Field,; ; +C /. / ; 01+ / +q -q unstable equilibrium /,(25%. / stable equilibrium 0 E/2 E / %. / %/ / F 12

10 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

11 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

12 Next time: Lecture 15: Magnetic Fields of Currents

13 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 2 1 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 bar. 17

14 Appendix II: Magnetic Dipole in Non-Uniform Magnetic Field 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