Short Version : 26. Magnetism: Force & Field

1. Short Version : 26. Magnetism: Force & Field

2. 26.1. What is Magnetism? Magnets exert (magnetic) forces on each other & materials like irons. Field description: A magnet produces a magnetic field B. Another interacts with B at its position. Ampere : Moving charges can produce B. Magnetic dipoles ~ current loops. Iron filings align with the magnetic field, tracing out the field of a bar magnet. Quantum mechanics: Intrinsic magnetic moment related to spin.

3. 26.2. Magnetic Force & Field Magnetic force on a charge q moving with velocity v in a field B: [B] = N  s / (C  m) = T (Tesla) = 10,000 G (Gauss) v B: greatest F BEarth ~ 1 G Bsmall magnet ~ 100 G BMRI ~ 1 T Bmagnetar ~ 1011 T v// B:

4. Example 26.1. Steering Protons Figure shows 3 protons entering a 0.10-T magnetic field. All three are moving at 2.0 Mm/s. Find the magnetic force on each. B 1 3 2 FB+ FE = 0 when v = E/B Electromagnetic force velocity selector.

5. 26.3. Charged Particles in Magnetic Fields  WB = 0 For v B, FB = q v B. Circular motion CW about B for q > 0. v = const

6. Example 26.2. Mass Spectrometer A mass spectrometer separates ions according to their ratio of charge to mass. Such devices are widely used to analyze unknown mixtures, and to separate isotopes of chemical elements. Figure below shows ions of charge q & mass m being first accelerated from rest through a potential difference V, then entering a region of uniform B pointing out of the page. Find an expression for the horizontal distance x.

7. The Cyclotron Frequency Period of particle in circular orbit in uniform B: Cyclotron frequency Motion // B not affected by it Charged particles frozen to B field lines. Trajectory in 3-D

8. Application: The Cyclotron Whole device in vacuum chamber. Small V across the dee’s, which alternates polarities at the cyclotron frequency. Particle injected at center of gap & spirals out. E ~ MeVs. Applications: Manufacture of radioactive isotopes. e.g., PET (Positron Emission Tomography). Higher energies: Relativity effects  Synchrotron (B also varies)

9. 26.4. The Magnetic Force on a Current Force on carrier in wire: Force on straight wire of cross section A & length L: F out of paper Prob 58 e moving left deflected upward … Fmag on + charge is also upward … resulting charge separation leads to upward force on whole wire

10. Conceptual Example 26.1. Power Line A power line runs along Earth’s equator, where B points horizontally from south to north; the line carries current flowing from west to east. What’s the direction of the magnetic force on the power line? F upward L

11. Making the Connection Earth’s equatorial field strength is 30 T, and the power line carries 500 A. What’s the magnetic force on a kilometer of the line. upward c.f. weight of wire is ~ 10 kN

12. The Hall Effect e moves to left & deflected upward Direction of FB depends on I, not on sign of charged carriers. Carriers of both signs are deflected upwards EHis upward Direction of E due to built-up charges depends on signs of charged carriers: Hall effect. EHis downward y z x Steady state, Fz = 0 :  p moves to right & deflected upward Hall potential: Hall coefficient:

13. 26.5. Origin of the Magnetic Field Biot-Savart Law: permeability constant Curl of finger gives B. C.f. Thumb // I

14. Example 26.4. Current Loop Find the magnetic field at an arbitrary point P on the axis of a circular loop of radius a carrying current I. dL r cos = a / r By symmetry, only Bx 0.

15. Example 26.5. Straight Line Find the magnetic field produced by an infinitely long straight wire carrying current I.

16. The Magnetic Force Between Conductors Field of I1 at I2 : Force on length L of I2 : Force per unit length on I2 : points toward I1 Hum from electric equipments are vibrations of transformers in response to AC. Definition: 1 A is the current in two long, parallel wire 1 m apart & exerting 2107 N per meter of length. 1C is the charge passing in 1s through a wire carrying 1A.

17. 26.6. Magnetic Dipoles Field on axis of current loop of radius a : C.f. electric dipole: Setting magnetic dipole  N-turn current loop: Far away, fields look similar … … but close in, they’re different. • Detailed calculations show: •  = I A valid for arbitrary loop. • Vector behavior of  similar to that of p for r >> a. Multi-turn loops = electromagnets Very strong field requires superconducting wires, e.g., MRI scanners. Orbiting e in atom  .

18. Detailed calculations show ： •  = I A valid for arbitrary loop • Vector behavior of  similar to that of p for r >> a. Multi-turn loops = electromagnets Very strong field requires superconducting wires, e.g., MRI scanners. Orbiting e in atom  .

19. Application: Magnetic Fields of Earth & Sun • Currents flows in Earth liquid-iron outer core (due to rotation) • dipole field with  = 8.0  1022 Am2 ( direction not exactly S-N ) • Field deflects harmful high E solar particles. Magnetic reversal: Earth (period ~ millions of yrs) : map for sea-floor spreading over. Sun (period ~ 11 yrs): coincides with period of sun-spot abundancy.

20. Dipoles & Monopoles Some elementary particle theories suggest existence of magnetic monopoles. But none was ever observed. • Microscopic origin of B: • Charged current. • Intrinsic spin. No magnetic charges : B lines always form loops, either encircling moving charges, or joining the 2 poles of a magnet. Gauss Law

21. Torque on a Magnetic Dipole Forces on top & bottom cancel. Forces on sides also cancel; but give net torque.  above plane beneath plane Torque on dipole: Associated energy:

22. Application: Electric Motors Rotating loop CW Commutator CCW Brushes Battery Always CCW

23. 26.7. Magnetic Matter Ferromagnetism: e.g., Fe, Ni, Co, & their alloys. Material divided into magnetic domains in which ’s are aligned. T > TC : orientations of  in different domains random  <  >  B (paramagnetic) T < TC : orientations of  in different domains aligned  <  >  0 even when B = 0 (ferromagnetic) Atomic current loops are CCW. Adjacent loops cancel No cancellation on boundaries: net I on surface.

24.  // Bapplied Field inside dipole > Bapplied Distant field similar. Internal field opposite. Field inside dipole < Eapplied p // Eapplied

25. Paramagnetism : Materials with randomly oriented permanent  and very weak -interaction. <  > =  B with  > 0  = magnetic susceptibility Diamagnetism : Materials with no intrinsic . <  > =  B with  < 0 ( induced  )

26. 26.8. Ampere’s law Field around long wire carrying current I : from Biot-Savart law  True for arbitrary closed paths & steady currents: Bdr = 0 Ampere’s law c.f. Gauss’ law net field from ALL sources

27. Example 26.7. Solar Currents The long dimension of the rectangular loop in figure is 400 Mm, and B near the loop has a constant magnitude of 2 mT. Estimate the total current enclosed by the rectangle. B  dr on the short segment  B  dr= 0 there. B // dr on the long segment  Direction: into the page 3D: around equator

28. Using Ampere’s Law STRATEGY 26.1 Ampère’s Law : Base on symmetry, choose the amperian loop such that B is either // or  to it.

29. Example 26.8. Outside & Inside a Wire • A long, straight wire of radius R carries a current I distributed uniformly over its cross section. • Find the magnetic field • outside and • inside the wire. By symmetry, B is azimuthal. Amperian loop is a circle. (a) (b)

30. Example 26.9. Current Sheet An infinite flat sheet carries current out of this page. The current is distributed uniformly along the sheet, with current per unit width given by JS . Find the magnetic field of this sheet. By symmetry, B is // to sheet & I. Amperian loop is a rectangle. Far out, B lines nearly circular Close in, B lines ~ infinite sheet

31. Fields of Simple Current Distributions For arbitrary distributions: Far away from any loop ~ dipole. Very near any wire ~ infinite straight wire. Very near any current sheet ~ infinite flat sheet.

32. Solenoids Solenoid: a long, tightly wounded coil. n turns per unit length. n L loops in L. encircled current = nLI. solenoid field Boutside = 0 Application: magnetic switches, valves, ….

33. Table 26.1. Fields of Ssome Simple Charge & Current Distributions Dipole Dipole r 3 Point / spherical r 2 NA r 1 Line Line const Sheet Sheet

34. Example 26.10. MRI Scanner The solenoid used in an MRI scanner is 2.4 m long & 95 cm in diameter. It’s wounded from superconducting wire 2.0 mm in diameter, with adjacent turns separated by an insulating layer of negligible thickness. Find the current that will produce a 1.5-T magnetic field inside the solenoid. wire diameter = 1/500 m n = 500 turns/m

35. Field of a solenoid is very similar to that of a bar magnet.