2. Chapter 28 Magnetic Fields

3. Magnetism The Magnetic Force x x x v F B q  v F B q   v F = 0 B q

4. Summary Introduction to Magnetic Phenomena Bar magnets & Magnetic Field Lines Source of Fields: Monopoles? Currents? Zip disks and refrigerators Magnetic forces: The Lorentz Force equation Motion of charged particle in a Constant Magnetic Field.

5. Magnetism Magnetic effects from natural magnets have been known for a long time. Recorded observations from the Greeks more than 2500 years ago. The word magnetism comes from the Greek word for a certain type of stone (lodestone) containing iron oxide found in Magnesia, a district in northern Greece. Properties of lodestones: could exert forces on similar stones and could impart this property (magnetize) to a piece of iron it touched. Small sliver of lodestone suspended with a string will always align itself in a north-south direction—it detects the earth’s magnetic field.

6. Bar Magnet Bar magnet... two poles: N and S Like poles repel; Unlike poles attract. Magnetic Field lines: (defined in same way as electric field lines, direction and density) Does this remind you of a similar case in electrostatics? You will map this field (and others) in lab !!

7. Magnetic Field Lines of a bar magnet Electric Field Lines of an Electric Dipole

8. Magnetic Monopoles Perhaps there exist magnetic charges, just like electric charges. Such an entity would be called a magnetic monopole (having + or - magnetic charge). How can you isolate this magnetic charge? Try cutting a bar magnet in half: Many searches for magnetic monopoles—the existence of which would explain (within framework of QM) the quantization of electric charge (argument of Dirac) No monopoles have ever been found: NS NNSS Even an individual electron has a magnetic “dipole”!

9. Source of Magnetic Fields? What is the source of magnetic fields, if not magnetic charge? Answer: electric charge in motion! –e.g., current in wire surrounding cylinder (solenoid) produces very similar field to that of bar magnet. Therefore, understanding source of field generated by bar magnet lies in understanding currents at atomic level within bulk matter. Orbits of electrons about nuclei Intrinsic “spin” of electrons (more important effect)

10. Magnetic Materials (a simple look at an advanced topic) Materials can be classified by how they respond to an applied magnetic field, B app. Paramagnetic (aluminum, tungsten, oxygen,…) Atomic magnetic dipoles (~atomic bar magnets) tend to line up with the field, increasing it. But thermal motion randomizes their directions, so only a small effect persists: B ind ~ B app 10 -5 Diamagnetic (gold, copper, water,…) The applied field induces an opposing field; again, this is usually very weak; B ind ~ -B app 10 -5 [Exception: Superconductors exhibit perfect diamagnetism  they exclude all magnetic fields] Ferromagnetic (iron, cobalt, nickel,…) Somewhat like paramagnetic, the dipoles prefer to line up with the applied field. But there is a complicated collective effect due to strong interactions between neighboring dipoles  they tend to all line up the same way. Very strong enhancement. B ind ~ B app 10 +5

11. Ferromagnets, cont. Even in the absence of an applied B, the dipoles tend to strongly align over small patches – “domains”. Applying an external field, the domains align to produce a large net magnetization. “Soft” ferromagnets The domains re-randomize when the field is removed “Hard” ferromagnets The domains persist even when the field is removed “Permanent” magnets Domains may be aligned in a different direction by applying a new field Domains may be re-randomized by sudden physical shock If the temperature is raised above the “Curie point” (770˚ for iron), the domains will also randomize  paramagnet Magnetic Domains

12. 1A Which kind of material would you use in a video tape? (a) diamagnetic (b) paramagnetic (c) “soft” ferromagnetic 1B How does a magnet attract screws, paper clips, refrigerators, etc., when they are not “magnetic”? Magnetism (d) “hard” ferromagnetic

13. 1A Which kind of material would you use in a video tape? (a) diamagnetic (b) paramagnetic (c) “soft” ferromagnetic Magnetism (d) “hard” ferromagnetic Diamagnetism and paramagnetism are far too weak to be used for a video tape. Since we want the information to remain on the tape after recording it, we need a “hard” ferromagnet. These are the key to the information age— cassette tapes, hard drives, ZIP disks, credit card strips,…

14. 1B How does a magnet attract screws, paper clips, refrigerators, etc., when they are not “magnetic”? Magnetism The materials are all “soft” ferromagnets. The external field temporarily aligns the domains so there is a net dipole, which is then attracted to the bar magnet. - The effect vanishes with no applied B field - It does not matter which pole is used. End of paper clip S N

15. A “bit” of history IBM introduced the first hard disk in 1957, when data usually was stored on tapes. It consisted of 50 platters, 24 inch diameter, and was twice the size of a refrigerator. It cost $35,000 annually in leasing fees (IBM would not sell it outright). It’s total storage capacity was 5 MB, a huge number for its time!

16. Magnetic Fields What is the "magnetic force"? How is it distinguished from the "electric" force? We know about the existence of magnetic fields by their effect on moving charges. The magnetic field exerts a force on the moving charge. a) magnitude:  to velocity of q b) direction:  to direction of q ’s velocity c) direction:  to direction of B B is the magnetic field vector q F v mag Let’s start with some experimental observations about the magnetic force:

17. Lorentz Force The force F on a charge q moving with velocity v through a region of space with a magnetic field B is given by: F x x x v B q  v B q F = 0  v B q F     Bvq F   What’s that??

20. The magnetic field of the Earth has a magnitude of 0.6 G is directed northward making an angle of 70 o with the horizontal. Calculate the force on a proton moving northward with 10 7 m/s. a) Using regular formula.

21. b) Using unit vectors

22. Three points are arranged in a uniform magnetic field. The B field points into the screen. 1) A positively charged particle is located at point A and is stationary. The direction of the magnetic force on the particle is: a) right b) left c) into the screen d) out of the screen e) zero 2) The positive charge moves from point A toward B. The direction of the magnetic force on the particle is: a) right b) left c) into the screen d) out of the screen e) zero Preflight :

23. 3) The positive charge moves from point A toward C. The direction of the magnetic force on the particle is: a) up and right b) up and left c) down and right d) down and left Preflight :

24. Magnetic Force: If v = 0  F = 0. If then F = qvB If v is up, and B is into the page, then F is to the left.

25. Magnetic Force Two protons each move at speed v (as shown in the diagram) in a region of space which contains a constant B field in the -z-direction. Ignore the interaction between the two protons. –What is the relation between the magnitudes of the forces on the two protons? (a) F 1 < F 2 (b) F 1 = F 2 (c) F 1 > F 2 (a) F 2x < 0 (b) F 2x = 0 (c) F 2x > 0 2B – What is F 2x, the x -component of the force on the second proton? 2A B x y z 1 2 v v (a) decreases (b) increases (c) stays the same 2C – Inside the B field, the speed of each proton:

26. Magnetic Force Two independent protons each move at speed v (as shown in the diagram) in a region of space which contains a constant B field in the -z- direction. Ignore the interaction between the two protons. –What is the relation between the magnitudes of the forces on the two protons? (a) F 1 < F 2 (b) F 1 = F 2 (c) F 1 > F 2 2A The magnetic force is given by:  θqvBFBvqFsin    In both cases the angle between v and B is 90  !! Therefore F 1 = F 2. B x y z 1 2 v v

27. Magnetic Force Two independent protons each move at speed v (as shown in the diagram) in a region of space which contains a constant B field in the -z- direction. Ignore the interaction between the two protons. – What is F 2x, the x-component of the force on the second proton? (a) F 2x < 0 (b) F 2x = 0 (c) F 2x > 0 2B To determine the direction of the force, we use the right-hand rule. As shown in the diagram, F 2x < 0. F1F1 F2F2 B x y z 1 2 v v

28. Magnetic Force Two protons each move at speed v (as shown in the diagram) in a region of space which contains a constant B field in the -z-direction. Ignore the interaction between the two protons. – Inside the B field, the speed of each proton: (a) decreases (b) increases (c) stays the same 2C B x y z 1 2 v v Although the proton does experience a force (which deflects it), this is always to. Therefore, there is no possibility to do work, so kinetic energy is constant and is constant.

29. Suppose charge q enters B-field with velocity v as shown below. What will be the path q follows? Trajectory in Constant B Field x x x v B q F F v R Force is always  to velocity and B. What is path? –Path will be circle. F will be the centripetal force needed to keep the charge in its circular orbit. Calculate R :

30. A Circulating Charged Particle: Consider a particle of charge magnitude |q| and mass m moving perpendicular to a uniform magnetic field B, at speed v. The magnetic force continuously deflects the particle, and since B and v are always perpendicular to each other, this deflection causes the particle to follow a circular path. Fig. 28-10 Electrons circulating in a chamber containing gas at low pressure (their path is the glowing circle). A uniform magnetic field, B, pointing directly out of the plane of the page, fills the chamber. Note the radially directed magnetic force F B ; for circular motion to occur, F B must point toward the center of the circle, (Courtesy John Le P.Webb, Sussex University, England)

31. Radius of Circular Orbit Lorentz force: qvBF  centripetal acc: R v a 2  x x x v F B q F v R Newton's 2nd Law: maF   R v mqvB 2   qB mv R  This is an important result, with useful experimental consequences !

32. Ratio of charge to mass for an electron 1) Turn on electron ‘gun’ qVmv  2 2 1 qB mv R  2) Turn on magnetic field B e-e- VV ‘gun’ R 3) Calculate B … next week; for now consider it a measurement 4) Rearrange in terms of measured values, V, R and B  22 2 BR V m q  m q Vv2 2  2 2        RB m q v and

33. Let’s Try It... 1) Do the expt. Adjust I and V to get a good circle 2) Measure R =.05 m 3) What was V ? V = 230 V 4) How about B ? B ~10 -3 T I r NI B   4 0 108.7 55 8  for our coils 22 2 BR V m q = ≈ 1.8  10 11 C/kg meme e ( = 1.76  10 11 C/kg )

34. The drawing below shows the top view of two interconnected chambers. Each chamber has a unique magnetic field. A positively charged particle is fired into chamber 1, and observed to follow the dashed path shown in the figure. 5) What is the direction of the magnetic field in chamber 1? a) Up b) Down c) Left d) Right e) Into page f) Out of page Preflight 12:

35. 6) What is the direction of the magnetic field in chamber 2? a) Up b) Down c) Left d) Right e) Into page f) Out of page Preflight 12:

36. In chamber 1, the velocity is initially up. Since the particle’s path curves to the right, the force is to the right as the particle enters the chamber. Three ways to figure out the direction of B from this: 1) Put your thumb in the direction of the F (right) and your fingers in the direction of v (up) The way that your fingers curl is the direction of B. 2) Put your palm in the direction of F (right), and your thumb in the direction of v (up), your fingers (keep them straight) point in the direction of B. 3) Keep your thumb, index and middle fingers at right angles from each other. Your thumb points in the direction of v (up), middle finger points towards F (right), then the index finger gives the the direction of B (out of page)

37. 8) Compare the magnitude of the magnetic field in chamber 1 to the magnitude of the magnetic field in chamber 2. a) B 1 > B 2 b) B 1 = B 2 c) B 1 < B 2

38. The magnetic force is always perpendicular to v. The force doesn’t change the magnitude of v, it only changes the particle’s direction of motion. The force gives rise to a centripetal acceleration. The radius of curvature is given by:

39. Work (a) W 1 < W (b) W 1 = W (c) W 1 > W L B B v v B B v1v1 v1v1 – Compare the work done by the magnetic field ( W for v, W 1 for v 1 ) to deflect the protons. A proton, moving at speed v, enters a region of space which contains a constant B field in the -z -direction and is deflected as shown. Another proton, moving at speed v 1 = 2v, enters the same region of space and is deflected as shown.

40. Work (a) W 1 < W (b) W 1 = W (c) W 1 > W Remember that the work done W is defined as: Also remember that the magnetic force is always perpendicular to the velocity: Therefore, the work done is ZERO in each case: – Compare the work done by the magnetic field ( W for v, W 1 for v 1 ) to deflect the protons. A proton, moving at speed v, enters a region of space which contains a constant B field in the -z -direction and is deflected as shown. Another proton, moving at speed v 1 = 2v, enters the same region of space and is deflected as shown. L B B v v B B v1v1 v1v1

41. Summary Lorentz force equation: –Static B-field does no work –Velocity-dependent force given by right hand rule formula Next time: magnetic forces and dipoles

42. The Hall Effect Which charges carry current? Positive charges moving counterclockwise experience upward force Upper plate at higher potential Negative charges moving clockwise experience upward force Upper plate at lower potential Equilibrium between electrostatic & magnetic forces: This type of experiment led to the discovery (E. Hall, 1879) that current in conductors is carried by negative charges (not always so in semiconductors). Can be used as a B-sensor; used in some ABS to detect shaft rotation speed – ferromagnetic rotating blades interupt the magnetic field  oscillating voltage

43. The Hall Effect Fig. 28-8 A strip of copper carrying a current i is immersed in a magnetic field. (a)The situation immediately after the magnetic field is turned on. The curved path that will then be taken by an electron is shown. (b) The situation at equilibrium, which quickly follows. Note that negative charges pile up on the right side of the strip, leaving uncompensated positive charges on the left. Thus, the left side is at a higher potential than the right side. (c) For the same current direction, if the charge carriers were positively charged, they would pile up on the right side, and the right side would be at the higher potential. A Hall potential difference V is associated with the electric field across strip width d, and the magnitude of that potential difference is V =Ed. When the electric and magnetic forces are in balance (Fig. 28-8b), where v d is the drift speed. But, Where J is the current density, A the cross-sectional area, e the electronic charge, and n the number of charges per unit volume. Therefore, Here, l=( A/d), the thickness of the strip.