Magnetic field

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2. UNIT 9 TIC FIELD Structure 9.1 Introduction Objectives 9.2 MagneticField Source of MagneticField Definitionof MagneticField 9.3 Gauss's Law for Magnetism 9.4 Biot andSavart Law 9.5 Force between Two Parallel Conductors(Definition of Ampere) 9.6 Ampere's Law ApplicationsofAmpere'sLaw Differential Form of Ampere'sLaw 9.7 Torqueon a Current Loop 9.8 Summary 9.9 TerminalQuestions 9.10 Solutions and Answers In Block1 of this course, you were introducedto theconcept of an electriccharge and studiedsome properties of charges at rest. You learnt that a static distribution of charge producesa staticelectric field. Similarly,steady flow of charge (i.e., a steady current) producesa static magnetic field, which is, infact,the topic of this unit. However, there . aresome major differences between the two fieldswhich you will discoverin this unit. Inthesciencelaboratory, during yourschool days, you must have been fascinated with magnets. Recall, when you tried to push two magnets togetherin a way they didn't want togo, you felt a mysterious force! In fact, magneticfields or the effect of such fields have been known since ancient times when the effect of the naturally occurring permanentmagnet (Fe304) was first observed. The northand south seeking properties of such materials played a large role in early navigationand exploration. Except for this application, magnetism was a little known phenomenoi~until the 19thcentury, when Oersted discovered that an electric current in a wire deflects a compass needle. This discoveryshowed that electriccurrent has something to do with the magnetic field becausea compass needle gets deflected and finally points inathenorth-southdirection only when placed in a magnetic field. In this Unit, weshall considerin detail the productionof the magneticfields due to steady currents,and the forces they exert on circuits carrying steady currentsand on isolated movingcharge. A good way of gaining a better understandingof the nature of fields is to know how they affect thecharged particles on which they act. Hence, in the next unit, you will study the behaviour of charged particles in bothelectricand magneticfields. Objectives Afterstudying this unit you should be able to: r understandwhat is meant by the magneticfield, the right hand rule, Biot-Savart + law, right hand method, Ampere'slaw, r define the magneticfield at a point in tenns of the force ona steady current element and alsoon a moving charged particle, r usethe formulaforthe force on asteady current eleclent-oron charged particle due toa magnetic field to calculate the force on a certainsimple current cirrrying ' circuits, and solve simple phblems, 3. BecQricCurrent and MagneticField Fig.9.1: ?Whenthe magmetis Precly suspended,rr p d c u h r endof it points no& lWsend o tUac magoletis defiid PO tbe northp B c (b) Fig.9.2: a) A mmpm needle poinls Inthe directiond themagneticfield.b) Magneticfie119lhesof a magnet drawnusingthe Pactthat acompass needle should line upalongthe Pieldlines show that the divergence of B vanishes, + use Biot-Savart law to describeand compute the magnetic field generated by a simplecurrent flow, s identify the natureof force(attractionor repulsion)on a given lengthof a long, straightcunent-carryingwire that islaid parallelto a similar currentcarryingwire, e use Atnpere's law to calculate the magneticfield Eromsteady current distributions having simplegeometries, e relate Ampere's law to its differential formvia Stokes theorem, compute the torqueexerted by asteady magneticfield up011closed current loops, + appreciatehow the forceson current-carryingconductors,placed in a magnetic field, are used to understand the working of galvanometersand motors. 9.2 MAGNETIC FIELD Wheneverwe speak of magneticfield, wespeak in termsof bar magnetssince this is theway the fields were first studied. You are already awareof the basicfeatures of the magneticfield from yourschool days. For example,you know that the polesof a bar magnetexperience forceswhen placed in a magnetic field. If a bar magnet issuspended by a delicate fibreas shownin Fig. 9.1, a particularend of the magnet will always point towards north. This end of the magnet is called the north pole of the magnet. The other end is thesouthpole. Do you recallthat this arrangement is a simplecompass? The north poles of two magnets repel each other. Thesouth poleof a magnet is always attracted by thenorthpole of another magnet. If one tries to break off the north or south pole from a simple bar magnet, then thisexercise proves to be futile. The broken magnet becomes two new bar magiletseach havinga north anda south pole.This shows that an isolated magneticpole does not exist. a In order to plot the magneticfield due to a bar magnet,we need only a compass needle. The directionin which thecompass needle points is taken to be directionof the magnetic field.In class XII, you must have used this fact to plot the magneticfield in the vicinity of the bar magnet as shownin Fig. 9.2a.The magneticfield lines are drawn in such a way that a compass needle placed on the line will align itself tangentially to the line. Fig. 9.2bshows the typical magnetic fieldfor the bar magnet. Noticethat the field lines emergefrom the north pole and enterthe south pole. Thesearesomequalitative features with which we areall familiar. 92.1 Source of Magnetic Field As you know, thespace neara ~ubbedglass rod (rubbed either by rubberor rabbit's fur) is characterisedby anelectricfield which is denoted by E. Similarly, a magnetic field around a magnet may be representedby thesymbol B.Inelectrostatics,the relation between the electricfield E and theelectriccharge is representedas follows: electriccharge s E electriccharge (9.1) Tbat is, tlie electricchargesset up anelectricfield and the field, in turn, exerts a force (electricin nature) on anotherelectric charge that may be placed in that field. Now, by analogy, can you set up a similar relation for magnetism. Yes, the relation will be as follows: magneticcharge 2 ]B ;;z magneticcharge (9.2) You knowthat the two poles, i.e., northand southalways, occur together. Asingle isolated pole is not knownto exist. This means that thereare no magneticcharges (also called magneticmonopoles). Howdoes, then, the magnetic field arise?The answer to this question you willfind in the followinglines. Let us considertwowires, runningparallel tooneanother, as shown inFig. 9.3 a. As soonas the circuit is closed, thecurrent in the two wires,flows in thesame direction, and the wires are found to attract. If the directionof one of thecurrentsis reversed, the wires repel each other. Thus the twosectionsof the wires in Fig.9.3 b, whichare part of same circuit, tend to moveapaft.If a sheet of metal is put between the two wires, the force with which wiresattractor repel is not at all affected(Fig.9.3 c). How explai~mthis? Does electrostatic force account for the attractionof parallel ones? No, the 4. force acting is not an electrostatic or Coulomb force. This is because(i) there is no net charge on the conductor (the charge density of conduclionelectrons just compensates for the positive charge on the lattice ions); (ii) the force is reversed in sign by revesir~g the directionof either current; (iii) the force ceasesas soon as thecircuit is broken; (iv) the forceis not affected in the presence of a simple medium; (v) the attraction and repulsionof the electric currents is contrary to the attractionor repulsion of the electric charges. Mngnelc Field (0' Fig. 9.3: n) Parallelwirescarrying currtntsin (be same direction w pulled logether.b) Pnralld wires carrying curnnls in opposik directions nrrpushedapart c) A sheet of metal bclweenLhetwo wires docsnot nlTectthese foms. 'The experimentsof Fig. 9.3show that there is an additionalforce associated with a movingcharge, which is different from the electrostaticforce. This new force that comes into play when charges are moving is called the magneticforce. A charge sets up an electricfield whether the charge is at rest or ismoving. However,a charge sets up a magnetic fieldonly if it is moving. You may ask asimple question. A bar magnet sets up a magnetic field in its vicinity, but whereare the movingelectriccharges in a bar magnet? Actually the spinning and circulatingelectrons in the ironatoms of the magnetic materialare responsible for its magnetism. You will learn more about it in Units 11 &12of this block. Hence. in magnetism, we can think in termsof the'following relation: .PI moving electric charge SB 3 movingelectric charge (9.3) As the movingcharges constitute an electric currentin a wire, Eq.(9.3) can be written Bs electriccurrent *B B electriccurrent (9.4) Eq. (9.3) or (9.4) tells us that (i) a moving charge or a current sets up a magnetic field and also (ii) if we place a movingcharge or a wire carrying a currentin a magnetic field. a force will'act on it. Now, let us define the magneticfield. But before doing this gtry to answer the follotvingSAQ. SAQ 1 L Fig. 9.41 A sLrrIgh1 wire CAIy h g cumnl andplaced inn mnguetic field experiences nforce. You have probablystudied about an electric motor in your school, and you may be knowing the principle on whichit works. Briefly explain how an electric motor illustrates Eq. (9.4). 9.2.2 Definition of Magnetic Field . In Block1,we defined the electric field E at a point in termsof the electricforce FEthat acted on a test charge q at rest at that point as follows: FE = q E (9.5) As suggested by Eq. (9.3), we can define the magnetic field in termsof the magnetic force exertedon a moving electric charge. It can also bedefined in termsof the force on 23 5. EkechicC u m tcmd MagneticField a current. Sincecurrent is a flowof electric charge, the two definitions are related. First, let us state the definitionin terms of fo