1、Chapter 25 & 26 Magnetic Field1. Magnetic Flux and Gauss Law2. The Magnetic Force on a Particle3. Magnetic Force on a Current-Carrying Wire4. Magnetic Field Due to Current5. Amperes Law6. Magnetic MaterialsMagnets used in: compass(指南針); refrigerator-door; microphone; computer disk; Magnetic phen

2、omena:25-1.The Definition of (P580)BIThe interacting force between moving charges is magnetic force which is transferred by magnetic field (運動電荷之間的相互作運動電荷之間的相互作用力用力-磁力磁力, ,由磁場傳遞由磁場傳遞) ).A electric charge set up an electric field that can affect other electric charges. Might you expect that a magneti

3、c charge set up a magnetic field that can then affect other magnetic charges? Such magnetic charges are called magnetic monopoles, which are predicted by certain theory, although their existence has not been confirmed.moving chargesmoving chargesmagnetic field BThe magnetic force is:BvqF Lorentzs fo

4、rcevqFB|max The magnitude of :BBThe direction of is along the zero-force.(P586)BvF +q In a similar way as the definition of electric field, we could define by the acting force of magnetic filed on the moving charges as:B)gauss(G10T14 Dimension: I-1MT2.地磁場地磁場Bearth 5 10-5 TiBBBB 21 iiBThe principle o

5、f superposition of : BSI unit: tesla (特斯拉特斯拉) (T), gauss or N/(Am).An equivalent way to define in terms of the force on electric currents is discussed in Sec.25-3 (P583).B25-2. Magnetic Field Lines and Flux (P581) No origin, no termination and no cross point( (無頭無尾不相交無頭無尾不相交, ,閉合曲線閉合曲線) ); Wrap arou

6、nd with current與與電流套連電流套連; ; Right-hand rule with current ( (與電流成右手螺旋關系與電流成右手螺旋關系) )1. The features of magnetic field lines2. Magnetic flux & Gauss lawBABddLet number of magnetic field line threading area dA, Bd surfaceclosed0dABB(Gauss law of magnetic field)In the absence of magnetic monopoles

7、the magnetic flux threading a closed surface is zero.ABABndddm nAd SSABddmmThe flux of magnetic field intersects a closed surface:25-3. Magnetic Force on Moving Charges1. Lorentzs force (P586)BvqF F=qvsin (when = 0, ,F=0),BvFq The charged particles moving in magnetic field acted by magnetic force:Th

8、e force acting on a charged particle moving with velocity through a magnetic field is always perpendicular to and .BFBvvBMagnetic force only change moving direction WB=0.BvqFqBvqFq , 0, 02. The motion of charged particle in uniform :BB+qvq-vvq+BRq0: , particle straightly move in constant velocity.0

9、FqBmvR Radius of track:(2) Particle moves perpendicular toB(1) Particle moves parallel toBvq-BRq0:F=qvB = C and radially directed.B Circular motion with constant speed in the plane of perpendicular to qBmvRT22 Period:v/= vcos , v=vsin qBmvR sin Radius:qBmvRT2sin2 Period:qBvmTvhcos2cosPitch(螺距螺距):Par

10、ticle follows a spiral path (螺旋線運動螺旋線運動) ).(3) Particle moves in an angle with (P587):BLook at example 25-6 of P588 to learn about the phenomenon of aurora(極光極光), please.h+q From it one can determine the motion of particles.amBvqEq It is often called the Lorentz equation and is considered one of the

11、 basic equation in physics.If a particle of charge q moves with velocity in the presence of both magnetic field and the electric field , it will feel a force EvB)(BvEqF If no other external forces, then(1) Discovery and properties of electron (P592)BBoth and can produce a force on a charged particle

12、:EFig.25-29 in P5933. Samples of practical applicationrvmevB2 Brvme evBeE BEv/ rBvme2 It can be used to measure the ratio m/q of the particles moving through Thomsons apparatus.電子比荷電子比荷(2) The Hall effect (霍爾效應霍爾效應) )(P594)載流導體放在磁場中，載流導體放在磁場中，磁場垂直于電流方向，磁場垂直于電流方向，則在磁場與電流垂直方則在磁場與電流垂直方向出現橫向電勢差。向出現橫向電勢差

13、。It is called Hall effect, the electric potential arisen from it is Hall voltage. BvqqE HBvE HBlvlEV HHldvqnI dIBRnqdIBVHH What relation between V & I ?qnldIv HHand1qdVIBnnqR Hall effect can be used to find B, I, density, positive or negative and drift speed of charged carriersBV+ + + + +- - - -

14、 - - - - - - - - -+IldOO半導體類型判別半導體類型判別BlvlEV HH25-4. on a Current-carrying Wire (P605)BFFreely choose a short length dl, cross area A, mass m, charge q, then the Lorentz force on each particle: sinqvBf N=nA dl)where(d)d(dqnAviBliBvqlnAfNF BliFFdd(force on a current)Note: where is external magnetic f

15、ield.B1. Magnetic force on a currentBBBi Example:A straight, horizontal length of copper wire has a current I=28A through it. What are magnitude and direction of the minimum magnetic field needed to suspend the wire (to balance ) on it? The linear density of wire is 46.6g/m.gFLgFBFBmgiLB sinSolution

16、:T106 . 1)/(sin2 igLmiLmgB This is about 160 times the strength of Earths magnetic field.A half-circular wire of radius R with current I is put in a uniform magnetic field as the figure shown, =30, find the magnetic force acted on the arc (此段圓弧電流受的磁力此段圓弧電流受的磁力) ).)()(dbaBlIFBlIba)()(dBabI uniform Ra

17、b2 BabIlId Solution: sinBabIF IBR direction of magnetic force:For this case,Example:BabINo magnetic force exerted on a current loop in a uniform magnetic field.Bl IBabIF From equationwe can learn:(1) The sum of magnetic force acted on a whole curved wire = the one of straight wire with same current

18、connected from start to end points (均勻磁均勻磁場中任意場中任意彎曲彎曲導線所受的磁場力導線所受的磁場力= =相同電流相同電流的的直直導線所受的磁場力導線所受的磁場力).(2) If points a and b overlapping, then l=0, F=0. (3) Magnetic force is always perpendicular to the plane defined by length vector and magnetic field.(均勻均勻磁場中磁場中任意閉合任意閉合導線所受的磁場力導線所受的磁場力=0)abbbaBlIF

19、 dddIBaa20 babldIIid20 baaabBlIFddabaldIIid20 Two long parallel wires carrying currents exert forces (平行載流導線平行載流導線) )on each other as the figure,2. Force between two parallel wires (P605)Think about how about the force between two antiparallel wire 平行反向載流導線之間的力怎樣平行反向載流導線之間的力怎樣( (方向方向、大小、大小)?)? IaIba

20、babFdbBbaFdaBdlbdladx repel each other!Parallel currents attract, and antiparallel currents repel.dIIlFlFbabbaaab 2dddd0 Acting force exerted on the ab wire in unit-length:It is the same for both wires of ab and cd.To find the force on a current-carrying wire due to a 2nd current-carrying wire, firs

21、t find the field due to 2nd wire at the site of 1st wire. Then find the force on 1st wire due to that field.The definition of the ampere (電流單位電流單位“安培安培”) ) (P607): 兩無限長平行載流導線相距兩無限長平行載流導線相距1m,1m,通通有相同電流有相同電流, ,單位長互作用力為單位長互作用力為2 2 10 10 -7 -7 N N時時, ,導線內電流為導線內電流為1A.1A.Proof: when Ia =Ib=1A, dIIlFba2dd

22、0 N/m1027 20 m)Wb/(A104N/A1047270 Hence,One ampere is defined as that current flowing in each of two long parallel conductors 1m apart, which results in a force of exactly of length of each conductor. N/m1027 Coulomb is then defined as being exactly one ampere-second: 1C = 1A s.aBIFFF 21 cosbBaI Tor

23、que (力矩力矩) : sinIBA 25-5. Torque on a Current Loop (P589)Magnetic force exerts on a current loop as the figure,bn This formula, derived here for a rectangular coil, is valid for any shape of flat coil. cosBAI BBnIAN nNIA Introducing the magnetic dipole moment of the coil (線圈的磁矩線圈的磁矩). ). It is consi

24、dered a vector:It is also right for any shape coil of N loops immersed in a uniform magnetic field.bn The direction of is perpendicular to the plane of the coil consistent with the right-hand rule Another way to measure magnetic field:the one of magnetic dipole moment when a coil is in stable-equili

25、brium.Direction of :BMagnitude of : B max B = TAm2 =Nm SI unit of the torque: Magnetic potential energy (P590):In order to rotate a current loop so as to increase . 21d W 21dsin B)cos(cos21 BWRecall that an electric dipole has potential energy when in ,EpUeEwe must do work against the force due to t

26、he magnetic field:or, the work done by a magnetic torque is:dsindBW(1),(B cosddBW dW can also be written as:)cosd(cosd SBIISB I dIWdd(2)That is,so, it can be seen that thework done by magnetic force on a current wire:W = I dsindBW(1)From equation:A dipole is capable of doing work and possesses U.As

27、a convenience, U=0 is chosen as the axis of dipole is perpendicular to the , i.e. . ThenBB dsind2/2/ BWU 2/| )(cosB That is,BBU cosLook at P590!The potential energy of magnetic dipole moment in the magnetic field (磁矩在磁場中勢能）磁矩在磁場中勢能）.The torque on the dipole is zero whether & are parallel or anti

28、parallel. So both orientations are equilibrium configurations. The parallel is one of stable equi., and the antiparallel is one of unstable. BExample:Solution:A current of 0.5A exists in a circular coil of wire situated in a uniform B=0.5T. The coil has a mean radius of 5cm and contains 250 turns of

29、 wire. How much work is done by magnetic forces in rotating the coil from a position where to a position where . B B/ 1 . .2 BThe magnitude of magnetic moment is:NIS 22mA982. 0)m05. 0()A5 . 0(250 To calculate the work done by magnetic forces, we may use one of following methods:IWMethod II:J491. 0BN

30、ISBMethod III:02cosinitialBBUBBBU0cosfinalfiUUUWJ491. 0Bdsin02/BWJ491. 0| )cos(2/BBMethod I:1 . .2 B)0(BSNILook at the examples given in page 590!25-6. Biot-Savarts Law & Applications (P613)1. Biot-Savarts lawElectric currents create magnetic fields. This discovery by Oersted in 1820 opened up a

31、 new field of research.Shortly after 1820 J. B. Biot and F. Savart (both are French physicists) developed what is known as the law of Biot and Savart gives a quantitative description of Oersteds magnetic field in terms of the electric current. A question arises: Is there a general relation between a

32、 current in a wire of any shape and the magnetic field around it?A/mT10470 kIn SI unit:270A/N104 220LMTI 30d4drrsIB IBd PrsIdThe current element Ids generates a magnetic field dB given by:Magnetic permeability constant in a vacuum (真空磁導率真空磁導率):):The principle of superposition of magnetic field, can

33、be applied for a complete circuit: 30d4drrsIBB (Biot-Savart law)(1) A long straight wire (載流長直導線的磁場載流長直導線的磁場 P614)P614) LLrsIBB20dsin4d The magnitude of the differential magnetic field produced at P by the current-length element Ids located a distance r from P is given by:20d4drrsIB 2. The applicati

34、on of Biot-Savart law r =R /sin , s = -R cot xPRssIdrLIBd 1 2 dsin4sind421020 RIrsIBL 210coscos4 RIBBIBIDiscussions:The magnetic field from all current element of straight wire are in the same direction and into the screen. For straight and infinite wire (無限無限長載流直導線長載流直導線) ), 1 1=0,=0, 2 2= = , ,the

35、n The magnitude of magnetic field of infinite straight wire at a point with distance R is in inverse proportion to the distance.RIB20 The case given in P614(2) In a circular arc of wire (載流圓線圈軸線上的磁場載流圓線圈軸線上的磁場) )30d4drrsIB 20d4drsIB As the figure, given I, R, x IBdBd /dB sIdpxxR0rBdThe direction of

36、is perpendicular to the plane made of Ids and r and . LBB/d sind4sin)d(d20/rsIBB LrsIB sind420 2/322202xRIR 20202sind4sinrIRsrIL I BdBd /dB sIdpxxR0r 2/322202xRIRB Discussions:RIB20 At center O, x=0,For a circular arc of wire (一段圓弧電流圓心處的磁感應強度一段圓弧電流圓心處的磁感應強度):):RIB40 The direction of is along the axi

37、s direction (right-hand rule) as right figure and Fig.26-18 in P615.BsIdBd0/d21 BBrsiSolution:RiRiB84)2/(003 (direct into the screen)RiBBBB80321 (into the screen)You may comparing with the calculation of electrostatic field.Example 26-11 of P616:One quarter of a circular loop of wire carries a curre

38、nt I, and two straight sections whose extensions intersect the center C of the arc. Find at point C?B132For parts 1 and 2,The number of turns per unit length of the solenoid( (螺線管螺線管, ,單位長匝數單位長匝數) ) n. As vertical cross section (縱剖面縱剖面) shown, (I, n, R, L, 1, 2 ) (3) A stretched-out solenoid (載流直螺線管

39、載流直螺線管軸線上軸線上的磁場的磁場) )IIThink about: how to choose a current-element dI? 2/322202/32220d2d2dxRxnIRxRIRBP .dxx . . . . . . . . . . . . . .L 1 2Pchoose dI=nIdxx=Rcot , dx=Rcsc2 d , R2+x2= R2 csc2 210dsin2 nIB 120coscos2 nIBSame direction!Explanations:For a “infinitely long” solenoid, The at the axial l

40、ine: :BnIB0 0, 21 5R5R0.439BxDirection inside the solenoid is along the axis and right-hand rule as Fig.26-12.B-x curve inside the solenoid (L=10R):25-7. Amperes Loop Law & ApplicationAndre Marie Ampere (17751836) French mathematician and physicist.安培分子環流假說安培分子環流假說 : The source of magnetism of a

41、ll natural-matter come from current（一切磁性（一切磁性的根源是電流）的根源是電流）.Ampere: All magnetic phenomena come from current (一切磁現象都起源于電流一切磁現象都起源于電流).1. Amperes loop law (P607)Any line integral of around a closed path was proportional to the current encircled by the path. B LilBenc0d (in SI units)在恒定磁場中，磁感強度在恒定磁場中，

42、磁感強度沿沿任一閉合環路的線積分任一閉合環路的線積分等于等于穿過穿過該環路所圍曲面該環路所圍曲面的的所有電流代數和所有電流代數和的的 倍倍B0It is only hold for closed steady (閉合恒定閉合恒定) current.BContributed by all current inspace.Freely choose a closed loop and assign a positive direction.LExplanation:enciThe net current encircled by the loop (與與L套連的電流套連的電流, ,回路所圍面積截得

43、回路所圍面積截得) )accord with right-hand rule (與與L繞行方向成右螺電流取正繞行方向成右螺電流取正).ldAny differential elements on the L, along the tangent direction to the loop.In the above case,)(d210iilBL Lld3I1I2IBLook at the description given in textbook of P608.BIt plays very important role in the theory of 電磁電磁場。場。One can al

44、so use Amperes law to calculate with symmetrical distribution of current.2. The application of Amperes lawMain steps:(1) Symmetry analyses (由由I I的分布，分析分布的對稱性的分布，分析分布的對稱性) ).(2) Choose a loop (使回路上各處使回路上各處B相等，方向特殊，從而相等，方向特殊，從而可從回路積分中提出可從回路積分中提出B) ).(3) Put into the law and calculate B.Find the inside

45、 and outside a long straight wire of radius R and uniformly distributed current I over a cross section of the wire 長直圓柱形載流圓柱體長直圓柱形載流圓柱體( (電流電流I I 均勻分布在圓柱的橫截面內均勻分布在圓柱的橫截面內) )內外的磁場內外的磁場. .BSolution:Cylindrical symmetryP609Choose a circular loop L1 with r R, Example:rBlBL2d Choose a circular loop L2 wi

46、th r R, rRiB20in2RrL2i L1r rriB20outLlBdenc0i220rRiirB02LlB dL2r rBCan you draw the curve of Distribution of in whole space| B? L2r renc02dirBlBL Bin=0IrBlBL02d rIB20out In the similar way one may find (長直圓柱長直圓柱面面載流導線載流導線內外的磁場內外的磁場) as figure:BBr0RriB20 Compare with produced by a current i in a long

47、 straight wire, BCan we find the result of finite long straight wire (有限長載流直導線有限長載流直導線) ) by using Ampares law?I I= 0Solution:abcd Plane symmetryllB 02 20 B LlBdIt is a uniform field, independent of position r| B在無限大均勻平面電流兩側的磁場均在無限大均勻平面電流兩側的磁場均為為均勻磁場均勻磁場，并且大小相等，并且大小相等, ,方向相反。方向相反。Example:R1R2Magneti

48、c field of a toroid (載流螺繞環內的磁場載流螺繞環內的磁場), given i, N, R1 , R2 .Solution:Example:NirBlBL02d From symmetry, the lines of form concentric circles inside the toroid, directed as figure,BChoose concentric (同心同心) ) circle loop L (R1 r R2 ),rNiB20 (Right-hand rule direction)niLNirNiB0002 Discussion:(1) In

49、contrast to the situation for a solenoid, B is not constant over cross section of a toroid. But if its cross area is relatively small, thenIt is as same as the expression of the one of long solenoid.(2) so the magnetic field of toroid is limited in its interior., 0out BFind inside a “infinitely long

50、” solenoid (載流長直螺線載流長直螺線管內部的磁場管內部的磁場)()(The number of turns per unit length of the solenoid n and total number of coils N).B0d lBat paths ad, bc, and dc,nlIBllBlBba0dd Example:Solution:B=0labcdBased on the feature of the distribution of , we choose a rectangular loop abcd,BLNInIB00 1. Classification

51、 of magnetic materialsrr000+, =BBBBBBBWhen the interior of a solenoid is filled with some material of relative permeability r 相對磁導率相對磁導率) ),1rFerromagnetism (鐵磁質鐵磁質) )Paramagnetism (順磁質順磁質) )1rDiamagnetism (抗磁質抗磁質) )1r25-8. Interaction between Magnetic Field and Dielectrics Magnetic Materials2. Magn

52、etization(磁化磁化) )of magnetic materialsCross area of solenoidWhen the interior of a solenoid is fully filled with some isotropic material r , it will be magnetized (長直螺線管內部充滿均勻的各向同性介質，將被均勻磁化長直螺線管內部充滿均勻的各向同性介質，將被均勻磁化) ). Taking the paramagnetic material as an example,each molecular in it is equivalent to a loop current: ITo express the extent to which a given medium is magnetized, a vector quantity, magnetization 磁化強度磁化強度, is introduced:MThe unit of is A/m.MVMiiv0limIt can be proved, LlMIdThe net bound current enclosed in a closed loop L equals to the circulation of mag