Sources of the Magnetic Field

Transcription

1 2.2 This is the Nerest One Hed 937 P U Z Z L E R All three of these commonplce items use mgnetism to store informtion. The cssette cn store more thn n hour of music, the floppy disk cn hold the equivlent of hundreds of pges of informtion, nd mny hours of television progrmming cn be recorded on the videotpe. How do these devices work? (George Semple) c h p t e r Sources of the Mgnetic Field Chpter Outline 30.1 The Biot Svrt Lw 30.2 The Mgnetic Force Between Two Prllel Conductors 30.3 Ampère s Lw 30.4 The Mgnetic Field of Solenoid 30.5 Mgnetic Flux 30.6 Guss s Lw in Mgnetism 30.7 Displcement Current nd the Generl Form of Ampère s Lw 30.8 (Optionl) Mgnetism in Mtter 30.9 (Optionl) The Mgnetic Field of the Erth 937

2 938 CHAPTER 30 Sources of the Mgnetic Field n the preceding chpter, we discussed the mgnetic force exerted on chrged prticle moving in mgnetic field. To complete the description of the mgnetic interction, this chpter dels with the origin of the mgnetic field moving chrges. We begin by showing how to use the lw of Biot nd Svrt to clculte the mgnetic field produced t some point in spce by smll current element. Using this formlism nd the principle of superposition, we then clculte the totl mgnetic field due to vrious current distributions. Next, we show how to determine the force between two current-crrying conductors, which leds to the definition of the mpere. We lso introduce Ampère s lw, which is useful in clculting the mgnetic field of highly symmetric configurtion crrying stedy current. This chpter is lso concerned with the complex processes tht occur in mgnetic mterils. All mgnetic effects in mtter cn be explined on the bsis of tomic mgnetic moments, which rise both from the orbitl motion of the electrons nd from n intrinsic property of the electrons known s spin. Properties of the mgnetic field creted by n electric current 30.1 THE BOT SAVART LAW Shortly fter Oersted s discovery in 1819 tht compss needle is deflected by current-crrying conductor, Jen-Bptiste Biot ( ) nd Félix Svrt ( ) performed quntittive experiments on the force exerted by n electric current on nerby mgnet. From their experimentl results, Biot nd Svrt rrived t mthemticl expression tht gives the mgnetic field t some point in spce in terms of the current tht produces the field. Tht expression is bsed on the following experimentl observtions for the mgnetic field db t point P ssocited with length element ds of wire crrying stedy current (Fig. 30.1): The vector db is perpendiculr both to ds (which points in the direction of the current) nd to the unit vector rˆ directed from ds to P. The mgnitude of db is inversely proportionl to r 2, where r is the distnce from ds to P. The mgnitude of db is proportionl to the current nd to the mgnitude ds of the length element ds. The mgnitude of db is proportionl to sin, where is the ngle between the vectors ds nd rˆ. db out P P r Figure 30.1 rˆ rˆ θ ds ds P ds rˆ P db in () (b) (c) () The mgnetic field db t point P due to the current through length element ds is given by the Biot Svrt lw. The direction of the field is out of the pge t P nd into the pge t P. (b) The cross product d s rˆ points out of the pge when rˆ points towrd P. (c) The cross product d s rˆ points into the pge when rˆ points towrd P.

3 30.1 The Biot Svrt Lw 939 These observtions re summrized in the mthemticl formul known tody s the Biot Svrt lw: d B 0 4 ds rˆ r 2 (30.1) Biot Svrt lw where 0 is constnt clled the permebility of free spce: Tm/A (30.2) Permebility of free spce t is importnt to note tht the field d B in Eqution 30.1 is the field creted by the current in only smll length element ds of the conductor. To find the totl mgnetic field B creted t some point by current of finite size, we must sum up contributions from ll current elements ds tht mke up the current. Tht is, we must evlute B by integrting Eqution 30.1: B 0 4 ds rˆ r 2 (30.3) where the integrl is tken over the entire current distribution. This expression must be hndled with specil cre becuse the integrnd is cross product nd therefore vector quntity. We shll see one cse of such n integrtion in Exmple Although we developed the Biot Svrt lw for current-crrying wire, it is lso vlid for current consisting of chrges flowing through spce, such s the electron bem in television set. n tht cse, ds represents the length of smll segment of spce in which the chrges flow. nteresting similrities exist between the Biot Svrt lw for mgnetism nd Coulomb s lw for electrosttics. The current element produces mgnetic field, wheres point chrge produces n electric field. Furthermore, the mgnitude of the mgnetic field vries s the inverse squre of the distnce from the current element, s does the electric field due to point chrge. However, the directions of the two fields re quite different. The electric field creted by point chrge is rdil, but the mgnetic field creted by current element is perpendiculr to both the length element ds nd the unit vector rˆ, s described by the cross product in Eqution Hence, if the conductor lies in the plne of the pge, s shown in Figure 30.1, db points out of the pge t P nd into the pge t P. Another difference between electric nd mgnetic fields is relted to the source of the field. An electric field is estblished by n isolted electric chrge. The Biot Svrt lw gives the mgnetic field of n isolted current element t some point, but such n isolted current element cnnot exist the wy n isolted electric chrge cn. A current element must be prt of n extended current distribution becuse we must hve complete circuit in order for chrges to flow. Thus, the Biot Svrt lw is only the first step in clcultion of mgnetic field; it must be followed by n integrtion over the current distribution. n the exmples tht follow, it is importnt to recognize tht the mgnetic field determined in these clcultions is the field creted by current-crrying conductor. This field is not to be confused with ny dditionl fields tht my be present outside the conductor due to other sources, such s br mgnet plced nerby.

4 940 CHAPTER 30 Sources of the Mgnetic Field EXAMPLE 30.1 Mgnetic Field Surrounding Thin, Stright Conductor Consider thin, stright wire crrying constnt current nd plced long the x xis s shown in Figure Determine the mgnitude nd direction of the mgnetic field t point P due to this current. Solution From the Biot Svrt lw, we expect tht the mgnitude of the field is proportionl to the current in the wire nd decreses s the distnce from the wire to point P increses. We strt by considering length element ds locted distnce r from P. The direction of the mgnetic field t point P due to the current in this element is out of the pge becuse ds rˆ is out of the pge. n fct, since ll of the current elements ds lie in the plne of the pge, they ll produce mgnetic field directed out of the pge t point P. Thus, we hve the direction of the mgnetic field t point P, nd we need only find the mgnitude. Tking the origin t O nd letting point P be long the positive y xis, with k being unit vector pointing out of the pge, we see tht ds rˆ k ds rˆ k(dx sin ) n expression in which the only vrible is. We cn now obtin the mgnitude of the mgnetic field t point P by integrting Eqution (4) over ll elements, subtending ngles rnging from 1 to 2 s defined in Figure 30.2b: 2 B sin d (30.4) 4 4 (cos 1 cos 2) 0 1 We cn use this result to find the mgnetic field of ny stright current-crrying wire if we know the geometry nd hence the ngles 1 nd 2. Consider the specil cse of n infinitely long, stright wire. f we let the wire in Figure 30.2b become infinitely long, we see tht 1 0 nd 2 for length elements rnging between positions x nd x. Becuse (cos 1 cos 2 ) (cos 0 cos ) 2, Eqution 30.4 becomes B (30.5) Equtions 30.4 nd 30.5 both show tht the mgnitude of where, from Chpter 3, ds rˆ represents the mgnitude of ds rˆ. Becuse rˆ is unit vector, the unit of the cross product is simply the unit of ds, which is length. Substitution into Eqution 30.1 gives ds = dx P y Becuse ll current elements produce mgnetic field in the k direction, let us restrict our ttention to the mgnitude of the field due to one current element, which is (1) db (db)k db To integrte this expression, we must relte the vribles, x, nd r. One pproch is to express x nd r in terms of. From the geometry in Figure 30.2, we hve (2) r csc sin Becuse tn from the right tringle in Figure 30.2 (the negtive sign is necessry becuse ds is locted t negtive vlue of x), we hve /(x) Tking the derivtive of this expression gives (3) dx csc 2 d Substitution of Equtions (2) nd (3) into Eqution (1) gives (4) db 0 4 x cot dx sin r 2 dx sin r 2 k csc 2 sin d 2 csc sin d Figure 30.2 rˆ ds θ θ 1 r θ x P O () (b) θ 2 () A thin, stright wire crrying current. The mgnetic field t point P due to the current in ech element ds of the wire is out of the pge, so the net field t point P is lso out of the pge. (b) The ngles 1 nd 2, used for determining the net field. When the wire is infinitely long, 1 0 nd x

5 30.1 The Biot Svrt Lw 941 the mgnetic field is proportionl to the current nd decreses with incresing distnce from the wire, s we expected. Notice tht Eqution 30.5 hs the sme mthemticl form s the expression for the mgnitude of the electric field due to long chrged wire (see Eq. 24.7). Exercise Clculte the mgnitude of the mgnetic field 4.0 cm from n infinitely long, stright wire crrying current of 5.0 A. Answer T. The result of Exmple 30.1 is importnt becuse current in the form of long, stright wire occurs often. Figure 30.3 is three-dimensionl view of the mgnetic field surrounding long, stright current-crrying wire. Becuse of the symmetry of the wire, the mgnetic field lines re circles concentric with the wire nd lie in plnes perpendiculr to the wire. The mgnitude of B is constnt on ny circle of rdius nd is given by Eqution A convenient rule for determining the direction of B is to grsp the wire with the right hnd, positioning the thumb long the direction of the current. The four fingers wrp in the direction of the mgnetic field. Figure 30.3 The right-hnd rule for determining the direction of the mgnetic field surrounding long, stright wire crrying current. Note tht the mgnetic field lines form circles round the wire. EXAMPLE 30.2 Mgnetic Field Due to Curved Wire Segment Clculte the mgnetic field t point O for the current-crrying wire segment shown in Figure The wire consists of two stright portions nd circulr rc of rdius R, which subtends n ngle. The rrowheds on the wire indicte the direction of the current. Solution The mgnetic field t O due to the current in the stright segments AA nd CC is zero becuse ds is prllel to rˆ long these pths; this mens tht ds rˆ 0. Ech length element ds long pth AC is t the sme distnce R from O, nd the current in ech contributes field element db directed into the pge t O. Furthermore, t every point on AC, ds is perpendiculr to rˆ; hence, ds rˆ ds. Using this informtion nd Eqution 30.1, we cn find the mgnitude of the field t O due to the current in n element of length ds: db 0 4 ds R 2 Figure 30.4 O θ R R The mgnetic field t O due to the current in the curved segment AC is into the pge. The contribution to the field t O due to the current in the two stright segments is zero. A C rˆ ds A C

6 942 CHAPTER 30 Sources of the Mgnetic Field Becuse nd R re constnts, we cn esily integrte this expression over the curved pth AC: rdins. The direction of B is into the pge t O becuse ds rˆ is into the pge for every length element. B 0 4R 2 ds 0 4R 2 s 0 4R (30.6) Exercise A circulr wire loop of rdius R crries current. Wht is the mgnitude of the mgnetic field t its center? where we hve used the fct tht s R with mesured in Answer 0 /2R. EXAMPLE 30.3 Mgnetic Field on the Axis of Circulr Current Loop Consider circulr wire loop of rdius R locted in the yz plne nd crrying stedy current, s shown in Figure Clculte the mgnetic field t n xil point P distnce x from the center of the loop. Solution n this sitution, note tht every length element ds is perpendiculr to the vector rˆ t the loction of the element. Thus, for ny element, ds rˆ (ds)(1) sin 90 ds. Furthermore, ll length elements round the loop re t the sme distnce r from P, where r 2 x 2 R 2. Hence, the mgnitude of db due to the current in ny length element ds is The direction of db is perpendiculr to the plne formed by rˆ nd ds, s shown in Figure We cn resolve this vector into component db x long the x xis nd component db y perpendiculr to the x xis. When the components db y re summed over ll elements round the loop, the resultnt component is zero. Tht is, by symmetry the current in ny element on one side of the loop sets up perpendiculr component of d B tht cncels the perpendiculr component set up by the current through the element dimetriclly opposite it. Therefore, the resultnt field t P must be long the x xis nd we cn find it by integrting the components db x db cos. Tht is, B B x i, where B x ds cos db cos x 2 R 2 nd we must tke the integrl over the entire loop. Becuse, x, nd R re constnts for ll elements of the loop nd becuse cos we obtin B x db 0 4 R /(x 2 R 2 ) 1/2, 0R ds rˆ r 2 4(x 2 R 2 ) 3/2 ds 0 4 0R 2 2(x 2 R 2 ) 3/2 (30.7) where we hve used the fct tht ds 2R (the circumference of the loop). To find the mgnetic field t the center of the loop, we set x 0 in Eqution At this specil point, therefore, 0 4 ds (x 2 R 2 ) (t x 0) (30.8) which is consistent with the result of the exercise in Exmple t is lso interesting to determine the behvior of the mgnetic field fr from the loop tht is, when x is much greter thn R. n this cse, we cn neglect the term R 2 in the denomintor of Eqution 30.7 nd obtin B B 0 2R 0R 2 2x 3 (for x W R) (30.9) Becuse the mgnitude of the mgnetic moment of the loop is defined s the product of current nd loop re (see Eq ) (R 2 ) for our circulr loop we cn express Eqution 30.9 s B 0 2 (30.10) This result is similr in form to the expression for the electric field due to n electric dipole, E k e (2q/y 3 ) (see Exmple z Figure 30.5 R ds y O θ ˆr Geometry for clculting the mgnetic field t point P lying on the xis of current loop. By symmetry, the totl field B is long this xis. x 3 x r P db y θ db x db x

7 30.2 The Mgnetic Force Between Two Prllel Conductors ), where 2q p is the electric dipole moment s defined in Eqution The pttern of the mgnetic field lines for circulr current loop is shown in Figure For clrity, the lines re drwn for only one plne one tht contins the xis of the loop. Note tht the field-line pttern is xilly symmetric nd looks like the pttern round br mgnet, shown in Figure 30.6c. N N S S () (b) (c) Figure 30.6 () Mgnetic field lines surrounding current loop. (b) Mgnetic field lines surrounding current loop, displyed with iron filings (Eduction Development Center, Newton, MA). (c) Mgnetic field lines surrounding br mgnet. Note the similrity between this line pttern nd tht of current loop THE MAGNETC FORCE BETWEEN TWO PARALLEL CONDUCTORS n Chpter 29 we described the mgnetic force tht cts on current-crrying conductor plced in n externl mgnetic field. Becuse current in conductor sets up its own mgnetic field, it is esy to understnd tht two current-crrying conductors exert mgnetic forces on ech other. As we shll see, such forces cn be used s the bsis for defining the mpere nd the coulomb. Consider two long, stright, prllel wires seprted by distnce nd crrying currents 1 nd 2 in the sme direction, s illustrted in Figure We cn determine the force exerted on one wire due to the mgnetic field set up by the other wire. Wire 2, which crries current 2, cretes mgnetic field B 2 t the loction of wire 1. The direction of B 2 is perpendiculr to wire 1, s shown in Figure According to Eqution 29.3, the mgnetic force on length of wire 1 is F 1 1 B 2. Becuse is perpendiculr to B 2 in this sitution, the mgnitude of F 1 is F 1 1 B 2. Becuse the mgnitude of B 2 is given by Eqution 30.5, we see tht F 1 1 B 2 1 (30.11) 2 2 The direction of F 1 is towrd wire 2 becuse B 2 is in tht direction. f the field set up t wire 2 by wire 1 is clculted, the force F 2 cting on wire 2 is found to be equl in mgnitude nd opposite in direction to F 1. This is wht we expect be B 2 Figure 30.7 F 1 Two prllel wires tht ech crry stedy current exert force on ech other. The field B 2 due to the current in wire 2 exerts force of mgnitude F 1 1 B 2 on wire 1. The force is ttrctive if the currents re prllel (s shown) nd repulsive if the currents re ntiprllel. 1 2

8 944 CHAPTER 30 Sources of the Mgnetic Field cuse Newton s third lw must be obeyed. 1 When the currents re in opposite directions (tht is, when one of the currents is reversed in Fig. 30.7), the forces re reversed nd the wires repel ech other. Hence, we find tht prllel conductors crrying currents in the sme direction ttrct ech other, nd prllel conductors crrying currents in opposite directions repel ech other. Becuse the mgnitudes of the forces re the sme on both wires, we denote the mgnitude of the mgnetic force between the wires s simply F B. We cn rewrite this mgnitude in terms of the force per unit length: F B (30.12) The force between two prllel wires is used to define the mpere s follows: Definition of the mpere When the mgnitude of the force per unit length between two long, prllel wires tht crry identicl currents nd re seprted by 1 m is N/m, the current in ech wire is defined to be 1 A. web Visit mpere.html for more informtion. Definition of the coulomb The vlue N/m is obtined from Eqution with A nd 1 m. Becuse this definition is bsed on force, mechnicl mesurement cn be used to stndrdize the mpere. For instnce, the Ntionl nstitute of Stndrds nd Technology uses n instrument clled current blnce for primry current mesurements. The results re then used to stndrdize other, more conventionl instruments, such s mmeters. The S unit of chrge, the coulomb, is defined in terms of the mpere: When conductor crries stedy current of 1 A, the quntity of chrge tht flows through cross-section of the conductor in 1 s is 1 C. n deriving Equtions nd 30.12, we ssumed tht both wires re long compred with their seprtion distnce. n fct, only one wire needs to be long. The equtions ccurtely describe the forces exerted on ech other by long wire nd stright prllel wire of limited length. Quick Quiz 30.1 For 1 2 A nd 2 6 A in Figure 30.7, which is true: () F 1 3F 2, (b) F 1 F 2 /3, or (c) F 1 F 2? Quick Quiz 30.2 A loose spirl spring is hung from the ceiling, nd lrge current is sent through it. Do the coils move closer together or frther prt? 1 Although the totl force exerted on wire 1 is equl in mgnitude nd opposite in direction to the totl force exerted on wire 2, Newton s third lw does not pply when one considers two smll elements of the wires tht re not exctly opposite ech other. This pprent violtion of Newton s third lw nd of the lw of conservtion of momentum is described in more dvnced tretments on electricity nd mgnetism.

9 30.3 Ampère s Lw AMPÈRE S LAW Oersted s 1819 discovery bout deflected compss needles demonstrtes tht current-crrying conductor produces mgnetic field. Figure 30.8 shows how this effect cn be demonstrted in the clssroom. Severl compss needles re plced in horizontl plne ner long verticl wire. When no current is present in the wire, ll the needles point in the sme direction (tht of the Erth s mgnetic field), s expected. When the wire crries strong, stedy current, the needles ll deflect in direction tngent to the circle, s shown in Figure 30.8b. These observtions demonstrte tht the direction of the mgnetic field produced by the current in the wire is consistent with the right-hnd rule described in Figure When the current is reversed, the needles in Figure 30.8b lso reverse. Becuse the compss needles point in the direction of B, we conclude tht the lines of B form circles round the wire, s discussed in the preceding section. By symmetry, the mgnitude of B is the sme everywhere on circulr pth centered on the wire nd lying in plne perpendiculr to the wire. By vrying the current nd distnce from the wire, we find tht B is proportionl to the current nd inversely proportionl to the distnce from the wire, s Eqution 30.5 describes. Now let us evlute the product B ds for smll length element ds on the circulr pth defined by the compss needles, nd sum the products for ll elements over the closed circulr pth. Along this pth, the vectors ds nd B re prllel t ech point (see Fig. 30.8b), so B ds B ds. Furthermore, the mgnitude of B is constnt on this circle nd is given by Eqution Therefore, the sum of the products B ds over the closed pth, which is equivlent to the line integrl of B ds, is B ds B ds where ds 2r is the circumference of the circulr pth. Although this result ws clculted for the specil cse of circulr pth surrounding wire, it holds 0 2r (2r) 0 Andre-Mrie Ampère ( ) Ampère, Frenchmn, is credited with the discovery of electromgnetism the reltionship between electric currents nd mgnetic fields. Ampère s genius, prticulrly in mthemtics, becme evident by the time he ws 12 yers old; his personl life, however, ws filled with trgedy. His fther, welthy city officil, ws guillotined during the French Revolution, nd his wife died young, in Ampère died t the ge of 61 of pneumoni. His judgment of his life is cler from the epitph he chose for his grvestone: Tndem Felix (Hppy t Lst). (AP Emilio Segre Visul Archive) B ds = 0 Figure 30.8 () () When no current is present in the wire, ll compss needles point in the sme direction (towrd the Erth s north pole). (b) When the wire crries strong current, the compss needles deflect in direction tngent to the circle, which is the direction of the mgnetic field creted by the current. (c) Circulr mgnetic field lines surrounding current-crrying conductor, displyed with iron filings. (b) (c)

10 946 CHAPTER 30 Sources of the Mgnetic Field for closed pth of ny shpe surrounding current tht exists in n unbroken circuit. The generl cse, known s Ampère s lw, cn be stted s follows: The line integrl of Bds round ny closed pth equls 0, where is the totl continuous current pssing through ny surfce bounded by the closed pth. Ampère s lw B ds 0 (30.13) Ampère s lw describes the cretion of mgnetic fields by ll continuous current configurtions, but t our mthemticl level it is useful only for clculting the mgnetic field of current configurtions hving high degree of symmetry. ts use is similr to tht of Guss s lw in clculting electric fields for highly symmetric chrge distributions. Quick Quiz 30.3 Rnk the mgnitudes of B ds for the closed pths in Figure 30.9, from lest to gretest. 1 A b c d 5 A 2 A Figure 30.9 Four closed pths round three currentcrrying wires. Quick Quiz 30.4 Rnk the mgnitudes of B ds for the closed pths in Figure 30.10, from lest to gretest. b c d Figure Severl closed pths ner single current-crrying wire.

11 30.3 Ampère s Lw 947 EXAMPLE 30.4 The Mgnetic Field Creted by Long Current-Crrying Wire A long, stright wire of rdius R crries stedy current 0 tht is uniformly distributed through the cross-section of the wire (Fig ). Clculte the mgnetic field distnce r from the center of the wire in the regions r R nd r R. Solution For the r R cse, we should get the sme result we obtined in Exmple 30.1, in which we pplied the Biot Svrt lw to the sme sitution. Let us choose for our pth of integrtion circle 1 in Figure From symmetry, B must be constnt in mgnitude nd prllel to ds t every point on this circle. Becuse the totl current pssing through the plne of the circle is 0, Ampère s lw gives B ds Bds B(2r) 0 0 B 0 0 2r (for r R) (30.14) which is identicl in form to Eqution Note how much esier it is to use Ampère s lw thn to use the Biot Svrt lw. This is often the cse in highly symmetric situtions. Now consider the interior of the wire, where r R. Here the current pssing through the plne of circle 2 is less thn the totl current 0. Becuse the current is uniform over the cross-section of the wire, the frction of the current enclosed by circle 2 must equl the rtio of the re r 2 enclosed by circle 2 to the cross-sectionl re R 2 of the wire: 2 0 Following the sme procedure s for circle 1, we pply Ampère s lw to circle 2: r 2 B ds B(2r) 0 0 R B 2R 2r r 2 R 2 r 2 R 2 0 (for r R) (30.15) This result is similr in form to the expression for the electric field inside uniformly chrged sphere (see Exmple 24.5). The mgnitude of the mgnetic field versus r for this configurtion is plotted in Figure Note tht inside the wire, B : 0 s r : 0. Note lso tht Equtions nd give the sme vlue of the mgnetic field t r R, demonstrting tht the mgnetic field is continuous t the surfce of the wire. B 1 0 B r 2 R B 1/r r ds R r Figure A long, stright wire of rdius R crrying stedy current 0 uniformly distributed cross the cross-section of the wire. The mgnetic field t ny point cn be clculted from Ampère s lw using circulr pth of rdius r, concentric with the wire. Figure Mgnitude of the mgnetic field versus r for the wire shown in Figure The field is proportionl to r inside the wire nd vries s 1/r outside the wire. EXAMPLE 30.5 The Mgnetic Field Creted by Toroid A device clled toroid (Fig ) is often used to crete n lmost uniform mgnetic field in some enclosed re. The device consists of conducting wire wrpped round ring ( torus) mde of nonconducting mteril. For toroid hv- ing N closely spced turns of wire, clculte the mgnetic field in the region occupied by the torus, distnce r from the center. 2 Another wy to look t this problem is to see tht the current enclosed by circle 2 must equl the product of the current density J nd the re r 2 0 /R 2 of this circle.

12 948 CHAPTER 30 Sources of the Mgnetic Field Solution To clculte this field, we must evlute B ds over the circle of rdius r in Figure By symmetry, we see tht the mgnitude of the field is constnt on this circle nd tngent to it, so B ds B ds. Furthermore, note tht B the circulr closed pth surrounds N loops of wire, ech of which crries current. Therefore, the right side of Eqution is 0 N in this cse. Ampère s lw pplied to the circle gives B ds B ds B(2r) 0N Figure ds A toroid consisting of mny turns of wire. f the turns re closely spced, the mgnetic field in the interior of the torus (the gold-shded region) is tngent to the dshed circle nd vries s 1/r. The field outside the toroid is zero. The dimension is the cross-sectionl rdius of the torus. r B 0N 2r (30.16) This result shows tht B vries s 1/r nd hence is nonuniform in the region occupied by the torus. However, if r is very lrge compred with the cross-sectionl rdius of the torus, then the field is pproximtely uniform inside the torus. For n idel toroid, in which the turns re closely spced, the externl mgnetic field is zero. This cn be seen by noting tht the net current pssing through ny circulr pth lying outside the toroid (including the region of the hole in the doughnut ) is zero. Therefore, from Ampère s lw we find tht B 0 in the regions exterior to the torus. EXAMPLE 30.6 Mgnetic Field Creted by n nfinite Current Sheet So fr we hve imgined currents through wires of smll cross-section. Let us now consider n exmple in which current exists in n extended object. A thin, infinitely lrge sheet lying in the yz plne crries current of liner current density J s. The current is in the y direction, nd J s represents the current per unit length mesured long the z xis. Find the mgnetic field ner the sheet. Solution This sitution brings to mind similr clcultions involving Guss s lw (see Exmple 24.8). You my recll tht x B z J s (out of pge) B the electric field due to n infinite sheet of chrge does not depend on distnce from the sheet. Thus, we might expect similr result here for the mgnetic field. To evlute the line integrl in Ampère s lw, let us tke rectngulr pth through the sheet, s shown in Figure The rectngle hs dimensions nd w, with the sides of length prllel to the sheet surfce. The net current pssing through the plne of the rectngle is J s. We pply Ampère s lw over the rectngle nd note tht the two sides of length w do not contribute to the line integrl becuse the component of B long the direction of these pths is zero. By symmetry, we cn rgue tht the mgnetic field is constnt over the sides of length becuse every point on the infinitely lrge sheet is equivlent, nd hence the field should not vry from point to point. The only choices of field direction tht re resonble for the symmetry re perpendiculr or prllel to the sheet, nd perpendiculr field would pss through the current, which is inconsistent with the Biot Svrt lw. Assuming field tht is constnt in mgnitude nd prllel to the plne of the sheet, we obtin B ds 0 0 J s 2B 0 J s Figure w End view of n infinite current sheet lying in the yz plne, where the current is in the y direction (out of the pge). This view shows the direction of B on both sides of the sheet. B 0 J s 2 This result shows tht the mgnetic field is independent of distnce from the current sheet, s we suspected.

13 30.4 The Mgnetic Field of Solenoid 949 EXAMPLE 30.7 The Mgnetic Force on Current Segment Wire 1 in Figure is oriented long the y xis nd crries stedy current 1. A rectngulr loop locted to the right of the wire nd in the xy plne crries current 2. Find the mgnetic force exerted by wire 1 on the top wire of length b in the loop, lbeled Wire 2 in the figure. Solution You my be tempted to use Eqution to obtin the force exerted on smll segment of length dx of wire 2. However, this eqution pplies only to two prllel wires nd cnnot be used here. The correct pproch is to Wire 1 F B Wire 2 1 y ds 2 x consider the force exerted by wire 1 on smll segment ds of wire 2 by using Eqution This force is given by df B ds B, where 2 nd B is the mgnetic field creted by the current in wire 1 t the position of ds. From Ampère s lw, the field t distnce x from wire 1 (see Eq ) is where the unit vector k is used to indicte tht the field t ds points into the pge. Becuse wire 2 is long the x xis, ds dx i, nd we find tht df B ntegrting over the limits x to x b gives F B x B 0 1 2x [i ( k)]dx b ln x j ( k) dx x j ln 1 b j b Figure The force points in the positive y direction, s indicted by the unit vector j nd s shown in Figure Exercise Wht re the mgnitude nd direction of the force exerted on the bottom wire of length b? Answer The force hs the sme mgnitude s the force on wire 2 but is directed downwrd. Quick Quiz 30.5 s net force cting on the current loop in Exmple 30.7? A net torque? 30.4 THE MAGNETC FELD OF A SOLENOD A solenoid is long wire wound in the form of helix. With this configurtion, resonbly uniform mgnetic field cn be produced in the spce surrounded by the turns of wire which we shll cll the interior of the solenoid when the solenoid crries current. When the turns re closely spced, ech cn be pproximted s circulr loop, nd the net mgnetic field is the vector sum of the fields resulting from ll the turns. Figure shows the mgnetic field lines surrounding loosely wound solenoid. Note tht the field lines in the interior re nerly prllel to one nother, re uniformly distributed, nd re close together, indicting tht the field in this spce is uniform nd strong. The field lines between current elements on two djcent turns tend to cncel ech other becuse the field vectors from the two elements re in opposite directions. The field t exterior points such s P is wek becuse the field due to current elements on the right-hnd portion of turn tends to cncel the field due to current elements on the left-hnd portion. Exterior P nterior Figure The mgnetic field lines for loosely wound solenoid.

14 950 CHAPTER 30 Sources of the Mgnetic Field N S Figure () () Mgnetic field lines for tightly wound solenoid of finite length, crrying stedy current. The field in the interior spce is nerly uniform nd strong. Note tht the field lines resemble those of br mgnet, mening tht the solenoid effectively hs north nd south poles. (b) The mgnetic field pttern of br mgnet, displyed with smll iron filings on sheet of pper. (b) A technicin studies the scn of ptient s hed. The scn ws obtined using medicl dignostic technique known s mgnetic resonnce imging (MR). This instrument mkes use of strong mgnetic fields produced by superconducting solenoids. f the turns re closely spced nd the solenoid is of finite length, the mgnetic field lines re s shown in Figure This field line distribution is similr to tht surrounding br mgnet (see Fig b). Hence, one end of the solenoid behves like the north pole of mgnet, nd the opposite end behves like the south pole. As the length of the solenoid increses, the interior field becomes more uniform nd the exterior field becomes weker. An idel solenoid is pproched when the turns re closely spced nd the length is much greter thn the rdius of the turns. n this cse, the externl field is zero, nd the interior field is uniform over gret volume. B w Figure Cross-sectionl view of n idel solenoid, where the interior mgnetic field is uniform nd the exterior field is zero. Ampère s lw pplied to the red dshed pth cn be used to clculte the mgnitude of the interior field.

15 30.5 Mgnetic Flux 951 We cn use Ampère s lw to obtin n expression for the interior mgnetic field in n idel solenoid. Figure shows longitudinl cross-section of prt of such solenoid crrying current. Becuse the solenoid is idel, B in the interior spce is uniform nd prllel to the xis, nd B in the exterior spce is zero. Consider the rectngulr pth of length nd width w shown in Figure We cn pply Ampère s lw to this pth by evluting the integrl of B ds over ech side of the rectngle. The contribution long side 3 is zero becuse B 0 in this region. The contributions from sides 2 nd 4 re both zero becuse B is perpendiculr to ds long these pths. Side 1 gives contribution B to the integrl becuse long this pth B is uniform nd prllel to ds. The integrl over the closed rectngulr pth is therefore B ds B ds B ds B pth 1 pth 1 The right side of Ampère s lw involves the totl current pssing through the re bounded by the pth of integrtion. n this cse, the totl current through the rectngulr pth equls the current through ech turn multiplied by the number of turns. f N is the number of turns in the length, the totl current through the rectngle is N. Therefore, Ampère s lw pplied to this pth gives QuickLb Wrp few turns of wire round compss, essentilly putting the compss inside solenoid. Hold the ends of the wire to the two terminls of flshlight bttery. Wht hppens to the compss? s the effect s strong when the compss is outside the turns of wire? B ds B 0N B 0 N 0n (30.17) Mgnetic field inside solenoid where n N/ is the number of turns per unit length. We lso could obtin this result by reconsidering the mgnetic field of toroid (see Exmple 30.5). f the rdius r of the torus in Figure contining N turns is much greter thn the toroid s cross-sectionl rdius, short section of the toroid pproximtes solenoid for which n N/2r. n this limit, Eqution grees with Eqution Eqution is vlid only for points ner the center (tht is, fr from the ends) of very long solenoid. As you might expect, the field ner ech end is smller thn the vlue given by Eqution At the very end of long solenoid, the mgnitude of the field is one-hlf the mgnitude t the center. web For more detiled discussion of the mgnetic field long the xis of solenoid, visit MAGNETC FLUX The flux ssocited with mgnetic field is defined in mnner similr to tht used to define electric flux (see Eq. 24.3). Consider n element of re da on n rbitrrily shped surfce, s shown in Figure f the mgnetic field t this element is B, the mgnetic flux through the element is B da, where da is vector tht is perpendiculr to the surfce nd hs mgnitude equl to the re da. Hence, the totl mgnetic flux B through the surfce is B B da (30.18) Definition of mgnetic flux

16 952 CHAPTER 30 Sources of the Mgnetic Field d A d A θ B d A B B Figure The mgnetic flux through n re element da is B d A BdA cos, where da is vector perpendiculr to the surfce. Figure () Mgnetic flux through plne lying in mgnetic field. () The flux through the plne is zero when the mgnetic field is prllel to the plne surfce. (b) The flux through the plne is mximum when the mgnetic field is perpendiculr to the plne. Consider the specil cse of plne of re A in uniform field B tht mkes n ngle with da. The mgnetic flux through the plne in this cse is B BA cos (30.19) f the mgnetic field is prllel to the plne, s in Figure 30.20, then 90 nd the flux is zero. f the field is perpendiculr to the plne, s in Figure 30.20b, then 0 nd the flux is BA (the mximum vlue). The unit of flux is the Tm 2, which is defined s weber (Wb); 1 Wb 1 Tm 2. (b) EXAMPLE 30.8 Mgnetic Flux Through Rectngulr Loop A rectngulr loop of width nd length b is locted ner long wire crrying current (Fig ). The distnce between the wire nd the closest side of the loop is c. The wire is prllel to the long side of the loop. Find the totl mgnetic flux through the loop due to the current in the wire. Solution From Eqution 30.14, we know tht the mgnitude of the mgnetic field creted by the wire t distnce r from the wire is B 0 2r The fctor 1/r indictes tht the field vries over the loop, nd Figure shows tht the field is directed into the pge. Becuse B is prllel to da t ny point within the loop, the mgnetic flux through n re element da is B B da 0 2r da dr r b (Becuse B is not uniform but depends on r, it cnnot be removed from the integrl.) To integrte, we first express the re element (the tn region in Fig ) s da b dr. Becuse r is now the only vrible in the integrl, we hve B 0b 2 0b 2 c c dr r ln c c 0b 2 ln r c c 0b 2 ln 1 c c Figure The mgnetic field due to the wire crrying current is not uniform over the rectngulr loop. Exercise Apply the series expnsion formul for ln(1 x) (see Appendix B.5) to this eqution to show tht it gives resonble result when the loop is fr from the wire reltive to the loop dimensions (in other words, when c W ). Answer B : 0.

17 30.6 Guss s Lw in Mgnetism GAUSS S LAW N MAGNETSM n Chpter 24 we found tht the electric flux through closed surfce surrounding net chrge is proportionl to tht chrge (Guss s lw). n other words, the number of electric field lines leving the surfce depends only on the net chrge within it. This property is bsed on the fct tht electric field lines originte nd terminte on electric chrges. The sitution is quite different for mgnetic fields, which re continuous nd form closed loops. n other words, mgnetic field lines do not begin or end t ny point s illustrted by the mgnetic field lines of the br mgnet in Figure Note tht for ny closed surfce, such s the one outlined by the dshed red line in Figure 30.22, the number of lines entering the surfce equls the number leving the surfce; thus, the net mgnetic flux is zero. n contrst, for closed surfce surrounding one chrge of n electric dipole (Fig ), the net electric flux is not zero. Guss s lw in mgnetism sttes tht the net mgnetic flux through ny closed surfce is lwys zero: B da 0 (30.20) Guss s lw for mgnetism This sttement is bsed on the experimentl fct, mentioned in the opening of Chpter 29, tht isolted mgnetic poles (monopoles) hve never been detected nd perhps do not exist. Nonetheless, scientists continue the serch be- N + S Figure The mgnetic field lines of br mgnet form closed loops. Note tht the net mgnetic flux through the closed surfce (dshed red line) surrounding one of the poles (or ny other closed surfce) is zero. Figure The electric field lines surrounding n electric dipole begin on the positive chrge nd terminte on the negtive chrge. The electric flux through closed surfce surrounding one of the chrges is not zero.

18 954 CHAPTER 30 Sources of the Mgnetic Field Pth P S 1 Q Q S 2 Figure Two surfces S 1 nd S 2 ner the plte of cpcitor re bounded by the sme pth P. The conduction current in the wire psses only through S 1. This leds to contrdiction in Ampère s lw tht is resolved only if one postultes displcement current through S 2. A 12.9 cuse certin theories tht re otherwise successful in explining fundmentl physicl behvior suggest the possible existence of monopoles DSPLACEMENT CURRENT AND THE GENERAL FORM OF AMPÈRE S LAW We hve seen tht chrges in motion produce mgnetic fields. When currentcrrying conductor hs high symmetry, we cn use Ampère s lw to clculte the mgnetic field it cretes. n Eqution 30.13, B ds 0, the line integrl is over ny closed pth through which the conduction current psses, nd the conduction current is defined by the expression dq/dt. (n this section we use the term conduction current to refer to the current crried by the wire, to distinguish it from new type of current tht we shll introduce shortly.) We now show tht Ampère s lw in this form is vlid only if ny electric fields present re constnt in time. Mxwell recognized this limittion nd modified Ampère s lw to include time-vrying electric fields. We cn understnd the problem by considering cpcitor tht is being chrged s illustrted in Figure When conduction current is present, the chrge on the positive plte chnges but no conduction current psses cross the gp between the pltes. Now consider the two surfces S 1 nd S 2 in Figure 30.24, bounded by the sme pth P. Ampère s lw sttes tht B ds round this pth must equl 0, where is the totl current through ny surfce bounded by the pth P. When the pth P is considered s bounding S 1, B ds is 0 becuse the conduction current psses through S 1. When the pth is considered s bounding S 2, however, B ds 0 becuse no conduction current psses through S 2. Thus, we rrive t contrdictory sitution tht rises from the discontinuity of the current! Mxwell solved this problem by postulting n dditionl term on the right side of Eqution 30.13, which includes fctor clled the displcement current d, defined s 3 Displcement current d 0 d E dt (30.21) where 0 is the permittivity of free spce (see Section 23.3) nd E E da is the electric flux (see Eq. 24.3). As the cpcitor is being chrged (or dischrged), the chnging electric field between the pltes my be considered equivlent to current tht cts s continution of the conduction current in the wire. When the expression for the displcement current given by Eqution is dded to the conduction current on the right side of Ampère s lw, the difficulty represented in Figure is resolved. No mtter which surfce bounded by the pth P is chosen, either conduction current or displcement current psses through it. With this new term d, we cn express the generl form of Ampère s lw (sometimes clled the Ampère Mxwell lw) s 4 Ampère Mxwell lw B d s 0( d ) 0 00 d E dt (30.22) 3 Displcement in this context does not hve the mening it does in Chpter 2. Despite the inccurte implictions, the word is historiclly entrenched in the lnguge of physics, so we continue to use it. 4 Strictly speking, this expression is vlid only in vcuum. f mgnetic mteril is present, one must chnge 0 nd 0 on the right-hnd side of Eqution to the permebility m nd permittivity chrcteristic of the mteril. Alterntively, one my include mgnetizing current m on the righthnd side of Eqution to mke Ampère s lw fully generl. On microscopic scle, m is s rel s.

19 30.7 Displcement Current nd the Generl Form of Ampère s Lw 955 Q E Q S 2 S 1 Figure Becuse it exists only in the wires ttched to the cpcitor pltes, the conduction current dq /dt psses through S 1 but not through S 2. Only the displcement current d 0 d E /dt psses through S 2. The two currents must be equl for continuity. We cn understnd the mening of this expression by referring to Figure The electric flux through surfce S 2 is E E da EA, where A is the re of the cpcitor pltes nd E is the mgnitude of the uniform electric field between the pltes. f Q is the chrge on the pltes t ny instnt, then E Q /0A (see Section 26.2). Therefore, the electric flux through S 2 is simply Hence, the displcement current through S 2 is d E EA Q 0 d E dt dq dt (30.23) Tht is, the displcement current through S 2 is precisely equl to the conduction current through S 1! By considering surfce S 2, we cn identify the displcement current s the source of the mgnetic field on the surfce boundry. The displcement current hs its physicl origin in the time-vrying electric field. The centrl point of this formlism, then, is tht mgnetic fields re produced both by conduction currents nd by time-vrying electric fields. This result ws remrkble exmple of theoreticl work by Mxwell, nd it contributed to mjor dvnces in the understnding of electromgnetism. 0 Quick Quiz 30.6 Wht is the displcement current for fully chrged 3-F cpcitor? EXAMPLE 30.9 Displcement Current in Cpcitor A sinusoidlly vrying voltge is pplied cross n 8.00-F cpcitor. The frequency of the voltge is 3.00 khz, nd the voltge mplitude is 30.0 V. Find the displcement current between the pltes of the cpcitor. Solution The ngulr frequency of the source, from Eqution 13.6, is 2f 2( Hz) s 1. Hence, the voltge cross the cpcitor in terms of t is V V mx sin t (30.0 V) sin( t) We cn use Eqution nd the fct tht the chrge on the cpcitor is Q C V to find the displcement current: d dq dt ( F) d dt [(30.0 V) sin( t)] d dt (C V ) C d dt (V ) (4.52 A) cos( t) The displcement current vries sinusoidlly with time nd hs mximum vlue of 4.52 A.

20 956 CHAPTER 30 Sources of the Mgnetic Field Optionl Section 30.8 MAGNETSM N MATTER The mgnetic field produced by current in coil of wire gives us hint s to wht cuses certin mterils to exhibit strong mgnetic properties. Erlier we found tht coil like the one shown in Figure hs north pole nd south pole. n generl, ny current loop hs mgnetic field nd thus hs mgnetic dipole moment, including the tomic-level current loops described in some models of the tom. Thus, the mgnetic moments in mgnetized substnce my be described s rising from these tomic-level current loops. For the Bohr model of the tom, these current loops re ssocited with the movement of electrons round the nucleus in circulr orbits. n ddition, mgnetic moment is intrinsic to electrons, protons, neutrons, nd other prticles; it rises from property clled spin. Figure An electron moving in circulr orbit of rdius r hs n ngulr momentum L in one direction nd mgnetic moment in the opposite direction. Orbitl mgnetic moment Angulr momentum is quntized L µ r The Mgnetic Moments of Atoms t is instructive to begin our discussion with clssicl model of the tom in which electrons move in circulr orbits round the much more mssive nucleus. n this model, n orbiting electron constitutes tiny current loop (becuse it is moving chrge), nd the mgnetic moment of the electron is ssocited with this orbitl motion. Although this model hs mny deficiencies, its predictions re in good greement with the correct theory, which is expressed in terms of quntum physics. Consider n electron moving with constnt speed v in circulr orbit of rdius r bout the nucleus, s shown in Figure Becuse the electron trvels distnce of 2r (the circumference of the circle) in time T, its orbitl speed is v 2r /T. The current ssocited with this orbiting electron is its chrge e divided by T. Using T 2/ nd we hve v/r, e T e The mgnetic moment ssocited with this current loop is is the re enclosed by the orbit. Therefore, A ev where A r 2 (30.24) 2rr evr Becuse the mgnitude of the orbitl ngulr momentum of the electron is L m e vr (Eq with 90 ), the mgnetic moment cn be written s e (30.25) 2m e L This result demonstrtes tht the mgnetic moment of the electron is proportionl to its orbitl ngulr momentum. Note tht becuse the electron is negtively chrged, the vectors nd L point in opposite directions. Both vectors re perpendiculr to the plne of the orbit, s indicted in Figure A fundmentl outcome of quntum physics is tht orbitl ngulr momentum is quntized nd is equl to multiples of where h is Plnck s constnt. The smllest nonzero vlue of the electron s mgnetic moment resulting from its orbitl motion is e (30.26) 2m e We shll see in Chpter 42 how expressions such s Eqution rise.!2 2 ev 2r A, h/ J s,

21 30.8 Mgnetism in Mtter 957 Becuse ll substnces contin electrons, you my wonder why not ll substnces re mgnetic. The min reson is tht in most substnces, the mgnetic moment of one electron in n tom is cnceled by tht of nother electron orbiting in the opposite direction. The net result is tht, for most mterils, the mgnetic effect produced by the orbitl motion of the electrons is either zero or very smll. n ddition to its orbitl mgnetic moment, n electron hs n intrinsic property clled spin tht lso contributes to its mgnetic moment. n this regrd, the electron cn be viewed s spinning bout its xis while it orbits the nucleus, s shown in Figure (Wrning: This clssicl description should not be tken literlly becuse spin rises from reltivistic dynmics tht must be incorported into quntum-mechnicl nlysis.) The mgnitude of the ngulr momentum S ssocited with spin is of the sme order of mgnitude s the ngulr momentum L due to the orbitl motion. The mgnitude of the spin ngulr momentum predicted by quntum theory is The mgnetic moment chrcteristiclly ssocited with the spin of n electron hs the vlue 2m e This combintion of constnts is clled the Bohr mgneton: B e S!3 2 Spin ngulr momentum spin e (30.27) J/T (30.28) 2m e Thus, tomic mgnetic moments cn be expressed s multiples of the Bohr mgneton. (Note tht 1 J/T 1 A m 2.) n toms contining mny electrons, the electrons usully pir up with their spins opposite ech other; thus, the spin mgnetic moments cncel. However, toms contining n odd number of electrons must hve t lest one unpired electron nd therefore some spin mgnetic moment. The totl mgnetic moment of n tom is the vector sum of the orbitl nd spin mgnetic moments, nd few exmples re given in Tble Note tht helium nd neon hve zero moments becuse their individul spin nd orbitl moments cncel. The nucleus of n tom lso hs mgnetic moment ssocited with its constituent protons nd neutrons. However, the mgnetic moment of proton or neutron is much smller thn tht of n electron nd cn usully be neglected. We cn understnd this by inspecting Eqution nd replcing the mss of the electron with the mss of proton or neutron. Becuse the msses of the proton nd neutron re much greter thn tht of the electron, their mgnetic moments re on the order of 10 3 times smller thn tht of the electron. Figure Bohr mgneton TABLE 30.1 Mgnetic Moments of Some Atoms nd ons Atom or on Mgnetic Moment (10 24 J/T) H 9.27 He 0 Ne 0 Ce Yb µµ spin Clssicl model of spinning electron. This model gives n incorrect mgnitude for the mgnetic moment, incorrect quntum numbers, nd too mny degrees of freedom. Mgnetiztion Vector nd Mgnetic Field Strength The mgnetic stte of substnce is described by quntity clled the mgnetiztion vector M. The mgnitude of this vector is defined s the mgnetic moment per unit volume of the substnce. As you might expect, the totl mgnetic field B t point within substnce depends on both the pplied (externl) field B 0 nd the mgnetiztion of the substnce. To understnd the problems involved in mesuring the totl mgnetic field B in such situtions, consider this: Scientists use smll probes tht utilize the Hll ef- Mgnetiztion vector M

22 958 CHAPTER 30 Sources of the Mgnetic Field Mgnetic field strength H fect (see Section 29.6) to mesure mgnetic fields. Wht would such probe red if it were positioned inside the solenoid mentioned in the QuickLb on pge 951 when you inserted the compss? Becuse the compss is mgnetic mteril, the probe would mesure totl mgnetic field B tht is the sum of the solenoid (externl) field B 0 nd the (mgnetiztion) field B m due to the compss. This tells us tht we need wy to distinguish between mgnetic fields originting from currents nd those originting from mgnetic mterils. Consider region in which mgnetic field B 0 is produced by current-crrying conductor. f we now fill tht region with mgnetic substnce, the totl mgnetic field B in the region is B B 0 B m, where B m is the field produced by the mgnetic substnce. We cn express this contribution in terms of the mgnetiztion vector of the substnce s B m 0M; hence, the totl mgnetic field in the region becomes B B 0 0M (30.29) When nlyzing mgnetic fields tht rise from mgnetiztion, it is convenient to introduce field quntity, clled the mgnetic field strength H within the substnce. The mgnetic field strength represents the effect of the conduction currents in wires on substnce. To emphsize the distinction between the field strength H nd the field B, the ltter is often clled the mgnetic flux density or the mgnetic induction. The mgnetic field strength is vector defined by the reltionship H B 0 / 0 (B/ 0) M. Thus, Eqution cn be written B 0(H M) (30.30) The quntities H nd M hve the sme units. n S units, becuse M is mgnetic moment per unit volume, the units re (mpere)(meter) 2 /(meter) 3, or mperes per meter. To better understnd these expressions, consider the torus region of toroid tht crries current. f this region is vcuum, M 0 (becuse no mgnetic mteril is present), the totl mgnetic field is tht rising from the current lone, nd B B 0 0H. Becuse B 0 0n in the torus region, where n is the number of turns per unit length of the toroid, H B 0 / 0 0n/ 0, or H n (30.31) n this cse, the mgnetic field B in the torus region is due only to the current in the windings of the toroid. f the torus is now mde of some substnce nd the current is kept constnt, H in the torus region remins unchnged (becuse it depends on the current only) nd hs mgnitude n. The totl field B, however, is different from tht when the torus region ws vcuum. From Eqution 30.30, we see tht prt of B rises from the term 0 H ssocited with the current in the toroid, nd prt rises from the term 0 M due to the mgnetiztion of the substnce of which the torus is mde. Oxygen, prmgnetic substnce, is ttrcted to mgnetic field. The liquid oxygen in this photogrph is suspended between the poles of the mgnet. Clssifiction of Mgnetic Substnces Substnces cn be clssified s belonging to one of three ctegories, depending on their mgnetic properties. Prmgnetic nd ferromgnetic mterils re those mde of toms tht hve permnent mgnetic moments. Dimgnetic mterils re those mde of toms tht do not hve permnent mgnetic moments. For prmgnetic nd dimgnetic substnces, the mgnetiztion vector M is proportionl to the mgnetic field strength H. For these substnces plced in n externl mgnetic field, we cn write M H (30.32)

23 30.8 Mgnetism in Mtter 959 TABLE 30.2 Mgnetic Susceptibilities of Some Prmgnetic nd Dimgnetic Substnces t 300 K Prmgnetic Dimgnetic Substnce Substnce Aluminum Bismuth Clcium Copper Chromium Dimond Lithium Gold Mgnesium Led Niobium Mercury Oxygen Nitrogen Pltinum Silver Tungsten Silicon where (Greek letter chi) is dimensionless fctor clled the mgnetic susceptibility. For prmgnetic substnces, is positive nd M is in the sme direction s H. For dimgnetic substnces, is negtive nd M is opposite H. (t is importnt to note tht this liner reltionship between M nd H does not pply to ferromgnetic substnces.) The susceptibilities of some substnces re given in Tble Substituting Eqution for M into Eqution gives or B 0(H M) 0(H H) 0(1 )H B mh (30.33) where the constnt m is clled the mgnetic permebility of the substnce nd is relted to the susceptibility by m 0(1 ) (30.34) Mgnetic susceptibility Mgnetic permebility m Substnces my be clssified in terms of how their mgnetic permebility m compres with 0 (the permebility of free spce), s follows: Prmgnetic m 0 Dimgnetic m 0 Becuse is very smll for prmgnetic nd dimgnetic substnces (see Tble 30.2), m is nerly equl to 0 for these substnces. For ferromgnetic substnces, however, m is typiclly severl thousnd times greter thn 0 (mening tht is very gret for ferromgnetic substnces). Although Eqution provides simple reltionship between B nd H, we must interpret it with cre when deling with ferromgnetic substnces. As mentioned erlier, M is not liner function of H for ferromgnetic substnces. This is becuse the vlue of m is not only chrcteristic of the ferromgnetic substnce but lso depends on the previous stte of the substnce nd on the process it underwent s it moved from its previous stte to its present one. We shll investigte this more deeply fter the following exmple.

24 960 CHAPTER 30 Sources of the Mgnetic Field EXAMPLE An ron-filled Toroid A toroid wound with 60.0 turns/m of wire crries current of 5.00 A. The torus is iron, which hs mgnetic permebility of m under the given conditions. Find H nd B inside the iron. Solution Using Equtions nd 30.33, we obtin H n 60.0 turns m (5.00 A) 300 Aturns m B m H H Tm A This vlue of B is times the vlue in the bsence of iron! Exercise Determine the mgnitude of the mgnetiztion vector inside the iron torus. Answer M A/m. Aturns 300 m 1.88 T Quick Quiz 30.7 A current in solenoid hving ir in the interior cretes mgnetic field B 0H. Describe qulittively wht hppens to the mgnitude of B s () luminum, (b) copper, nd (c) iron re plced in the interior. Figure () B 0 (b) () Rndom orienttion of tomic mgnetic moments in n unmgnetized substnce. (b) When n externl field B 0 is pplied, the tomic mgnetic moments tend to lign with the field, giving the smple net mgnetiztion vector M. Ferromgnetism A smll number of crystlline substnces in which the toms hve permnent mgnetic moments exhibit strong mgnetic effects clled ferromgnetism. Some exmples of ferromgnetic substnces re iron, coblt, nickel, gdolinium, nd dysprosium. These substnces contin tomic mgnetic moments tht tend to lign prllel to ech other even in wek externl mgnetic field. Once the moments re ligned, the substnce remins mgnetized fter the externl field is removed. This permnent lignment is due to strong coupling between neighboring moments, coupling tht cn be understood only in quntum-mechnicl terms. All ferromgnetic mterils re mde up of microscopic regions clled domins, regions within which ll mgnetic moments re ligned. These domins hve volumes of bout to 10 8 m 3 nd contin to toms. The boundries between the vrious domins hving different orienttions re clled domin wlls. n n unmgnetized smple, the domins re rndomly oriented so tht the net mgnetic moment is zero, s shown in Figure When the smple is plced in n externl mgnetic field, the mgnetic moments of the toms tend to lign with the field, which results in mgnetized smple, s in Figure 30.28b. Observtions show tht domins initilly oriented long the externl field grow lrger t the expense of the less fvorbly oriented domins. When the externl field is removed, the smple my retin net mgnetiztion in the direction of the originl field. At ordinry tempertures, therml gittion is not sufficient to disrupt this preferred orienttion of mgnetic moments. A typicl experimentl rrngement tht is used to mesure the mgnetic properties of ferromgnetic mteril consists of torus mde of the mteril wound with N turns of wire, s shown in Figure 30.29, where the windings re represented in blck nd re referred to s the primry coil. This pprtus is sometimes referred to s Rowlnd ring. A secondry coil (the red wires in Fig ) connected to glvnometer is used to mesure the totl mgnetic flux through the torus. The mgnetic field B in the torus is mesured by incresing the current in the toroid from zero to. As the current chnges, the mgnetic flux through

25 30.8 Mgnetism in Mtter 961 the secondry coil chnges by n mount BA, where A is the cross-sectionl re of the toroid. As we shll find in Chpter 31, becuse of this chnging flux, n emf tht is proportionl to the rte of chnge in mgnetic flux is induced in the secondry coil. f the glvnometer is properly clibrted, vlue for B corresponding to ny vlue of the current in the primry coil cn be obtined. The mgnetic field B is mesured first in the bsence of the torus nd then with the torus in plce. The mgnetic properties of the torus mteril re then obtined from comprison of the two mesurements. Now consider torus mde of unmgnetized iron. f the current in the primry coil is incresed from zero to some vlue, the mgnitude of the mgnetic field strength H increses linerly with ccording to the expression H n. Furthermore, the mgnitude of the totl field B lso increses with incresing current, s shown by the curve from point O to point in Figure At point O, the domins in the iron re rndomly oriented, corresponding to B m 0. As the incresing current in the primry coil cuses the externl field B 0 to increse, the domins become more ligned until ll of them re nerly ligned t point. At this point the iron core is pproching sturtion, which is the condition in which ll domins in the iron re ligned. Next, suppose tht the current is reduced to zero, nd the externl field is consequently eliminted. The B versus H curve, clled mgnetiztion curve, now follows the pth b in Figure Note tht t point b, B is not zero even though the externl field is B 0 0. The reson is tht the iron is now mgnetized due to the lignment of lrge number of its domins (tht is, B B m ). At this point, the iron is sid to hve remnent mgnetiztion. f the current in the primry coil is reversed so tht the direction of the externl mgnetic field is reversed, the domins reorient until the smple is gin unmgnetized t point c, where B 0. An increse in the reverse current cuses the iron to be mgnetized in the opposite direction, pproching sturtion t point d in Figure A similr sequence of events occurs s the current is reduced to zero nd then incresed in the originl (positive) direction. n this cse the mgnetiztion curve follows the pth def. f the current is incresed sufficiently, the mgnetiztion curve returns to point, where the smple gin hs its mximum mgnetiztion. The effect just described, clled mgnetic hysteresis, shows tht the mgnetiztion of ferromgnetic substnce depends on the history of the substnce s well s on the mgnitude of the pplied field. (The word hysteresis mens lgging behind. ) t is often sid tht ferromgnetic substnce hs memory becuse it remins mgnetized fter the externl field is removed. The closed loop in Figure is referred to s hysteresis loop. ts shpe nd size depend on the proper- QuickLb You ve probbly done this experiment before. Mgnetize nil by repetedly drgging it cross br mgnet. Test the strength of the nil s mgnetic field by picking up some pper clips. Now hit the nil severl times with hmmer, nd gin test the strength of its mgnetism. Explin wht hppens in terms of domins in the steel of the nil. ε Figure R S G A toroidl winding rrngement used to mesure the mgnetic properties of mteril. The torus is mde of the mteril under study, nd the circuit contining the glvnometer mesures the mgnetic flux. B b c O f H d e Figure mteril. Mgnetiztion curve for ferromgnetic

26 962 CHAPTER 30 Sources of the Mgnetic Field B B H H Figure Hysteresis loops for () hrd ferromgnetic mteril nd (b) soft ferromgnetic mteril. () (b) B H ties of the ferromgnetic substnce nd on the strength of the mximum pplied field. The hysteresis loop for hrd ferromgnetic mterils is chrcteristiclly wide like the one shown in Figure 30.31, corresponding to lrge remnent mgnetiztion. Such mterils cnnot be esily demgnetized by n externl field. Soft ferromgnetic mterils, such s iron, hve very nrrow hysteresis loop nd smll remnent mgnetiztion (Fig b.) Such mterils re esily mgnetized nd demgnetized. An idel soft ferromgnet would exhibit no hysteresis nd hence would hve no remnent mgnetiztion. A ferromgnetic substnce cn be demgnetized by being crried through successive hysteresis loops, due to decresing pplied mgnetic field, s shown in Figure Figure Demgnetizing ferromgnetic mteril by crrying it through successive hysteresis loops. Quick Quiz 30.8 Which mteril would mke better permnent mgnet, one whose hysteresis loop looks like Figure or one whose loop looks like Figure 30.31b? The mgnetiztion curve is useful for nother reson: The re enclosed by the mgnetiztion curve represents the work required to tke the mteril through the hysteresis cycle. The energy cquired by the mteril in the mgnetiztion process origintes from the source of the externl field tht is, the emf in the circuit of the toroidl coil. When the mgnetiztion cycle is repeted, dissiptive processes within the mteril due to relignment of the domins result in trnsformtion of mgnetic energy into internl energy, which is evidenced by n increse in the temperture of the substnce. For this reson, devices subjected to lternting fields (such s c dpters for cell phones, power tools, nd so on) use cores mde of soft ferromgnetic substnces, which hve nrrow hysteresis loops nd correspondingly little energy loss per cycle. Mgnetic computer disks store informtion by lternting the direction of B for portions of thin lyer of ferromgnetic mteril. Floppy disks hve the lyer on circulr sheet of plstic. Hrd disks hve severl rigid pltters with mgnetic cotings on ech side. Audio tpes nd videotpes work the sme wy s floppy disks except tht the ferromgnetic mteril is on very long strip of plstic. Tiny coils of wire in recording hed re plced close to the mgnetic mteril (which is moving rpidly pst the hed). Vrying the current through the coils cretes mgnetic field tht mgnetizes the recording mteril. To retrieve the informtion, the mgnetized mteril is moved pst plybck coil. The chnging mgnetism of the mteril induces current in the coil, s we shll discuss in Chpter 31. This current is then mplified by udio or video equipment, or it is processed by computer circuitry.

27 30.8 Mgnetism in Mtter 963 Prmgnetism Prmgnetic substnces hve smll but positive mgnetic susceptibility (0 resulting from the presence of toms (or ions) tht hve permnent mgnetic moments. These moments interct only wekly with ech other nd re rndomly oriented in the bsence of n externl mgnetic field. When prmgnetic substnce is plced in n externl mgnetic field, its tomic moments tend to line up with the field. However, this lignment process must compete with therml motion, which tends to rndomize the mgnetic moment orienttions. Pierre Curie ( ) nd others since him hve found experimentlly tht, under wide rnge of conditions, the mgnetiztion of prmgnetic substnce is proportionl to the pplied mgnetic field nd inversely proportionl to the bsolute temperture: V 1) M C B 0 (30.35) T This reltionship is known s Curie s lw fter its discoverer, nd the constnt C is clled Curie s constnt. The lw shows tht when B 0 0, the mgnetiztion is zero, corresponding to rndom orienttion of mgnetic moments. As the rtio of mgnetic field to temperture becomes gret, the mgnetiztion pproches its sturtion vlue, corresponding to complete lignment of its moments, nd Eqution is no longer vlid. When the temperture of ferromgnetic substnce reches or exceeds criticl temperture clled the Curie temperture, the substnce loses its residul mgnetiztion nd becomes prmgnetic (Fig ). Below the Curie temperture, the mgnetic moments re ligned nd the substnce is ferromgnetic. Above the Curie temperture, the therml gittion is gret enough to cuse rndom orienttion of the moments, nd the substnce becomes prmgnetic. Curie tempertures for severl ferromgnetic substnces re given in Tble Dimgnetism When n externl mgnetic field is pplied to dimgnetic substnce, wek mgnetic moment is induced in the direction opposite the pplied field. This cuses dimgnetic substnces to be wekly repelled by mgnet. Although dimgnetism is present in ll mtter, its effects re much smller thn those of prmgnetism or ferromgnetism, nd re evident only when those other effects do not exist. We cn ttin some understnding of dimgnetism by considering clssicl model of two tomic electrons orbiting the nucleus in opposite directions but with the sme speed. The electrons remin in their circulr orbits becuse of the ttrctive electrosttic force exerted by the positively chrged nucleus. Becuse the mgnetic moments of the two electrons re equl in mgnitude nd opposite in direction, they cncel ech other, nd the mgnetic moment of the tom is zero. When n externl mgnetic field is pplied, the electrons experience n dditionl force qv B. This dded force combines with the electrosttic force to increse the orbitl speed of the electron whose mgnetic moment is ntiprllel to the field nd to decrese the speed of the electron whose mgnetic moment is prllel to the field. As result, the two mgnetic moments of the electrons no longer cncel, nd the substnce cquires net mgnetic moment tht is opposite the pplied field. M s 0 M Ferromgnetic Figure TABLE 30.3 Curie Tempertures for Severl Ferromgnetic Substnces Substnce Prmgnetic T T Curie Mgnetiztion versus bsolute temperture for ferromgnetic substnce. The mgnetic moments re ligned below the Curie temperture T Curie, where the substnce is ferromgnetic. The substnce becomes prmgnetic (mgnetic moments unligned) bove T Curie. T Curie (K) ron Coblt Nickel 631 Gdolinium 317 Fe 2 O web Visit dimgnetism_www/index.html for n experiment showing tht grpes re repelled by mgnets!

28 964 CHAPTER 30 Sources of the Mgnetic Field Figure A smll permnent mgnet levitted bove disk of the superconductor YB 2 Cu 3 O 7 cooled to liquid nitrogen temperture (77 K). web For more detiled description of the unusul properties of superconductors, visit As you recll from Chpter 27, superconductor is substnce in which the electricl resistnce is zero below some criticl temperture. Certin types of superconductors lso exhibit perfect dimgnetism in the superconducting stte. As result, n pplied mgnetic field is expelled by the superconductor so tht the field is zero in its interior. This phenomenon of flux expulsion is known s the Meissner effect. f permnent mgnet is brought ner superconductor, the two objects repel ech other. This is illustrted in Figure 30.34, which shows smll permnent mgnet levitted bove superconductor mintined t 77 K. EXAMPLE Sturtion Mgnetiztion Estimte the sturtion mgnetiztion in long cylinder of iron, ssuming one unpired electron spin per tom. Solution The sturtion mgnetiztion is obtined when ll the mgnetic moments in the smple re ligned. f the smple contins n toms per unit volume, then the sturtion mgnetiztion M s hs the vlue M s n where is the mgnetic moment per tom. Becuse the molr mss of iron is 55 g/mol nd its density is 7.9 g/cm 3, the vlue of n for iron is toms/m 3. Assuming tht ech tom contributes one Bohr mgneton (due to one unpired spin) to the mgnetic moment, we obtin M s toms Am2 m tom A/m This is bout one-hlf the experimentlly determined sturtion mgnetiztion for iron, which indictes tht ctully two unpired electron spins re present per tom. Optionl Section 30.9 THE MAGNETC FELD OF THE EARTH When we spek of compss mgnet hving north pole nd south pole, we should sy more properly tht it hs north-seeking pole nd south-seeking pole. By this we men tht one pole of the mgnet seeks, or points to, the north geogrphic pole of the Erth. Becuse the north pole of mgnet is ttrcted towrd the north geogrphic pole of the Erth, we conclude tht the Erth s south mgnetic pole is locted ner the north geogrphic pole, nd the Erth s north mgnetic pole is locted ner the south geogrphic pole. n fct, the configurtion of the Erth s mgnetic field, pictured in Figure 30.35, is very much like the one tht would be chieved by burying gigntic br mgnet deep in the interior of the Erth.

29 30.9 The Mgnetic Field of the Erth 965 Figure Geogrphic equtor Mgnetic equtor South mgnetic pole South geogrphic pole North geogrphic pole North mgnetic pole The Erth s mgnetic field lines. Note tht south mgnetic pole is ner the north geogrphic pole, nd north mgnetic pole is ner the south geogrphic pole. S f compss needle is suspended in berings tht llow it to rotte in the verticl plne s well s in the horizontl plne, the needle is horizontl with respect to the Erth s surfce only ner the equtor. As the compss is moved northwrd, the needle rottes so tht it points more nd more towrd the surfce of the Erth. Finlly, t point ner Hudson By in Cnd, the north pole of the needle points directly downwrd. This site, first found in 1832, is considered to be the loction of the south mgnetic pole of the Erth. t is pproximtely mi from the Erth s geogrphic North Pole, nd its exct position vries slowly with time. Similrly, the north mgnetic pole of the Erth is bout mi wy from the Erth s geogrphic South Pole. Becuse of this distnce between the north geogrphic nd south mgnetic poles, it is only pproximtely correct to sy tht compss needle points north. The difference between true north, defined s the geogrphic North Pole, nd north indicted by compss vries from point to point on the Erth, nd the difference is referred to s mgnetic declintion. For exmple, long line through Florid nd the Gret Lkes, compss indictes true north, wheres in Wshington stte, it ligns 25 est of true north. N QuickLb A gold ring is very wekly repelled by mgnet. To see this, suspend 14- or 18-krt gold ring on long loop of thred, s shown in (). Gently tp the ring nd estimte its period of oscilltion. Now bring the ring to rest, letting it hng for few moments so tht you cn verify tht it is not moving. Quickly bring very strong mgnet to within few millimeters of the ring, tking cre not to bump it, s shown in (b). Now pull the mgnet wy. Repet this ction mny times, mtching the oscilltion period you estimted erlier. This is just like pushing child on swing. A smll force pplied t the resonnt frequency results in lrge-mplitude oscilltion. f you hve pltinum ring, you will be ble to see similr effect except tht pltinum is wekly ttrcted to mgnet becuse it is prmgnetic. () (b) The north end of compss needle points to the south mgnetic pole of the Erth. The north compss direction vries from true geogrphic north depending on the mgnetic declintion t tht point on the Erth s surfce.

30 966 CHAPTER 30 Sources of the Mgnetic Field Quick Quiz 30.9 f we wnted to cncel the Erth s mgnetic field by running n enormous current loop round the equtor, which wy would the current hve to flow: est to west or west to est? Although the mgnetic field pttern of the Erth is similr to the one tht would be set up by br mgnet deep within the Erth, it is esy to understnd why the source of the Erth s mgnetic field cnnot be lrge msses of permnently mgnetized mteril. The Erth does hve lrge deposits of iron ore deep beneth its surfce, but the high tempertures in the Erth s core prevent the iron from retining ny permnent mgnetiztion. Scientists consider it more likely tht the true source of the Erth s mgnetic field is chrge-crrying convection currents in the Erth s core. Chrged ions or electrons circulting in the liquid interior could produce mgnetic field just s current loop does. There is lso strong evidence tht the mgnitude of plnet s mgnetic field is relted to the plnet s rte of rottion. For exmple, Jupiter rottes fster thn the Erth, nd spce probes indicte tht Jupiter s mgnetic field is stronger thn ours. Venus, on the other hnd, rottes more slowly thn the Erth, nd its mgnetic field is found to be weker. nvestigtion into the cuse of the Erth s mgnetism is ongoing. There is n interesting sidelight concerning the Erth s mgnetic field. t hs been found tht the direction of the field hs been reversed severl times during the lst million yers. Evidence for this is provided by bslt, type of rock tht contins iron nd tht forms from mteril spewed forth by volcnic ctivity on the ocen floor. As the lv cools, it solidifies nd retins picture of the Erth s mgnetic field direction. The rocks re dted by other mens to provide timeline for these periodic reversls of the mgnetic field. SUMMARY The Biot Svrt lw sys tht the mgnetic field db t point P due to length element ds tht crries stedy current is Tm/A db 0 4 ds rˆ r 2 (30.1) where is the permebility of free spce, r is the distnce from the element to the point P, nd ˆr is unit vector pointing from ds to point P. We find the totl field t P by integrting this expression over the entire current distribution. The mgnetic field t distnce from long, stright wire crrying n electric current is B (30.5) 2 The field lines re circles concentric with the wire. The mgnetic force per unit length between two prllel wires seprted by distnce nd crrying currents 1 nd 2 hs mgnitude F B (30.12) 2 The force is ttrctive if the currents re in the sme direction nd repulsive if they re in opposite directions

31 Questions 967 Ampère s lw sys tht the line integrl of B ds round ny closed pth equls 0, where is the totl stedy current pssing through ny surfce bounded by the closed pth: B ds 0 (30.13) Using Ampère s lw, one finds tht the fields inside toroid nd solenoid re B 0N 2r (toroid) (30.16) B 0 N 0n (solenoid) (30.17) where N is the totl number of turns. The mgnetic flux B through surfce is defined by the surfce integrl B B da (30.18) Guss s lw of mgnetism sttes tht the net mgnetic flux through ny closed surfce is zero. The generl form of Ampère s lw, which is lso clled the Ampère-Mxwell lw, is B ds 0 d E 00 (30.22) dt This lw describes the fct tht mgnetic fields re produced both by conduction currents nd by chnging electric fields. QUESTONS 1. s the mgnetic field creted by current loop uniform? Explin. 2. A current in conductor produces mgnetic field tht cn be clculted using the Biot Svrt lw. Becuse current is defined s the rte of flow of chrge, wht cn you conclude bout the mgnetic field produced by sttionry chrges? Wht bout tht produced by moving chrges? 3. Two prllel wires crry currents in opposite directions. Describe the nture of the mgnetic field creted by the two wires t points () between the wires nd (b) outside the wires, in plne contining them. 4. Explin why two prllel wires crrying currents in opposite directions repel ech other. 5. When n electric circuit is being ssembled, common prctice is to twist together two wires crrying equl currents in opposite directions. Why does this technique reduce stry mgnetic fields? 6. s Ampère s lw vlid for ll closed pths surrounding conductor? Why is it not useful for clculting B for ll such pths? 7. Compre Ampère s lw with the Biot Svrt lw. Which is more generlly useful for clculting B for currentcrrying conductor? 8. s the mgnetic field inside toroid uniform? Explin. 9. Describe the similrities between Ampère s lw in mgnetism nd Guss s lw in electrosttics. 10. A hollow copper tube crries current long its length. Why does B = 0 inside the tube? s B nonzero outside the tube? 11. Why is B nonzero outside solenoid? Why does B 0 outside toroid? (Remember tht the lines of B must form closed pths.) 12. Describe the chnge in the mgnetic field in the interior of solenoid crrying stedy current () if the length of the solenoid is doubled but the number of turns remins the sme nd (b) if the number of turns is doubled but the length remins the sme. 13. A flt conducting loop is positioned in uniform mgnetic field directed long the x xis. For wht orienttion of the loop is the flux through it mximum? A minimum? 14. Wht new concept does Mxwell s generl form of Ampère s lw include? 15. Mny loops of wire re wrpped round nil nd then connected to bttery. dentify the source of M, of H, nd of B.

32 968 CHAPTER 30 Sources of the Mgnetic Field 16. A mgnet ttrcts piece of iron. The iron cn then ttrct nother piece of iron. On the bsis of domin lignment, explin wht hppens in ech piece of iron. 17. You re strnded on plnet tht does not hve mgnetic field, with no test equipment. You hve two brs of iron in your possession; one is mgnetized, nd one is not. How cn you determine which is which? 18. Why does hitting mgnet with hmmer cuse the mgnetism to be reduced? 19. s nil ttrcted to either pole of mgnet? Explin wht is hppening inside the nil when it is plced ner the mgnet. 20. A Hindu ruler once suggested tht he be entombed in mgnetic coffin with the polrity rrnged so tht he would be forever suspended between heven nd Erth. s such mgnetic levittion possible? Discuss. 21. Why does M 0 in vcuum? Wht is the reltionship between B nd H in vcuum? 22. Explin why some toms hve permnent mgnetic moments nd others do not. 23. Wht fctors contribute to the totl mgnetic moment of n tom? 24. Why is the mgnetic susceptibility of dimgnetic substnce negtive? 25. Why cn the effect of dimgnetism be neglected in prmgnetic substnce? 26. Explin the significnce of the Curie temperture for ferromgnetic substnce. 27. Discuss the differences mong ferromgnetic, prmgnetic, nd dimgnetic substnces. 28. Wht is the difference between hrd nd soft ferromgnetic mterils? 29. Should the surfce of computer disk be mde from hrd or soft ferromgnetic substnce? 30. Explin why it is desirble to use hrd ferromgnetic mterils to mke permnent mgnets. 31. Would you expect the tpe from tpe recorder to be ttrcted to mgnet? (Try it, but not with recording you wish to sve.) 32. Given only strong mgnet nd screwdriver, how would you first mgnetize nd then demgnetize the screwdriver? 33. Figure Q30.33 shows two permnent mgnets, ech hving hole through its center. Note tht the upper mgnet is levitted bove the lower one. () How does this occur? (b) Wht purpose does the pencil serve? (c) Wht cn you sy bout the poles of the mgnets on the bsis of this observtion? (d) Wht do you suppose would hppen if the upper mgnet were inverted? Figure Q30.33 Mgnetic levittion using two cermic mgnets. PROBLEMS 1, 2, 3 = strightforwrd, intermedite, chllenging = full solution vilble in the Student Solutions Mnul nd Study Guide WEB = solution posted t = Computer useful in solving problem = nterctive Physics = pired numericl/symbolic problems Section 30.1 The Biot Svrt Lw 1. n Niels Bohr s 1913 model of the hydrogen tom, n electron circles the proton t distnce of m with speed of m/s. Compute the mgnitude of the mgnetic field tht this motion produces t the loction of the proton. 2. A current pth shped s shown in Figure P30.2 produces mgnetic field t P, the center of the rc. f the rc subtends n ngle of 30.0 nd the rdius of the rc is m, wht re the mgnitude nd direction of the field produced t P if the current is 3.00 A? 3. () A conductor in the shpe of squre of edge length m crries current 10.0 A (Fig. P30.3). Clculte the mgnitude nd direction of the mgnetic P 30.0 Figure P30.2 field t the center of the squre. (b) f this conductor is formed into single circulr turn nd crries the sme current, wht is the vlue of the mgnetic field t the center?