大學(xué)物理 穩(wěn)恒磁場(chǎng)

1、2022-2-1512022-2-1522022-2-153uVacuum: without medium.uMagnetic Field : a vector field which exerts a magnetic force on moving electric charges uSteady Magnetic Field : invariable with time/ produced by steady current. unstable magnetic field.2022-2-154uPerceptual Knowledge例22022-2-155INSMagnetic fi

2、eld lines surrounding a current loop. The field pattern is axially symmetric.uPictorial Representation of Magnetic field2022-2-156IMagnetic field lines surrounding a long and straight wires 2022-2-157IIMagnetic field lines for a tightly wound solenoid of finite length carrying a steady current.2022-

3、2-1582022-2-159The strength of the field is proportional to the density of the lines.The direction of magnetic field lines is defined to be the direction that the north end of a compass needles points, i.e., it is tangent to the field line at any point in space.uProperties of Magnetic Field Lines202

4、2-2-1510Magnetic field lines can never cross, meaning the field is unique at any point in space.The magnetic field lines are continuous closed loops without beginning or end. （therefore, the north and south poles cannot be separated . It is a distinct difference from electric field lines, which begi

5、n and end on positive and negative charges. If magnetic monopoles existed, then magnetic field lines would begin and end on them.）2022-2-1511MAIN CONTENTSnMagnetic field produced by currents.nRepresentation ：nRules ：nBiot-savarts lawnGauss theoremnAmperes circulation theorem (Amperes Law)B0SdB034uId

6、lrBriiLIlB0dvFmaxqvFBmax2022-2-1512nAffect of magnetic field force on currentsnForce on a moving chargenForce on a current-carrying wirenForce on current-carrying loopBlIdFdBPMmBvqfL2022-2-1513Methods for StudynContrast: electrostatic field.2022-2-151413-1 Magnetic Induction uHow to quantify the mag

7、netic field ? BElectric field: investigate the result of the force exerted on a test point chargeMagnetic field: investigate the result of force exerted on a moving test point charge (Lorentz force)/EF qDemand? 2022-2-1515Results BvqqmFvB the Lorentz force acted on the charge is related with its ele

8、ctric quantity , instantaneous position, and its velocity. The force acted on the moving point charge by magnetic field is called Lorentz force. 2022-2-1516vBLorentz Force:sinFqvBsinFBqvFqFvBFqv BThe coefficient B is the function of position, it describes the natural quality of magnetic field at the

9、 position. 0 0F， The right hand rule2022-2-1517Fqv BThe electric force is always parallel or antiparallel to the direction of the electric field, whereas the magnetic force is perpendicular to the magnetic field.uDISCUSSIONThe magnetic force acts on a charged particle only when the particle is in mo

10、tion.2022-2-1518The electric force does work in displacing a charged particle, whereas the magnetic force associated with a steady magnetic field does no work when a charged particle is displaced.Fqv B2022-2-1519 Magnitude : sinmFBqvvBmFuMagnetic InductionBDirection : Unit : tesla (T) (brilliant inv

11、entor Nikola Tesla,1856-1943 )111 /NNTC m sA mAnother smaller unit: gauss (G), 41 10GT2022-2-1520Magnetic Force Due to the Earths Field is usually small.The earths magnetic field on its surface is only about 0.5G.For example, there is a moving point charge , CQ810210/vm s55 10BT 1110FNMagnetic force

12、 due to the earths field is usually negligible on any macroscopic object.2022-2-1521uSuperposition of Magnetic InductioniiBB2022-2-152213-2 Biot-Savart Law uSTRATEGYElectrostatic field : QdQdEEdESteady magnetic field :BdBBdlIdI2022-2-1523uCurrent element IlIdrPDirection: the direction of the current

13、Magnitude :Idl2022-2-1524uDifference Between the Current Element and the Point Charge:The point charge can exist independently. However, the current element cant.(Because we must have a complete circuit for a current.)2022-2-1525u rthe displacement from toward P.Idlu dBthe contribution of to the mag

14、netic induction at point P.Idlu Bthe magnetic field of I at point P.IlIdrPBdBuBiot-Savart law says that the magnetic field at point P created by an element of infinitesimal length dl of the wire.dB2022-2-1526I30d4drrlIB 270104NA magnetic permeability of vacuum:20sind4drlIB rPlIdu Biot-Savarts law rl

15、Id Magnitude:Direction :right hand rule2022-2-1527LBBdiBB304drrlIduBBLL uSuperposition Principle of Magnetic Induction2022-2-1528304drrlIduBBLL uDISCUSSIONThe integral is a vector operation and must reduce it to a scalar integral. is a vector directed from the current element toward the field point.

16、r zzyyxxzyxzyxdBBdBBdBBkBjBiBBkdBjdBidBBd 2022-2-1529We can obtain the magnetic induction of magnetic field generated by any current in principle according to Biot-Savart law .2022-2-1530uDISCCUSIONA point charge can generate electric field at any point around it. Similarly , can a current element p

17、roduce magnetic field at any point surrounding it ?2022-2-1531u Application Find magnetic field of a motion point chargel dvSCurrent elementrlIdThe magnetic field is generated by moving charges.2022-2-153220sin4rIdldB20sin4rqnvSdldB204rni svqdNdBBqnvSI 304rrvquB of a motion point chargeB2022-2-1533v

18、r0q0qvrPP304rrvquB u directed from the charged particle to the field point.r2022-2-1534uExampleBA charge Q does circular motion of radius R and angular speed .Find the magnetic induction at the centre O due to the motion charge .2022-2-1535two methods to find BuSUMMARYthe magnetic field is generated

19、 by currents.03 B4uIdlrdBdBrthe magnetic field is generated by the motion point charges.03 4uqvrBr2022-2-1536uExampleFind the magnetic induction at a point by a finite straight current-carrying wire. IaP02022-2-1537l 2 1IarP0lId30d4drrlIB 20sind4drlIB ar sin sinar ctgal-l 2sinddal Direction : Soluti

20、on: 2022-2-153821dsin40aIB 210coscos4 aIl 2 1IaP0usubstitute a with r: 210coscos4rIB2022-2-1539DISCCUSIONuThe magnetic induction at a point by an infinite straight current-carrying wire L , 1=0 2= rIB202022-2-1540？，Br 0u 21,2rIB40u the half infinite straight current-carrying wire :u at the points of

21、 the extended line:oB rp2022-2-1541uExampleThere is steady current I distributed uniformly on a plate of width a and infinite length. Point P is in the same plane with the plate and is away from right side of the plate by b.IaxPb0 xFind magnetic induction at P. 2022-2-1542IaxPb0 xdxaIdIxdIB2d0babaxd

22、xIB20axdxI20bbaaIln202022-2-1543uExamplexRIFind magnetic field on the axis of a circular current loop.2022-2-1544uReasoning lIdrxBdBd/dBRxIBNote that any element is perpendicular to . lIdrAll elements around the loop are at the same distance r from the P. can be resolved into a component ,along the

23、x axis, and , which is perpendicular to the x axis. Bd/dBBd2022-2-1545LdlrI204sin lIdrxBdBd/dB0dBB30d4drrlIB 20d4drlIB cosddBB sindd/BB /dBBRR 2x22sinxRRrR 2322202xRiIRB IB2022-2-1546002IxBR2032IRBx302 xIS BmPuAt the centre of the loop: )0(xatuWhen x is large compared with R, for xRDISCCUSION2022-2-

24、1547InmpInmpMagnetic Moment of a Current Loop Right hand rule.2022-2-1548RIB20 BI I RIRIB 42200 2 Central angle: Arbitrary central angle: I IB （Current-carrying circular arc）RIB40 O OI IRRIB80 IO RRIRIB 2400 ORI OIR32 )(RIRIB231600 BuFind the magnetic induction at point O.Find magnetic induction1. M

25、agnetic induction of current element and superposition principle.SUMMARYFind electric intensity1.Electric intensity of point charge and superposition principle.iiirrQE3 30 04 4 QrdQrE3 30 04 4 （separated）（continuous）30d4drrlIB LBBdTypical Magnetic Induction304rrqErE0230d4drrlIB 210coscos4rIBrIB20)co

26、s(cos4210Edirection:Typical Electric Intensitycurrent element:point charge:a finite straight current carrying wire:uniformly charged straight line:direction:an infinite straight current carrying wire:Infinitely uniformly charged straight line: 2322202xRiIRB 2 23 32 22 20 04 4RxixQEE/ RIBx2000 x0EnIS

27、PmlQpeon the axis of a circular current loop:on the axis of a charged circular loop:at the center of the loop:at the center of the loop:magnetic dipole moment:electric dipole moment:2022-2-1553IrLI uIf the magnetic field is created by currents which are mirror-image symmetric , its direction must be

28、 perpendicular to the symmetric plane.2022-2-1554 I.uThe Magnetic Field Created by a Long Current-Carrying Plane.2022-2-155513-3 Gauss Law of Magnetic Field The integral of magnetic induction on a curve surface in magnetic field is called the magnetic flux of the curve surface, written as the follow

29、ing: uDefinition: magnetic fluxcosmSSB dSBdSUnit: Wb (weber) S SBSm dScosBSdBm dScosBSdBm SBn dS SBBB cosBSSBm dS nn2022-2-1557ItrB，SSB0dSBuGauss Law of Magnetic Field2022-2-1558DISCCUSIONuIt shows that magnetic field is a non-divergent field .uThis statement is based on the experimental fact that i

30、solated magnetic poles (or monopoles) have never been detected and perhaps do not even exist.2022-2-1559021 SS 021 )RB(S 21RBS RO1S2SB1. Find the magnetic flux of a semi-sphere surface of radius R .uExample2022-2-1560SBm iS)ji( 23S3 32 ( )BijTXOYZSnB2.Find the magnetic flux of YOZ plane.2022-2-15611

31、3-4 Ampere Circulation Theorem 0 l dEIrlB l dB?2022-2-1562uAmpere Circulation Theorem in the Vacuum The line integral of around any closed path equals . 0B dliiIis the total steady current passing through any surface bounded by the closed path. Amperes law is valid only for steady currents and is us

32、eful only for calculating of current configuration having a high degree of symmetry.B2022-2-1563I1I2I3L2L13201dIIlBL 2102dIIlBL 2022-2-1564uEXPLANATIONILrOMSituation 1 : the integral along a closed curve L (it is just the magnetic field line )in the plane isAn infinite straight current-carrying wire

33、 intersects perpendicularly with plane M at point O, the direction of current I is up. satisfy the right hand principle .IB and 002LLIB dldlIr2022-2-1565002LLIB dldlIr（ ）ILrOdont satisfy the right hand principle .IB and Situation 2:uIn above equation, if the direction of current I with the direction

34、 of closed curve L satisfies the right hand principle, then 0I, otherwise . 0I2022-2-1566 LILoOThe integral is along an arbitrary closed curve L in the plane .Situation 3 2022-2-1567 220I dlBl dBcos dlrI cos20 rdrI20Il dB0 . dBl dr I2022-2-1568.0 l dBSituation 4 The current-carrying wire is not inte

35、rlinked with the closed curve .0 l dEElectrostatic Field iiIl dB0 0 SdB isqSdE01 non-conservative fieldconservative fielddivergence field non-divergent field Steady Magnetic Field2022-2-1570uApplication: IIlBiiL00dCalculate magnetic induction of configurations with high degrees of symmetry.BcosdlBl

36、dBI0B2022-2-1571How to determine a path L of integration (an amperian loop) The value of the magnetic field can be argued by symmetry to be constant over the path. The dot product can be expressed as a simple algebraic product because and are parallel.BdlBdl The dot product is zero because and are p

37、erpendicular.Bdl The magnetic field can be argued to be zero at all points on the path.2022-2-1572uExample: axis symmetryA long, straight wire of radius R carrying a sready current I uniformly distributed across the wire. Calculate the magnetic field created by the wire.IR2022-2-1573ILRrChoose a cir

38、cle of r concentric with the wire.Amperes law applied to the circular path gives.rBBdll dB 2 IldB0 rIB 20 Rr IrB02 2022-2-1574220rRI rBBdlldB 2 IldB 0 202 RIrB Rr IR0 I rB2022-2-1575IBBRI 20BROr020 (for)2(for)2IrrRRBIrRr2022-2-1576There are two coaxial cylindrical surfaces as shown in figure. The cu

39、rrent in the inner and the outer metal cylinder is both I, but the direction is opposite. Calculate the magnetic induction at points in the following uEXAMPLE1RI2RI2(1),rRB1(3),rRB12(2),RrRB2022-2-15770,)1(2 BRr0,)3(1 BRrrIBRrR 2,)2(0212022-2-1578The current density on an infinite uniform current-ca

40、rrying plate in vacuum is j, Find the magnetic induction. uExample: plane symmetry I2022-2-1579Solution: The infinite current-carrying plate may be considered as the set of many infinite straight thin current-carrying wires. The direction of magnetic induction at each point out of the plate is paral

41、lel to the plate and is perpendicular to current, is equal in magnitude. Draw a rectangle circulation .dabc.cross-sectional view2022-2-1580 baBdll dB0cos cbBdl2cos adBdl2cos dccosBdl0cdBabB abB 2dabc.jabl dB020jB2022-2-1581uExample: Find magnetic field of an infinite straight solenoid with steady cu

42、rrent I.An ideal solenoid (an infinite straight solenoid )The solenoids turns are closely spaced and its length is large compared with its radius. For an ideal solenoid, the field outside the solenoid is zero, and the field inside is uniform.2022-2-1582 . . . . . . I BCross-sectional view of an idea

43、l solenoid.2022-2-1583Consider a rectangular path as shown in figure.abB baBdll dB0cos cbBdl2cos adBdl2cos dcBdl cosApply Amperes law to this path by calculating the integral of over each of the four sides of the rectangle.B dlB. I dabc2022-2-1584nabIldB0 0 inside0 outsidenIBn: the number of turns p

44、er unit length N： the total number of turnsNnl2022-2-15850 inside0 outsidenIBThe field near each end is smaller than the value given by above equation. At the every end of a long solenoid, the magnitude of the field is about one-half of the field at the centre.nI: the current of per unit length.BxLL

45、O2022-2-1586uEXAMPLEIA torus consisting of many turns of wire wrapped around a doughnut-shaped structure. uFor a torus having N closely spaced turns of wire and air in the torus, calculate the magnetic field in the region occupied by the torus, a distance r from the centre.2022-2-1587ReasoningCross-

46、sectional view of the torus is shown in figure.rR1R2.+.I.Select a circular path of radius r.The magnetic field is constant in magnitude on this circle and tangent to it.2022-2-1588rR1R2.+.I.Note that the closed path surrounds a circular area through which N loops of wire pass, each of which carries

47、a current I.2022-2-1589SolutionAmperes law applied to the circular path givesrBBdll dB 2 0NI0122120,NIRrRBrrRRrnonuniform within the torus.BrO2R1R.1221RRRR、nIB0 12 RNn rR1R2.+.Discussion 2022-2-1591Discussion Ih2R1RAI7 . 1 1000N 6 .112 RRcmh0.5 Find the magnetic flux of the cross section.m2022-2-159

48、22100212ln2RRNIB dShdrrNIhRRIh1R2R2022-2-1593 13-5 Ampere Force and Magnetic TorqueA magnetic force is exerted on a single charged particle when it is moving through an external magnetic field.A current represents a collection of many charged particles in motion.The resultant magnetic force on the w

49、ire is due to the sum of the individual magnetic forces on the charged particles.uAmpere Force2022-2-1594BLorentz force on a moving particle:fqv BI Current element in the wire includes motion point charges of electric quantity q and velocity . vdNCurrent ElementBvl dBvdNqFd2022-2-1595nSqvI BvdNqFdVn

50、dNdlnSdBvlqnSFddCurrent ElementBvl d is called Ampere force on current element by the magnetic field. dF2022-2-1596 sinIdlBdF )B, lIdarcsin( Ampere force on a current element:Direction:Magnitude:Left hand rule.dI l B 2022-2-1597Ampere force on the current-carrying wire of length L and current I is F

51、The integral is over the wire.When this integration is carried out, the magnetic field and may vary from point to point.BdlI2022-2-159811dlI22dlIr22201114rdlIdlIdF02 dF21dFdF ? Does Ampere force between two current elements follow the Newton third law? 11dlIAmpere force on : 22I dlAmpere force on :

52、2022-2-1599IBFdlIdQuick QuizFind magnetic force on a current-carrying wire due to uniform magnetic field. sinBIdldF Direction: Magnitude: LBILBIdlF sinsin2022-2-15100As shown in figure, a curve wire of current I is set in cross section of the uniform magnetic field, the distance of two ends of the c

53、urve wire is L . The magnitude of magnetic induction is B and its direction is perpendicular down to the page plane, find the Ampere force on the wire. EXAMPLEIlB2022-2-15101BLIjIBLiIBdyjIBdxiIBdyjIBdxBjdyi xIBlIFL000)()()d(dIlBXYConclusion:2022-2-15102 BRabcIBIRf2 B IO0f 2022-2-15103 BBILLLBORRRRF=

54、 F= F= EXERCISES2022-2-15104EXAMPLEThe current in the long, straight wire CD is .The wire AB carries and its length is L. The wire AB lies in the plane perpendicular to CD. Their distance is d. 1I2IFind the net force exerted on the AB by the magnetic field created by the wire CD.LdBA1I2ICD2022-2-151

55、05Lxdba1I2Ifdl dI2Solution dlBIdf2 LdffdxxII 2210 dLdII ln2210 LdddxxII 22102022-2-15106Find the Ampere force of per unit length on two parallel infinite straight current-carrying wires.EXERCISESI1I2r2022-2-15107rIB 22012121112ddBlIF121012d2dlrIIF Direction: rIB 21021212221ddBlIF212021d2dlrIIF Direc

56、tion I1I2r11lId22lIdB21B12 12Fd21Fd2022-2-15108When the direction of I1 and I2 is identical, two wires attract each other; when the direction is opposite, they repel each other. rIIf 22102111212dldFfrII 2210I1I2rB21B12 12Fd21Fd2022-2-15109uTorque on a Current Loop in a Uniform Magnetic Field (Moment

57、 of Magnetic Force )Proving :the torque on a plane rigid current-carrying coil of arbitrary given shape in uniform magnetic field is where ,PNISN is number of turns of the coil, is the area vector of the coil.S2022-2-15110l1l2abcdIBdafbcfcdfBnabfl1Edge View of the Loop Sighting Down2022-2-15111cdfBn

58、abfl1In uniform magnetic field: 0F cdabff The two forces create a torque that trends to rotate the loop counter clockwise. 2022-2-151120M cos1lfMcd cos12lIBl sinSBPMm2,BPmM=ISB/ , for 0 or /2mPB M = 0cdfBnabfl12022-2-15113DISCUSSIONAlthough the torque expression was derived for a rectangular loop, t

59、he result is valid for a loop of any shape.The position where the moment of magnetic force on a coil equals zero is called the equilibrium position of the coil .0 the steady equilibrium position. (- farther away to the position ） the unsteady equilibrium position. (-return to the position ）2022-2-15

60、114Under the action of the torque, the magnetic moment always tries to turn to the direction of external magnetic field.2022-2-15115EXAMPLEZoXYBIAs shown in figure , a coil of half circle is set in the uniform magnetic field parallel to the coil plane in the beginning instant, the radius of coil R=0