Lesson 9 Dipoles and Magnets Lawence B. Rees 007. You may make a single copy of this document fo pesonal use without witten pemission. 9.0 Intoduction In this chapte we will lean about an assotment of topics that at fist glance seem dissimila, but which ae actually quite closely elated. In the fist pat, we will lean about the foce on a cuent-caying wie, electic and magnetic dipoles, and how dipoles behave in unifom and nonunifom fields. In the second half, we will talk about diffeent ways in which matte can espond to extenal magnetic fields. We ll lean about paamagnetism, diamagnetism, and feomagnetism and discuss some popeties of pemanent magnets. 9.1 The Foce on a Cuent-Caying Wie We know fom the Loentz Foce Law that a chage moving in a magnetic field expeiences a foce F = qv B. In a cuent-caying wie, the positive ions ae stationay (elative to an obseve at est with espect the wie), but the electons ae moving. This means that the electons in the wie expeience a magnetic foce, although the positive ions do not. To see how this affects a wie, let us think of the aangement shown below. B + L Figue 9.1 A section of wie in a magnetic field. When the switch is closed, chages (think of positive chages) begin moving fom the positive teminal of the battey. As the chages pass though the egion whee the magnetic field is located, they expeience a foce into the sceen. The chages move feely in the wie, pushed along to the ight by the battey and to the back of the wie by the magnetic field. Howeve, the chages can only move a bit towad the back of the wie befoe they ae epelled by othe chages collecting nea the back side of the wie. At this point, the foce exeted on the chages is tansfeed to the ions in the wie. That is, the entie wie feels a foce equal to the sum of the foces acting upon each chage on the wie. 1
Think About It Would the foce on the wie be any diffeent if we took into consideation the fact that the cuent is actually caused by electons moving to the left though the egion of the magnetic field? ow let s do a little algeba. Let be the numbe of chage caies (valence electons) pe unit length in the wie, L be the length of the egion whee the magnetic field is located, and be the total numbe of electons in this egion. Then = λl and the total foce on the wie is F = evb = λlevb whee v is the dift velocity of electons in the wie. ow, let T be the time it takes the electons to go a distance L in the wie, so v = L / T. The cuent in the wie is then Putting this all togethe e λle i = = = λev. T T F = λ LevB = We can genealize this to the fomula to something that is equivalent to the Loentz Foce Law but fo cuents athe than individual paticles: ilb (9.1) F = il B whee F is the foce on a segment of wie in newtons (). L is the length of the wie segment in metes (m). The diection of L is the diection of the cuent. i is the cuent in the wie in ampees (A). A little bad calculus can be used as a mnemonic device to help emembe this elationship: dl dq df = dq v B = dq B = dl B = idl B dt dt
Hee dl is the bit of foce on a small chage dq, and dl is a little segment of wie pointing in the diection of the cuent. Things to emembe: The foce on a segment of cuent-caying wie is F = il B 9. Magnetic Dipoles Many times an entie cicuit is located within a magnetic field. To see how the field affects a cicuit, let us take a paticulaly simple case of a cuent taveling aound a squae loop within a unifom magnetic field. The length of each side of the loop is taken to be a. F F F I Figue 9. A cuent-caying loop of wie in a unifom magnetic field. F uch a uch a loop, with a magnetic field passing pependiculaly though the loop, is shown in Fig. 9.. By applying Eq. (9.1) to each section of the loop, we find that the foce in each segment of the wie is diected away fom the cente of the loop. The total foce on the loop is zeo. Futhemoe, thee is no toque tending to otate the loop. F I a sin F Figue 9.3 A cuent-caying loop tilted at an angle 3 to the magnetic field.
ow let s look at a simila loop, but with the magnetic field passing though the loop at an angle. We call the angle between the magnetic field and the nomal to the loop,. A view of the loop seen edge-on is shown in Fig. 9.3. In this case, the foces on the sides of the loop point again away fom the cente of the loop. (These foces ae not shown in the figue.) Hence, they cancel out as befoe. The foces on the top and bottom segments of the wie ae still in opposite diections, as shown, but they ae no longe diected away fom the cente of the loop. The net esult is that the total foce on the loop is still zeo, but now thee is a net toque on the loop, tending to otate it in a counteclockwise diection. The magnitude of the toque fom each foce is the poduct of the moment am, a/ sin, and the foce F. The total toque is just twice this amount: a τ = sin F = a sin iab = ia Bsin. We now define a quantity called the magnetic dipole moment of the cuent loop. This is a vecto epesented by the Geek lette mu, µ. The magnitude of the magnetic dipole moment is just the poduct of the cuent in the loop and the aea of the loop. The diection is taken to be the nomal to the loop; howeve, this is still ambiguous as thee ae two diffeent possibilities of diections fo the nomal. Fo example, in Fig. 9., the nomal could be eithe into the sceen o out of the sceen. In ode to esolve this ambiguity, we establish a ule. Right-hand Rule fo the Magnetic Dipole Moment Cul the finges of you ight hand in the diection cuent flows aound the loop. The diection of the magnetic dipole moment is in the diection you thumb points. µ I Figue 9.4 The ight-hand ule fo the magnetic dipole 4
We can wite the magnetic moment as: Magnetic Dipole Moment (9.) = i A µ whee µ is the magnetic dipole moment in ampee-metes (Am ). i is the cuent in ampees (A). A is the diected ae in squae metes (m ). The diection of A is given by the ight hand ule above. In tems of the magnetic dipole moment, we see that we can easily wite the toque on a cuent loop as: Toque on a Magnetic Dipole (9.3) τ = µ B whee τ is the toque in newton-metes (m). µ is the magnetic dipole moment in ampee-metes (Am ). B is the magnetic field in tesla (T). This equation summaizes thee impotant facts: 1) The toque on a dipole in a unifom field is popotional to the magnitude of the dipole moment and to the magnitude of the magnetic field. ) The toque on a dipole in a unifom field is geatest when the dipole is pependicula to the field and zeo when the dipole is paallel o antipaallel to the field. 3) The toque on the dipole tends to otate the dipole moment into the diection of the magnetic field. If a dipole is oiented so that its dipole moment is in the diection of the field, it expeiences no toque. To otate the dipole by an angle, a toque is equied and wok must be done. In analogy with wok whee W = Fdx, we have W = τ d = µ Bsin d = µ B 1 1 ( cos ) cos 1 If we otate the dipole at a constant speed, the wok done to the dipole esults only in a change in potential enegy. We can then define the potential enegy of a magnetic dipole in a unifom magnetic field by: U = µ B cos + C 5
whee C is a constant of integation. Fo convenience, we can let the constant be equal to zeo. Then we can wite this in the simple fom: Enegy of a Magnetic Dipole (9.4) U = µ B whee U is the potential enegy of a magnetic dipole in joules (J). µ is the magnetic dipole moment in ampee-metes (Am ). B is the magnetic field in tesla (T). ote that the potential enegy is zeo when the dipole is pependicula to the field. This, howeve, does not mean that the potential enegy is a minimum when dipole moment and the field ae pependicula! The minimum value of the potential enegy is U = B when the dipole is paallel to the field, at = 0. The maximum value of the potential enegy is U = + B when the dipole is antipaallel to the field, at = π. o go back and eead this paagaph many students get confused by this! F B I µ F Figue 9.5 A cuent-caying loop of wie in a nonunifom magnetic field. µ is paallel to B at the cente of the loop. o fa, we have looked at what happens to magnetic dipoles when they ae placed in unifom magnetic fields. What happens when the field is not unifom? Conside a cicula cuent loop shown edge-on in Fig. 9.5. Cuent goes into the sceen at the bottom of the loop and comes out of the sceen at the top of the loop. The diection of the magnetic field vaies somewhat acoss the loop; howeve, at the cente of the loop the dipole moment and the magnetic field ae paallel. The foces on the loop at the top and bottom ae shown in the figue. If we add the foces fom all the sections of the loop, we see that the net foce must be to the left. In geneal, it can be shown that when the dipole moment and the field ae paallel, the dipole 6
expeiences a foce in the diection of inceasing magnetic field. On the othe hand, if the dipole moment is antipaallel to the field, the net foce is towad smalle fields. Howeve, if the dipole moment is neithe paallel no antipaallel to the magnetic field, the dipole also expeiences a toque that tends to align the dipole moment with the field. In geneal, the motion can be vey complicated, but usually, thee ae fictional foces that damp the oscillatoy motion that can ensue, and the dipole eventually aligns with the extenal field and is attacted into the egion whee the field is stonge. µ Figue 9.6 The inteaction of two pemanent magnets. At this point, you ae pobably asking youself why we bothe leaning about what happens to cuent loops in magnetic fields. The best answe is that natue is full of tiny cuent loops: electons obiting nuclei in atoms. Pemanent magnets ae a collection of many dipoles. The foces between pemanent magnets can be quite complicated because thee ae so many dipoles inteacting; howeve, in pinciple these foces can be undestood in tems of dipoles. The dipole moment of a pemanent magnet goes fom the noth pole to the south pole (inside the magnet), as shown in Fig. 9.6. When we place the noth pole of one magnet nea anothe magnet, 7
the dipole moment of the second magnet aligns with the field of the fist magnet, so the south pole of the second magnet faces the noth pole of the fist magnet. The second magnet then feels a foce in the diection of inceasing magnetic field, towad the fist magnet. Things to emembe: The magnetic dipole moment is given by = i A µ whee the diection of A is given by the ighthand ule: finges ae I, thumb is A. The toque on a magnetic dipole in a magnetic field is τ = µ B. The potential enegy of a magnetic dipole in a magnetic field is U = µ B. 9.3 Electic Dipoles We have aleady encounteed electic dipoles in Lesson 6 when we leaned how dipoles in dielectic mateials can align to decease the electic field and theeby incease the capacitance of a capacito. The simplest fom of an electic dipole is a chage +q joined by a igid od to a chage q. We will call the distance between the chages. If an electic dipole is placed in a unifom electic field, as shown in Fig.9.7, the two chages feel equal foces in opposite diections. The net foce on the dipole is zeo. Howeve, thee is a net toque on the dipole. The toque on each chage is given by the poduct of the foce, F = qe, and the moment am, sin /, whee is the angle between the od and the field, as shown in the figue. The total toque on the dipole is l τ = F sin = qelsin. E F l sin + q p l q F Figue 9.7 An electic dipole in a unifom electic field. We define an electic dipole moment p as vecto of magnitude negative chage to the positive chage, as shown in Fig. 9.7. Thus: p = ql pointing fom the 8
whee Electic Dipole Moment p = ql p is the electic dipole moment in units of Cm, q is the magnitude of the chage on each end of the dipole, in C l is the vecto distance fom the negative chage to the positive chage, in m In tems of p, the toque on the dipole is: Toque on an Electic Dipole (9.5) τ = p E. ote that this equation is vey simila to Eq. (9.3). Also note that the diection of p is chosen so that the toque tends to otate p into E. Just as in the case of magnetic dipoles, the potential enegy of an electic dipole is given by the wok equied to otate it in the electic field. W = τ d = pe sin d = pe 1 1 ( cos ) cos 1 This leads to the elationship Potential Enegy of an Electic Dipole (9.6) U = p E which esembles Eq. (9.4) fo magnetic dipoles. The behavio of electic dipoles in non-unifom fields also has similaities to the behavio of magnetic dipoles in non-unifom fields. The dipole expeiences a toque that tends to align the dipole moment with the electic field, and when aligned, the dipole expeiences a net foce in the diection the field is inceasing. 9
E F q p + q F Figue 9.8 An electic dipole in a non-unifom electic field. Things to emembe: The electic dipole moment is defined by p = q l whee the diection of l is fom q to +q. The toque on a magnetic dipole in a magnetic field is τ = p E. The potential enegy of a magnetic dipole in a magnetic field is U = p E. 9.4 Atomic Dipoles, Domain Alignment, and Themal Disalignment In the ealy 0 th Centuy, scientists discoveed that individual electons also behave like little magnetic dipoles. They initially thought of the electon as a vey tiny sphee of chage spinning like the eath about an axis. Hence, electons wee said to have spin. Howeve, thee wee inconsistencies between the spinning sphee model and expeimental obsevations. In paticula, as fa as we can now tell, the electon is a point paticle a paticle with no size at all. We now think of spin as an intinsic chaacteistic of an electon, one that obeys the ules of angula momentum. But we eally don t know what spin is and we eally don t know why electons behave like little magnets. We do know that many othe paticles also have magnetic dipole moments. Even neutons, which ae neutal, behave like little magnets. In fact, the magnetic dipole moment of a neuton suggests that it has positive chage towad the cente and negative chage on the outside. ince nomal matte is made up of potons, neutons, and electons, each of which ae little magnets themselves, it is not supising that atoms ae little magnets as well. But in addition to the magnetic dipole moment that aises fom paticle spin, thee is also a contibution to the total dipole moment of an atom fom the obital angula momentum of electons. That is, each electon in an atom poduces a tiny cuent loop that adds to the oveall magnetic dipole moment of the atom. As it tuns out, potons, neutons, and electons ae all in obitals occupied, whee possible, by two paticles with opposite spins. The magnetic dipole moments of these paied paticles then cancel. Futhemoe, the obital angula momentum of electons in diffeent obitals also tends to cancel. This geneally leaves atoms with vey small, but non-zeo, magnetic dipole moments. 10
When solids ae aanged in cystals, thee ae two competing effects that usually dominate a mateial s esponse to magnetic fields. These ae domain alignment and themal disalignment. To undestand domain alignment, we can think of atoms as little magnets. If we take a stack of magnets, noth poles attact south poles, and the magnets tend to align as shown in Fig, 9.9. The dipole moments of atoms in a cystal pefe to be aligned with all thei dipole moments pointing in the same diection. In mateials whee this effect pedominates, lage numbes of atoms ae aligned in domains whee the dipole moments ae all paallel. uch mateial ae called feomagnets. Often thee ae many small domains in a cystal with the diection of the dipole moment in diffeent domains andomly oiented, as illustated in Fig. 9.10. An example of such a mateial would be an ion nail. If the domains, howeve, have a pedominant diection as shown in Fig. 9.11, the mateial is a pemanent magnet. Figue 9.9 Magnets tend to align with thei dipole moments in the same diection. On the othe hand, atoms in cystals have vibational enegy. As atoms vibate, the oientation of thei magnetic dipole moments tends to be andomized. This is called themal disalignment. In almost all mateials, magnetic domains can not be established because themal disalignment oveshadows domain alignment. Even feomagnetic mateials cease to be feomagnetic when heated enough. The tempeatue at which a given feomagnetic substance ceases to have aligned domains is call the Cuie Tempeatue of the mateial. 11
Figue 9.10 Magnetic domains in non-magnetized soft ion. Figue 9.11 Magnetic domains in a pemanent magnet Things to emembe: Elementay paticles have magnetic dipole moments associated with thei spin. Atoms have dipole moments that aise fom spin and the obital motion of electons. Atomic dipoles tend to align with one anothe. Themal agitation tends to disalign the dipoles. 9.5 Matte in Extenal Magnetic Fields When matte is placed in an extenal magnetic field, its behavio falls into one of thee categoies: feomagnetic, paamagnetic, o diamagnetic. We have aleady seen that feomagnetic mateials ae those in which domains can emain aligned in spite of themal effects. We will conside chaacteistics of feomagnetic mateials in moe detail in the next section. When a small pemanent magnet is placed in a unifom magnetic field, it feels a toque that tends to align its magnetic moment with the extenal field. Once the magnetic dipole moment points in the diection of the extenal field, otational inetia will cay the magnet past the position of minimum enegy. omewhat like a pendulum o a mass attached to a sping, the dipole will continue to oscillate until fictional foces damp out the oscillatoy motion. At that point, the dipole will emain aligned in the field, as illustated in Fig. 9.1. Figue 9.1 A pemanent magnet o a paamagnetic atom in an extenal magnetic field. Most mateials have atoms that do pecisely this, except that themal effects egulaly knock the diection of the dipole aound. Even in spite of themal effects, the atoms do align 1 µ B
pefeentially with thei dipole moments in the diection of the extenal field. uch mateials ae called paamagnetic. We can chaacteize just how much the atoms o molecules of a mateial tend to align with the extenal field by a quantity called the magnetic susceptibility of the mateial. Befoe we define magnetic susceptibility, howeve, we need to clealy undestand the oigin of what we call the intenal magnetic field. All the atoms in a mateial contibute to the total magnetic dipole moment of a sample. The magnetic dipole moment pe unit volume is a quantity called the magnetization of a mateial. We may wite this as: M µ i i= 1 = Volume The intenal magnetic field of a sample is elated to the magnetization by the elationship (9.7) B = M. int µ 0 The magnetic susceptibility elates the stength of the intenal field to the extenal field. That is, (9.8 Magnetic susceptibility) Bint = χ Bext whee P (chi) is the magnetic susceptibility. It is dimensionless. The total magnetic field inside of a mateial is the sum of the intenal and extenal magnetic fields. The magnetic susceptibilities of paamagnetic mateials ange fom about +10 5 to +10 3. As you can see by these numbes, paamagnetism is a typically a vey weak effect. Because paamagnetic mateials ae weak magnetic dipoles, they ae weakly attacted fom egions of smalle magnetic field into egions of lage magnetic field. (ee Fig. 9.6.) It tuns out that othe mateials behave in exactly the opposite way in extenal magnetic fields. That is, the magnetic dipole moments pefe to align in the diection opposite to the magnetic field, as shown in Fig. 9.13. uch mateials ae called diamagnetic. This behavio is adequately explained only by teating atoms quantum mechanically. 13
µ B Figue 9.13 A diamagnetic atom in an extenal magnetic field. Diamagnetic mateials have susceptibilities that ae negative and vey small in size, anging geneally fom about 10 6 to 10 4. Because the diamagnetic magnets ae aligned opposite to the field, they ae epelled weakly fom egions of lage magnetic fields into egions of smalle magnetic field. Things to emembe: Magnetic susceptibility is defined by B = ee the table at the end of the next section. 9.6 Feomagnetism χ B int ext. Feomagnetic mateials have by fa the most inteesting behavio in extenal magnetic fields. Even in modeate extenal fields, the domain boundaies shift to allow domains aligned with the extenal field to incease in size while domains aligned in othe diections shink in size. Because of this, the intenal field can become much lage than the extenal field. Fo feomagnetic mateials, magnetic susceptibilities ae oughly 10 +3 to 10 +5. 14
B int ( T ) B int = esidual magnetization B ext = coecive foce B ext (mt) Figue 9.14 A typical hysteesis cuve fo a pemanent magnet with points showing esidual magnetization and coecive foce labeled. Howeve, susceptibilities of feomagnetic mateials ae not simple constants as they ae in the paamagnetic and diamagnetic cases. To see why, let us take a piece of unmagnetized feomagnetic mateial. We make a plot of the intenal field vs. the extenal field. uch a plot is called a hysteesis cuve as shown in Fig. 9.14. ote that the scales on the two axes ae not the same, as the intenal field is seveal odes of magnitude lage than the extenal field. Let's begin with no extenal field and no intenal field, so we stat at the oigin of the coodinate system. Then we slowly incease the extenal field and measue the intenal field as we do so. As the extenal field inceases in the positive diection, the intenal field ises apidly until all of the domains ae aligned with the extenal field. Once all the domains ae aligned, the intenal field can no longe incease. Then let's educe the extenal field, which is still positive. ow the aligned domains do not disalign easily, so the mateial follows a diffeent cuve as the extenal field deceases. When the extenal field goes to zeo, thee is still an intenal field that emains. This field is called the esidual magnetization. At this point, we evese the diection of the extenal field and incease its magnitude. Eventually, the intenal field of the magnet is foced to zeo. The extenal field equied to do this is called, athe inappopiately, the coecive foce. 15
If we continue to incease the extenal field stength in the negative diection, we begin to magnetize the mateial in the negative diection, as indicated by the continuation of the hysteesis cuve into the fouth quadant of the figue. Diffeent feomagnetic mateials have hysteesis cuves with diffeent chaacteistics. A pemanent magnet must have a lage esidual magnetization. The best mateials fo pemanent magnets also have a lage coecive foce. oft ion, on the othe hand, has lage intenal fields, but a small esidual magnetization and a small coecive foce. Thus, soft ion is easy to magnetize in an extenal field, but it doesn t keep its magnetization once the extenal field is gone. Things to emembe: paamagnetic 5 3 χ = + 10 to + 10 weakly attacted diamagnetic 6 4 χ 10 to 10 weakly epelled feomagnetic + 3 + 5 χ = + 10 to + 10 stongly attacted The elationship between the intenal field and extenal fo feomagnetic mateials is descibed by a hysteesis cuve. The esidual magnetization is the intenal field emaining in a magnetic mateial when the extenal field is educed to zeo. The coecive foce is the extenal field that must be applied to bing the intenal field to zeo. 16