Electricity and Magnetism Magnetic Dipole Moment Magnetism in Matter

Transcription

1 Electricity and Magnetism Magnetic Dipole Moment Magnetism in Matter Lana Sheridan De Anza College Nov 12, 2015

2 Last time the Hall effect particle accelerators

3 Overview force on a wire with a current in a B-field torque on a wire loop in a B-field motors relating a current loop to a magnet magnetic dipole moment torque and potential energy of magnetic dipole magnetism of matter

4 Magnetic Force on a Current Carrying Wire Charged particles moving in a magnetic field experience a force. Note that this e carrying a curr to the wire. B If the ma magnetic forc or L i v d F B L : Here is a l segment in the x x where f is th that of the cro Fig A close-up view of a section tity. Equation A wire carryingof athe current wire of also Fig. experiences 28-14b.The current a force, direc- since there is a force on each moving tion is upward, chargewhich confined means to that the electrons wire. drift downward. A magnetic field that emerges from the plane of the page causes L : by vectors a Equation defining equa

5 Magnetic Force on aa Current current through Carrying Wire The directionfig. of the force A depends flexible on wire the passes direction be- of thenow current. to the left. i = 0 A force acts on a B field. B B B (a) (b) (c) i i 28-8 Magne We have alread exerts a sidewa transmitted to sideways out of In Fig ends, extends t The magnetic fi 28-14b,a curren In Fig c, w the left. Figure 28-1 of the conducti Equation 28-3, tude ev d B must must be directe ence a force to t If, in Fig. 28 or the direction i i

6 Magnetic Force on a Current Carrying Wire The force on the wire in a uniform magnetic field is given by: F = I L B where L is a distance vector that points along the length of the wire in the direction of the conventional current I and is as long as the part of the wire inside the field is. By considering the force on an individual charge, we can motivate this equation.

7 Magnetic Force on a Current Carrying Wire The force on an individual conduction electron is F B = ( e) v d B. The total force will be the sum of the force on all the moving charges together. How much conduction charge is in the wire?

8 Magnetic Force on a Current Carrying Wire The force on an individual conduction electron is F B = ( e) v d B. The total force will be the sum of the force on all the moving charges together. How much conduction charge is in the wire? q = env where n is the volume density of charge carriers, and V is the volume of the wire. Also, this charge is negative, since the flowing charges are electrons.

9 Magnetic Force on a Current Carrying Wire F B = env v d B v d = I nea IL = envv d since V = AL where L is the length of the wire. If the wire is straight and in a uniform field and we define L to be a vector of length L pointed in the direction of the conventional current, then: F B = I L B

10 wire t. The n the Magnetic Force on The a force Current is perpendicular Carrying Wire to both the field and the length. 8-28) ts by lated intro-. i F B B φ L Fig A wire carrying current i makes an angle f with magnetic field B :. The wire has F length = L in the field and length vector L : I L B (in the direction of the cur- : : :

11 Torque on a Loop of Wire with a Current Or, how to turn electricity into motion. ON A CURRENT-CARRYING WIRE PART Consider two wires in a magnetic field with currents flowing in opposite directions MAGNETIC FORCE ON A CURRENT-CARRYING WIRE wire The force is perpendicular ht. The or the field is not to both uniform, the field we can and imagine the length. the wire The force is perpendicula n ght the segments and apply Eq to each segment. The to both the field and the l hole is then the vector sum of all the forces on the F In the differential limit, we B can write F 8-28) B B df : B i dl : B :, (28-28) ts by φ B i ltant force on any given arrangement of currents by φ i that arrangement. L L ear lated in mind that there is no such thing as an isolated ntro- ent of length dl.there must always be a way to introegment at one end and take it out at the other end. Fig A wire carrying current i They makes will an experience angle f with forces magnetic in opposite field B : Fig A wire carrying cur. directions. makes an angle f with magnetic fi The wire has length L in the field and The wire has length L in the field length vector (in the direction of the cur- length vector L : L : Fig A wire carrying curr B φ L FB i (in the direction o : :

12 Torque on a Loop of Wire with a Current This is the situation 752 that occurs when a loop of wire is placed in a CHAPTER 28 MAGNETIC FIELDS B-field. N Fig The elements of an electric motor.a rectangular loop of wire, carrying a These opposing forces on opposite sides of the loop creates action. a current and free to rotate about a fixed axis, torque on the loop. is placed in a magnetic field. Magnetic forces on the wire produce a torque that rotates it.a commutator (not shown) reverses the direction of the current every half-revoi F B F i S 28-9 Torque on Much of the world s are the magnetic fo forces that a magneti Figure sho loop immersed in a duce a torque on th many essential detai of a magnetic field o Figure 28-19a sh i through uniform m its long sides, labeled into the page), but it rent into and out of t

13 Torque on a Loop of Wire with a Current The current on the two sides away from the axle gives an upward 752 force on the left and downward CHAPTER on the 28 right. MAGNETIC FIELDS 28-9 Torque on F i Much of the world s are the magnetic fo forces that a magneti Figure sho N i S loop immersed in a B F duce a torque on th many essential detai Fig The elements of an electric of a magnetic field o motor.a rectangular loop of wire, carrying a On the two ends that connect to the axle, the force is action. zero when current and free to rotate about a fixed axis, the loop lays flat Figure 28-19a sh is placed parallel in a to magnetic the B-field. Magnetic i through uniform m forces on the wire produce a torque that ro- When the loop rotates, it.a commutator the forces(not on shown) those reverses two ends are itsequal long sides, and labeled the direction of the current every half-revolution so that the torque always acts in the rent into and out of t into the page), but it opposite. same direction. To define the or

14 B Torque on a Loop (b) of Wire with a Current ular loop, of carrying a curorm magnetic lign the normal on of the field. (a) ing in the direc-. (b) A perspechow the righttion of n :, which is ne of the loop. (c) rom side 2. ated. (c) F 1 i b Rotation Side 1 θ Side 2 Side 3 F 3 n B τ F = r F ; τ net = i τ i τ net = r 1 F 1 + r 2 F 2

15 Torque on a Loop of Wire with a Current B (b) F 1 = Ia B = iab j = F 3 ular loop, of carrying a curorm magnetic lign the normal on of the field. (a) ing in the direc-. (b) A perspechow the righttion of n :, which is ne of the loop. (c) rom side 2. ated. (c) F 1 i b Rotation Side 1 θ Side 2 Side 3 F 3 n B τ net = r 1 F 1 + r 3 F 3

16 Torque F 3 onb a Loop of Wire with a Current a (b) 9 A rectangular loop, of nd width b and carrying a curcatedin a uniform magnetic rque t acts to align the normal ith the direction of the field. (a) as seen by looking in the direcmagnetic field. (b) A perspecloop showing how the rightgives the direction of n :, which is ular to the plane of the loop. (c) w of the loop, from side 2. rotates as indicated. (c) F 1 i b Rotation Side 1 θ Side 2 Side 3 F 3 n B τ net = r 1 F 1 + r 2 F ( ) 2 ( ) b b = (IaB) sin θ + (IaB) sin θ 2 2 [cw in diag.] Noting that the area of the loop A = ab: τ = IAB sin θ

17 as discussed with regard to Figure and is zero when the field is ormal Torque to the on plane a Loop of the loop of (u Wire 5 0). with a Current S F 2 b 2 u b sin u 2 S A O u B S ure An edge view e loop in Figure the normal to the loop n angle u with respect to magnetic field. S F 4 When the normal to the loop makes an τ angle = IAB u with sinthe θ magnetic field, the moment arm for the torque is (b/2) sin u. We can make this expression more compact by defining A = Aˆn where ˆn is normal to the loop plane. τ = IA B

18 ick Quiz loop (seen s the great- (c)? Which test net a b i) Rank the magnitudes of the torques acting on the rectanguand Torque (c) shown onedge-on a Loopin of Figure Wire Question from highest to lowest. ntical Which and carry of the the rectangular same current. loops(ii) hasrank the largest the magnitudes of the ting on net the force rectangular acting on loops it? shown in Figure from highc (A) a ent of a Coil (B) b consists (C) cof 25 turns of wire and carries a current of 15.0 ma. ne of (D) the coil. all the same moment 1 Serway of the & coil. Jewett, 9th ed.

19 ick Quiz loop (seen s the great- (c)? Which test net a b i) Rank the magnitudes of the torques acting on the rectanguand Torque (c) shown onedge-on a Loopin of Figure Wire Question from highest to lowest. ntical Which and carry of the the rectangular same current. loops(ii) hasrank the largest the magnitudes of the ting on net the force rectangular acting on loops it? shown in Figure from highc (A) a ent of a Coil (B) b consists (C) cof 25 turns of wire and carries a current of 15.0 ma. ne of (D) the coil. all the same moment 1 Serway of the & coil. Jewett, 9th ed.

20 ick Quiz loop (seen s the great- (c)? Which test net a b i) Rank the magnitudes of the torques acting on the rectanguand Torque (c) shown onedge-on a Loopin of Figure Wire Question from highest to lowest. ntical Rank and carry the magnitudes the same current. of the torques (ii) Rank acting the onmagnitudes the rectangular of ting on loops the from rectangular highest to loops lowest. shown in Figure from highc (A) a, b, c ent of a Coil (B) b, a, c consists (C) c, of b, 25 aturns of wire and carries a current of 15.0 ma. ne of (D) the coil. c, a, b moment 1 Serway of the & coil. Jewett, 9th ed.

21 ick Quiz loop (seen s the great- (c)? Which test net a b i) Rank the magnitudes of the torques acting on the rectanguand Torque (c) shown onedge-on a Loopin of Figure Wire Question from highest to lowest. ntical Rank and carry the magnitudes the same current. of the torques (ii) Rank acting the onmagnitudes the rectangular of ting on loops the from rectangular highest to loops lowest. shown in Figure from highc (A) a, b, c ent of a Coil (B) b, a, c consists (C) c, of b, 25 aturns of wire and carries a current of 15.0 ma. ne of (D) the coil. c, a, b moment 1 Serway of the & coil. Jewett, 9th ed.

22 Torque on a Coil of Wire with a Current τ = IA B Remarkably, that equation also holds for other shapes of loop as long as they are flat (in one plane). A is the area of the loop. For a coil of N loops stacked together, the effect of each loop adds up: τ = NIA B

23 Electric Motors This effect can be used to turn electricity into mechanical work. 1 Figure from hyperphysics.phys-arstr.gsu.edu

24 Electric Motors Either direct current (DC) or alternating current (AC) can be used for a motor. 1 Figure from hyperphysics.phys-arstr.gsu.edu

25 Torque on a Loop 752 ofchapter Wire with 28 MAGNETIC a Current FIELDS 28-9 Torque on F i Much of the world s are the magnetic fo forces that a magneti Figure sho N i S loop immersed in a B F duce a torque on th many essential detai Fig The elements of an electric of a magnetic field o motor.a rectangular loop of wire, carrying a action. current and free to rotate about a fixed axis, Figure 28-19a sh is placed in a magnetic τ = IAB field. sinmagnetic θ i through uniform m forces on the wire produce a torque that rotates We can make this expression it.a commutator more(not compact shown) reverses by definingitsa long = Aˆn sides, labeled where ˆn is normal the direction to the loop of the plane. current every half-revolution so that the torque always acts in the rent into and out of t into the page), but it same direction. To define the or τ = IA B vector n : that is per aright-hand rule for right hand in the dire

26 Magnetic Moment for a Current Loop For a current loop, we can define the magnetic moment of the loop as µ = IA And for a coil of wire carrying a current: µ = IA Then the expression for the torque can be written τ = µ B

27 Magnetic Dipole Moment Recall our definition for the Electric dipole moment: dipole moment: p = q d z where d is a vector pointing from the negative charge to the positive charge, and its magnitude d is the separation of the Up here the +q charges and each charge in the dipole has magnitude q. field dominates. + that z d.at such larg proximation, we can neg The product qd, w dipole, is the magnitude p : of the dipole. (The un Torque on a electric dipole in an electric field: Dipole center p τ = p E Potential energy: U = p E p : The direction of is t dipole, as indicated in

28 Current Loop vs Bar Magnet RRYING COIL AS A MAGNETIC DIPOLE PART A loop of wire with a current in it produce a similar magnetic field as a bar magnet. etic dipole. space? The seful; so we sider only a central axis, e magnetic i N µ i (29-26) S B the point in f the magoment : of Fig A current loop produces a

29 Magnetic Dipole Moment For a pair of magnetic charges at either end of a779 thin bar magnet, this would be: µ = q m d. T-CARRYING COIL AS A MAGNETIC DIPOLE PART 3 magnetic dipole. ding space? The aw useful; so we consider only a ular central axis, of the magnetic i N µ i (29-26) S B e of the point in ion of the magle moment : of Fig A current loop produces a magnetic field like that of a bar magnet and

30 Magnetic Dipole Moment magnetic dipole moment, µ The quantity relating an external magnetic field that a magnet or coil of wire is in to the torque on the magnet or coil due to that field. τ = µ B For a magnet, it is a vector pointing from the south pole of a magnet to the north pole, that is proportional to the strength of the B-field produced by the magnet itself. For a coil, it is defined according the the right hand rule for current in a wire loop and is proportional to the coil area and current.

31 Potential Energy of a Dipole in a B-Field The magnetic moment vector attempts to align with the magnetic field. τ = µ B B µ Us netic fi in whic W i µ Highest energy The energy Fig. can be foundthe by integrating orientations the torque of highest over the angle of rotation. and lowest energy of a magnetic dipole (here a coil carrying U = current) µ B in an external magnetic field. The direction of the cur- B : Lowest energy i which exerte In each the vec A

32 are shown in Table Question CHECKPOINT 5 The figure shows four orientations, at angle θ, of a magnetic dipole : The figure moment shows µ in four a magnetic orientations, field. at angle Rank u, the of orientations a magnetic dipole according moment to in a magnetic field. Rank the orientations according to (a) the magnitude of the torque on the magnitude of the torque on the dipole, greatest first. the dipole and (b) the orientation energy of the dipole, greatest first. 1 µ µ 2 θ θ θ θ B 4 µ µ 3 (A) 1 and 2, 3 and 4 (B) 1 and 4, 2 and 3 (C) 3, 2, 1, 4 (D) all the same 1 Halliday, Resnick, Walker, 9th ed, page 745.

33 are shown in Table Question CHECKPOINT 5 The figure shows four orientations, at angle θ, of a magnetic dipole : The figure moment shows µ in four a magnetic orientations, field. at angle Rank u, the of orientations a magnetic dipole according moment to in a magnetic field. Rank the orientations according to (a) the magnitude of the torque on the magnitude of the torque on the dipole, greatest first. the dipole and (b) the orientation energy of the dipole, greatest first. 1 µ µ 2 θ θ θ θ B 4 µ µ 3 (A) 1 and 2, 3 and 4 (B) 1 and 4, 2 and 3 (C) 3, 2, 1, 4 (D) all the same 1 Halliday, Resnick, Walker, 9th ed, page 745.

34 are shown in Table Question CHECKPOINT 5 The figure shows four orientations, at angle θ, of a magnetic : The figure dipole shows moment four µ orientations, in a magnetic at angle field. u, Rank of a magnetic the orientations dipole moment in a magnetic field. Rank the orientations according to (a) the magnitude of the torque on according to the orientation energy of the dipole, greatest first. the dipole and (b) the orientation energy of the dipole, greatest first. 1 µ µ 2 θ θ θ θ B 4 µ µ 3 (A) 1 and 2, 3 and 4 (B) 1 and 4, 2 and 3 (C) 3, 2, 1, 4 (D) all the same 1 Halliday, Resnick, Walker, 9th ed, page 745.

35 are shown in Table Question CHECKPOINT 5 The figure shows four orientations, at angle θ, of a magnetic : The figure dipole shows moment four µ orientations, in a magnetic at angle field. u, Rank of a magnetic the orientations dipole moment in a magnetic field. Rank the orientations according to (a) the magnitude of the torque on according to the orientation energy of the dipole, greatest first. the dipole and (b) the orientation energy of the dipole, greatest first. 1 µ µ 2 θ θ θ θ B 4 µ µ 3 (A) 1 and 2, 3 and 4 (B) 1 and 4, 2 and 3 (C) 3, 2, 1, 4 (D) all the same 1 Halliday, Resnick, Walker, 9th ed, page 745.

36 Electric Dipole and Magnetic Dipole electric dipole magnetic dipole torque τ τ = p E τ = µ B potential energy U U = p E U = µ B

37 Magnetism in Matter Ordinary matter exhibits magnetic properties. Now we will see why. This is covered in Chapter 32 of the book, but we will not go into full detail.

38 ctrons do not n an electron n meaning Atomsof and subatomic particles also have magnetic moments! ics. Why? Consider a classical model of a hydrogen atom. One electron orbits the nucleus. Magnetism in Matter: Magnetic Moment of Atoms ows, in which s that is much owever, the ich we need µ = IA z cular path of the negative i (of positive de of the orom Eq i e r L orb µ orb A v (32-33)

39 Magnetic Moment of Atoms The current is the rate of charge flow with time: I = e T = e v 2πr assuming an orbital radius of r, speed v. µ = IA = e v 2πr (πr 2ˆn) = evr 2 ˆn

40 Magnetic Moment of Atoms The current is the rate of charge flow with time: I = e T = e v 2πr assuming an orbital radius of r, speed v. µ = IA = e v (πr 2ˆn) 2πr = evr 2 ˆn Recall that for a particle of mass m orbiting at a radius r, velocity v, the angular momentum is: L = mvr µ = e 2m e L

41 Electron Spin Angular Momentum Electrons also 872have another CHAPTER kind of 32 angular MAXWELL S momentum: EQUATIONS; intrinsic angular momentum. This is also called spin. Spin is an inherent property For an electron, of all electrons. the spinit cannot be Substituting understood with classical is opposite mechanics, the magnetic but also contributes a magnetic moment. dipole moment. B S where the pl to the z axis, The qua : µ s Spin magnet be expressed component o

42 particle is meaningless. Rotation about its axis a applies only to a rigid object, with sical interpreta an extent in space, as in Chapter spin is on the 10. Spin angular momentum is You might imagine an electron as a rigid charge sphere spinning tum S actually a relativistic effect. L on due to an axis through its center... of an electron Electron Spin Angular Momentum S m S spin The magnetic the value This combinat Figure Classical model of...but really, it s not. a spinning electron. We can adopt this model to remind ourselves that electrons have an intrinsic

43 For an electron, the spin is opposite the magnetic dipole moment. Electron Spin Angular Momentum B S where the pl to the z axis, The qua Electron s Fig. spin magneticthe dipole spin moment:, spin magnetic dipole moment, and magnetic dipole field B : of an electron µ s = represented g e S as a microscopic 2m e sphere. where g 2. : s S : µ s Spin magnet be expressed component o (The quantu veals that m s,z When an be associated :

44 Magnetic Moment of Atoms In atoms with many electrons, the electrons tend to cancel out each other s magnetic moments, but outer-shell, unpaired electrons can contribute a significant magnetic moment. The particles in the nucleus also have magnetic moments, but they are much smaller. Most of an atom s magnetic moment comes from unpaired electons. These tiny magnetic moments add up to big effects in bulk materials.

45 Three Types of Bulk Magnetism ferromagnetism paramagnetism diamagnetism

46 Ferromagnetism Atoms of ferromagnetic materials have non-zero magnetic moments. Interactions between outer electrons in different atoms causes alignment of each atom s magnetic moment. Magnetic moments reenforce each other and will tend to spontaneously align within domains.

47 Ferromagnetism Atoms of ferromagnetic materials have non-zero magnetic moments. Interactions between outer electrons in different atoms causes alignment of each atom s magnetic moment. Magnetic moments reenforce each other and will tend to spontaneously align within domains. Examples of ferromagnetic materials: iron nickel cobalt gadolinium dysprosium

48 the atomic magnetic dipoles are Ferromagnetism randomly oriented. ple as c, the very etiza- agita-. itical agned the ation tance ances No external B-field a

49 larger, giving the sample a net Ferromagnetism magnetization. interof an ernal ment mize Applied external B-field weak sing gnegnet- b da S S B

50 c netic the external field become very Ferromagnetism small. tion, n an magstatic antinetic elechat is Strong external B-field S B S B

51 Paramagnetism Atoms of paramagnetic materials have non-zero dipole moments, but electrons of different atoms do not interact with each other. They can interact with a strong magnetic field, and will align with the field. Paramagnetic effects tend to be much smaller than ferromagnetic ones.

52 Paramagnetism Atoms of paramagnetic materials have non-zero dipole moments, but electrons of different atoms do not interact with each other. They can interact with a strong magnetic field, and will align with the field. Paramagnetic effects tend to be much smaller than ferromagnetic ones. Examples of paramagnetic materials: Tungsten Cesium Aluminium Lithium Magnesium Sodium

53 Paramagnetism Liquid oxygen stream deflected in a strong magnetic field. The stream collects in the field. 1 Image created by Pieter Kuiper.

54 Diamagnetism Diamagnetism occurs in all materials, but is a weak effect, so it is drowned out if a material is ferro- or paramagnetic. It is the dominant (but weak) effect when the net magnetic moment of a material s atoms is zero. The field polarizes the atoms and the resulting magnetic moments oppose the external magnetic field.

55 Diamagnetism Diamagnetism occurs in all materials, but is a weak effect, so it is drowned out if a material is ferro- or paramagnetic. It is the dominant (but weak) effect when the net magnetic moment of a material s atoms is zero. The field polarizes the atoms and the resulting magnetic moments oppose the external magnetic field. Examples of diamagnetic materials: Pyrolytic carbon Bismuth Mercury Silver diamond (form of Carbon) water Also superconductors can be said to exhibit extreme diamagnetism.

56 Diamagnetism 1 Levitating pyrolytic carbon on neodymium magnets. Image by Splarka.

57 Diamagnetism 1 Magnet photo by Mai-Linh Doan, Wikipedia; Frog photo by Lijnis Nelemans/High Field Magnet Laboratory/Radboud University Nijmeg.

58 Summary force on a wire in a magnetic field torque on a wire loop in a magnetic field relating a current loop to a magnet magnetic dipole moment torque and potential energy of magnetic dipole magnetism in matter Homework Halliday, Resnick, Walker: PREVIOUS: Ch 28, Questions: 3; Problems: 13, 15, 27, 33, 35, 39, 41 NEW: Ch 28, Problems: 49, 54, 55, 57, 61, 65