DOCUMENT RESUME

ED 042 608 SE 009 196

AUTHOR Phillips, MelbaTITLE Electricity and Magnetism 2, Magnetostatics.INSTITUTION Commission on Coll. Physics, College Park, Md.SPANS AGENCY National Science Foundation, Washington, D.C.PUB DATE 66NOTE 37p.; Monograph written for the Conference on the

New Instructional Materials in Physics (Universityof Washington, Seattle, 1965)

EDRS PRICEDESCRIPTORS

ABSTRACT

EDRS Price MF-$0.25 HC-$1.95*College Science, Electric Circuits, *Electricity,*Instructional Materials, *Magnets, *Physics,Theories

This monograph was written for the conference on theNew Instructional Materials in Physics, held at the University ofWashington in summer, 1965. The approach is phenomenological andmicroscopic, and is intended for college students who are notpreparing to become professional physicists. The monograph has threesections. 3qction I includes a short review of the discovery ofmagnetic phenomena, and a discussion of the concepts of magneticpoles, magnetic field intensity, magnetic moment and magnetic flux.Section II deals with the magnetic interactions of steady currents.Basic calculus is used to reformulate the lajor concepts ofmagnetostatics in section Ili. Some problems for discussion areincluded in sections I and II. (LC)

COO%ID

CNJ

LU

CI

E. I , 14 I r1.1t..1

U.S. Of minim. 101,0114111 WItiAtfOfiKI Of OVATION

N1510(001 miS illy 14110DIX10 WM AS iffiirED NON TNI111i01 01 0161M/IM OttmAIN t. OolS Of vilW 01010001SP1TI1 NOT liffIS41Mt1 KARIM OfICAL OfIKI Of DOOM009141 01 Mk,.

MELBA PHYLLIPS

University of Chicago

.:11NO 1N9NA10) 1111

io NOKS 'au %NAM MUSA )111 1111

1041110 100110044R 111111111 1011Y)001 101)1110 TI1 114.1 HUM S1111011119Y 114N11

emumo soouviorreao CNY )W101

aatrioj mpg' Al

01111Y19 N111 SIN DAMN 01111911110101

S1N1 1X10014111 01 N0ISSIN111.,

.*1'11!a5 1( TS.11Ac!t Cllr;

GENERAL PREFACE

This monograph was written for the Conference on the Now Instructional

Materials in Physics, held at the University of Washington in the sum-

mer of 1965. The general purpose of the conference was to create effec-

tive ways of presenting physics to college students who are not pre-

paring to become profess.onal physicists. Such an audience might include

prospective secohuary school physics teachers, prospective practitioners

of other sciences, and those who wish to learn physics as one component

of a liberal education.

At the Conference some 40 physicists and 12 filmmakers and design-

ers worked for periods ranging from four to nine weeks. The central

tank, certainly the one in which most physicists participated, WAS the

writing of monographs.

Although there was no consensus on a single approach, many writers

felt that their presentations ought to put more than the customary

emphasis on physical insight and synthesis. Moreover, the treatment waa

to be "multi-levol" --- that is, each monograph would consist of sev-

eral sections arranged in increasing older of sophistication. Such

papers, it was hoped, could be readily introduced into existing courses

or provide the basis for new kinds of courses.

Monographs were written in four content areas: Forces and Fields,

Quantum Mechanics, Thermal and Statistical Physics, and the Structure

and Properties of Matter. Topic selections and gener:li outlines were

only loosely coordinated within each area in order to leave authors

free to invent new approaches. In point of fact, however, a number of

monographs do relate to others in complementary ways, a result of their

authors' close, informal interaction.

Because of stringent time limitations, few of the monographs have

been completed, and none has been extensively rewritten. Indeed, most

writers feel that they are barely more than clean first drafts. Yet,

because of the highly experimental nature of the undertaking, it is

essential that these manuscripts bemade available for careful review

..

by other physicists and for trial use with students. Much effort,

therefore, has gone into publishing them in a readable format intended

to facilitate serious consideration.

So many people have contributed to the project that complete

acknowledgement is not possible. The National Science Foundation sup-

ported the Conference. The staff of the Commission on College Physics,

led by E. Leonard Jossem, and that of the University of Washington

physics department, lei by Ronald Geballe and Ernest M. Henley, car-

ried the heavy burden of organization. Walter C. Michels, Lyman G.

Parrat., and George M. Volkoff read and criticized manuscripts at a

critical stage in the writing. Judith Bregman, Edward Gerjuoy, Ernest

M. Henley, and Lawrence Wilets read manuscripts editorially. Martha

Ellis ano Margery Lang did the technical editing; Ann Widditsch

supervised the initial typing and Issembled the final drafts, James

Grunbaum designed the format and, assisted in Seattle by Roselyn Pape,

directed the art preparation. Richard A. Mould has helped in all phases

of readying manuscripts for the printer. Finally, and crucially, Jay F.

Wilson, of the D. Van Nostrand Company, served as Managing Editor. For

the hard work and steadfast support of all these persons and many

others, 1 am deeply grateful.

Edward D. LambeChairman, Panel on theNew Instructional MaterialsCommission on College Physics

MAG;IETOSTATICS

PREFACE

This fragmentary and preliminar/ mate-rial fits into an outline of "mlli-level monographs" covering those as-pects of electromagnetism which in ourview an undergraduate physics majorshould come to know best. The anproachis phenemeaological and macroscopic,designed to take advantage of priorexperience; we begin magnelostaticswith magnets, for example. The mate-rial is planned on two levels to lent,through the four fundamental empiricallaws of electricity and magnetism toelectromagnetic radiation as a climax.The propagation of electromagnetic dis-turbances with velocity c, reached inthe "first course" material withoutuse of the calculus and equivalent tothe homogeneous wave equation, waswritten in an elementary way by OliverHeaviside (Electromagnetic Theory,London, 1912, Vol. 111, p. 3), butonly recently has appeared in the reg-ular pedagogical literature. In our

treatment we have tried to stress thephysical foundations of Maxwell's greatsynthesis, slating in words the argu-ment corresponding to each mathemati-cal step. This results in a consider-ably larger proportion of expositorywriting relative to mathematics thanis customarily found in derivations ofthe wave equation from Maxwell's equa-tions in their usual form. On theother hand, expression of the laws indifferential form seems essential fortracing radiation to its sources in aphysically meaningful way; the presentChapter 3 of Magnetostatics could befollowed almost immediately by Chap-ter 3 of Monograph III, which vouldtrace radiation fields to retardationeffects. We regret having not suffi-cient time to write such a chapter, aswell as thl omission of what shouldhave been Chapter 3 of Magnetostatics,an elementary treatment of magneticmaterials.

OUTLINE OF MONOGRAPHS ON ELECTRICITY AND MAGNETISM

1

i

I. ELECTROSTATICS II. MAGNETOSTATICSIII. CIRCULATION LAMS

AND THEIRCONSEQUENCES

r

1. Electric Forces 1. Magnets and 1. Faraday's Law of' and Fields Magnetism Induction

FIRST2. Electric Energy 2. Interaction of 2. Ampere's Law

COURSE and Potential Steady Currents Modified

MATERIAL 3. Electrical Proper-tits of Matter

"Magnetic Proper-tics of Matter

3. Propagation ofElectromagneticDisturbances

UPPER .

DIVISlov

4. Liect...ostatics

Prior uiated3, Magnotostatics

itotorm9lated

'Maxwell's Equa-Lions and PlaneWaves

CutiRSE'Radiation Fields

MATERIAL

'No textual material rat prepared In the summer of 1965 for these chapters.

We have assumed no knowledge ofspecial relativity, but have empha-sized the necessity for choosing aframe of reference in which to defineelectric and magnetic field quantities,thus laying a foundation for the his-torical development of relativity the-ory. Unlike mechanics, vacuum electro-dynamics needs no modification becauseof special relativity except in inter-pretation, so that an excursion intorelativity theory could be made beforeor after study of the present material.

The experiments leading to thefour fundamental laws are described atsome length, but in use this writtenmaterial should be accompanied by dem-onstrations and laboratory work. Thebasic experiments should come to be apart of genuine experience for stu-dents, but a laboratory monographshould be written as an extension ofthe present outline. Ohm's law and cir-cuitry, for example, do not al: ap-

preciable role in any other projected

booklets. We cannot. overemphasize theimportance of laboratory work, althoughwe were nut ale to undertake detailedconsideration o Its content.

We assume that students will havestudied mechanics, that they know New-ton's laws, the definition of work,the meaning of the symbol, and hai.e

ol elementary wee-ler algebra before our material is In-

troduced. (We do define the vectorcross product as if for the firsttime.) In the material designed forupper-class work we assume basic cal-culus. All vector calculus is developedas needed, but we attempt throughout tostress the physics, not the mathemat-ics, and attempt no mathematical rigor.

The first chapters of Monographs I,II, and III should be studied in thatorder. The few discussion exercises weinclude can only indicate a type ofproblem we consider desirable. Numeri-cal problems which we have made no effort to provide, are also necessary.

M. Phillips

R. T. Mara

CONTENTS

PREFACE

1 MAGNETS AND MAGNETISM 1

2 THE MAGNETIC INTERACTIONS OF STEADY CURRENTS 8

i MAGNETOSTATICS REFORMULATED 20

1 MAGNETS AND MAGNETISM

Magnetic iron oro, known to us as lode-stone or magnetite, is found in manyparts of the world, and its propertyof attracting iron was noted by morethan one civilization early in ituIron Age. The property remained merelya curiosity, even in the intellectualclimate of the Golden Age of Greece,and the tendency of a magnet to orientitself along the earth's meridian es-caped notice. These directive effectswere probably first discovered inChina, but there is no conclusive evi-dence that they were put to practicaluse. The origin of the mariner's com-paus is shrouded in mystery which maynever be dispelled, but by the en,1 ofthe 12th century the compass was wellknown in the Western world as a help-ful device for sailors when the starswere obscured.

Myths, legends, and superstitionsabout magnets multiplied from ancienttimes through the Middle Ages, andeven later. Magnets were employed inmedicine, especially for the healingof wounds, and once the directionalproperties were recognized, in the oc-cult science° such as astrology. Yet

1

magnetism began to be a genuine sci-ence during the Middle Ages; The firstaccount of the magnet that we wouldcall scientific was surprisingly early,a letter dated August 12, )26P. Theepistle of Peter Perigrinus (Peter thePilgrim), born Pierre de Maricourt inPicardy, sets forth a number of funda-mental properties of magnetism.

It was Perigrinus who discoveredpoles and distinguished precisely twokinds. His method is of interest: Se-lect a good piece of magnetite, shapeit into a sphere and polish it. Nowplace on it a needle or sliver ofiron, and mark on the surface of thesphere the direction taken by theneedle. Repeat the procedure at manydifferent positions on the sphere. Atthe end it is found that the lines"will run together in two points, justas all the meridian circles of theworld run together in two oppositepoles of the world" (Fig. 1.1). Onlyone of these poles points north if themagnet is free to turn. Thus two oppo-site magnetic pole.' were introduced,and Perigrinus noted that unlike polesattract. He went further to shot, thatif a magnet is cut (see Fig. 1.2), twopoles persist in every separpted part,and that if two fragments are put to-gether as before the new poles vanish.He did not notice repulsion. Perigrinusnamed the poles north and south, withthe north pole that which points to thenorth. At that time magnetism was at-tributed to the celestial sphere, not

Fig. 1.2

2 MACNETOSTATICS

NORTHS POLE

N POLE

SOUTH

Fag. 1.3

to the earth itself, and Perigrinusdid not think of the earth as a magnet.

This last step was taken by Wil-liam Gilbert, physician to Queen Eliza-beth, who repeated and extended theexperiments of Perigrinus and others.Once the earth is taken as a magnet,the action of a compass is simply anexample of the behavior of all magnets:The north magnetic pole of the earth isa "south" pole, which attrazts the"north" poles of all compass needles(see Fig. 1.3). Gilbert's great accoi-

qm

Fig. 1.4

plishment was to extract existing factsand laws of magnetism from a wealth ofspeculation and superstition, and todiscover new properties a 1 relations.Pis theories, including the supposedrelation of magnetism to gravitation,need not concern us here, although weshould note that they influenced Kep-ler, and that Newton found them sug-gestive in the development of his ownideas. Gilbert's book. be Magneto, pub-lished in 1600, Is still a classicpresentation of many qualitative as-pects of magnetism.

Magnets exert forces on each otherand on iron withoot being in contact.It was known to Ginert and those whofollowed him that the effect of a mag-net decreases as the distance fromthe magnet increases, but the quanti-tative relationship was first discov-ered by the Reverend John Michell in1750. The same relation was found byCoulomb in 1785. The torsion balance,which facilitated these experiments,was invented independently by Michelland by Coulomb, Lnd Michell's balancewas later used by Henry Cavendish forhis famous measurement of the constantin Newton's law of gravitation.

In long magnetized needles, oreiff wires of hardened iron, the lag-netic effect is well concentrated utthe ends, or poles (Fig. 1.4). Michellestablished experimentally that thetwo poles are opposite and of equalstrength for any or magLet, and thatrepulsion and attraction between thepoles of two magnets are of equal mag-nitude it distances of separation arekept the same. lie also found that theforce exerted by the pole of a longmagnetized wire is the same in all di-rections. He then determined the forcebetween poles for various distances,and found its strength was inverselyproportional to the square of the dis-tance between poles.

In order to write the law ofMichell and Coulomb in mathematicalform, we must assume some quantitativemeasure of pole Ptrength, and it isconvenient to call north-seeking polespositive (4) and south - seeking; poles

MAGNETS AND MAGNETISM

negative (-), by analogy to the desig-nation of electric charge as positiveand negative. 11 we designate polestrength by qm., the force between twopoles of strengths qm and qm' has amagnitude given by

F - kgmqW/r2,

with k an arbitrary constant determinedby the units employed. In this formulaa negative force signifies attraction,but force is actually a vector quan-tity, and the force exerted on qm byqm' at distance r may be written

= kqmqm'f/r2,

where r is a unit vector in the direc-tion from qm' toward qm. The force isdirected as in Fig. 1.5 if qm. and qm'are of like sign.

This law has exactly the sameform as Coulomb's law for the interac-tion of electric charges, and a formalanalogy between the interaction of mag-nets and electrostatic interactionscan be carried further. The magneticfield intensity may be defined at anypoint in the vicinity of magnet asthe force per unit positive pole atthat point. The conventional designa-tion for magnetic field intensity is H,so that

F = qm11,

where if is the field intensity at theposition of qm. The field intensityproduced by a single pole qm' at dis-tance r from the pole is

= (kqm1/r2)T,

where the unit vector F is directedfrom qm' to the point where It is to bedetermined. If qm' is negative (an Spole) the field intensity H is thendirected toward qm'. The field inten-sity produced by two poles is the vec-tor sum of the contributions of the twopoles taken separately - the principleof superposition applies. The fact thatfor real magnets the poles are not

-9'm

Fig. 1.5

m

points, that pole strength is actuallydistributed over the surface or volumeof a magnet, causes no difficulty: Alleffects are additive and can be summedover elementary surfaces or volumes atvarious distances from the point atwhich the field is to be computed, justas in electrostatics. In other words,the principle of superposition appliesto magnetic forces, just as it does toelectric forces.

We shall find that we have littleoccasion to work with poles as such inthe further development of magnetism,although the magnetic field intensityremains an important concept, We shouldnote that the mks unit of pole strengthis called the richer, so that II it:: in

newtons per weber. (To anticipate Cnap-ter 2: The weber is defined in termsof the ampere.) The constant k requiredto give the force in newtons if polestrength is measured in webers is (107/1672) newton-meter2/weber2, or roughly6 x 104. Clearly the mks system is notdesigned to be convenient in workingwith poles, but the weber remains use-ful in other connections. It was namedfor Wilhelm Eduard Weber (1804-1891),who collaborated witi, the great mathe-matician Gauss in putting the wholequestion of electromagnetic units on arational basis.

In principle the magnetic fieldintensity in the vicinity of magnetscould be computed at every point froma knowledge of the strength and posi-tion of all poles, but the numericalresults would be hard to visualize. Onthe other hand, the direction and somenotion of the strength of H can be dem-onstrated very easily with iron filings

4 MAGNETOSTATICS

'\1.11;i1...:''...?,%* , 0.

, ;'';'..141

...

11.41.111i

%. I-'..:, .'t%'.. %,

Fig. 1.6U y pe rint 1.1 on Phyylt Seim... Stmly CI onn Ice

(Fig. 1.0) or a large number of verysmall compasses. By definition a com-pass sets itself in the direction ofthe field intensity. A field line is aline so drawn that its tangent is inthe direction of the field at eachpoint, and to a good approximation afield line can be traced along itslength by a small compass. The numberof lines drawn does not matter, but wesee from Fig. 1.7 that the lines tendto converge in regions of high fieldintensity, and to become more widelyspaced where the field intensity isweak. We are restricted to the regionoutside the magnet, and even then showonly a two-dimensional cross sectionof space, but it is clear that thelines traced out in this way are smoothand continuous,

Iron filings and small compassneedles are not entirely equivalent,although both are oriented along mag-netic field lines. Soft iron, of whichthe filings are made, shows little orno residual magnetism; a small sliver

Fig. 1.7ay pornnion Allyn and Dat ono Inc.

of iron has no poles of its own, andorients itself equally readily ifturned through 1800 in the field of amagnet. The magnetism a piece of ironexhibits owing to the presence of mag-netite or other permanent magnets issaid to be induced.

The tiny compass needles withwhich lines of magnetic field intensitycan be traced out are each completewith two opposite poles of equal magni-tude. That the tiny magnet tends toalign itself along the lines of if isto be expected, since the force on thepositive pole is equal an opposite tothe force on the nearly coincident neg-ative pole. But poles are well local-ized only on long needles such as thoseused by Michell and Coulomb. (Coulomb'sneedles were 25 inches long.) For shortneedles it is practically impossible todetermine a point position equivalentto the actual distribution of polestrength. A more convenient propertyby which the strength of a magnet canbe measured is its magnetic moment, or,more precisely, its magnetic dipolemoment. The dipole moment can be deter-mined even for magnets which a.:e en-tirely inaccessible for direct exami-nation.

The dipole moment of a long mag-net with poles at the ends is definedas a vector directed from the negativetoward the positive (north-seeking)pole, whose magnitude is the polestrength qm times the distance betweenthe poles. Let us call the magneticmoment of such a magnet m, If the mag-net is placed in a uniform field, asindicated in Fig. 1.8, the forces onit are equal and opposite, but it willexperience a torque of magnitude mHsin U, which tends to bring it intoalignment with the lines of H. (Here eis the angle between the dipole momentm and the direction of H.) For macro-scopic magnets this torque can bemeasured even if the pole strength isso diffuse that its exact position haslittle or no meaning. For microscopiclinear magnets the dipo2e moment is theonly accessible measure of magneticstrength.

MAGNETS AND MAGNETISM 5

Even the torque may be difficultto measure for a magnet so small as tobe mechanically inaccessible with a de-vice such as a torsion balance. The di-pole moment can still be determined byfinding the energy involved in liningup such magnets in the direction of thefield lines, or by finding the energyrequired to reverse taeir positions.We note that if the magnet is initiallyin the direction of H, the work re-quired to turn it through 90°, so thatit becomes perpendicular to the field,is mil. (Proof of this statement is leftto the problems.) It is conventional tosay that the magnetic dipole has zeroenergy when it is perpendicular to thelines of H. The energy is equal to-mH cos 0 d -m 11 for any other orien-tation. This is potential energy, sinceit is determined by the position of themagnet relative to the field, and thenegative sign makes the most stable po-sition (alignment with the field lines)that of the lowest potential energy.

The description of magnetic inter-actions in terms of an inverse squarelaw of force between poles seems to govery smoothly, but it has a distinctweakness: It does not include anystatement of the experimental fact thatpoles cannot be isolated from eachother. This could, of course, be adoedin words. Coulomb went further thanMichell: After establishing the in-verse square law and the direct de-pendence on magnetic pole strength, hepostulated that the "molecules of mag-netic fluid" are themselves elementarymagnets, complete with two poles ofequal strength. Chains of such "mole-cules" would then cancel each otherexcept at the ends, which appear asmagnetic poles. This would explain thefact that two poles appear when a mag-net is cut, equal and opposite instrength and each equal and oppositeto the original pole still attached toit. This is very much like the mcdernview, except that we now view elemen-tary magnetism as a property of matteritself, not a separate fluid.

Wo have no proof that isolatedmagnetic poles do not exist somewhere,

Fig. 1.8

A

and there are no known reasons why theyshould nut exist. Nevertheless themathematical description of magnetismas we find it should reflect the empir-ical fact that no isolated magneticpole has ever been detected. This de-scription would have to be amended ifisolated poles are ever discovered,but in the meantime it would includean essential fact of magnetism aspresently experienced.

The impossibility of separatingpoles can be stated by saying that inany volume cut off physically from re-maining space by a bounding surface,there is no net pole strength; in cut-ting through a magnet, you create apole equal and opposite to one you weretrying to surround. Now in electrostat-ics we are able to write a simple rela-tion between the electric field inten-sity and its sources within a particu-lar volume in terms of the flux of thefield intensity. We can, similarly,define the flux of the magnetic fieldintensity 17 through an element of sur-face AS as II AS, and find that mathe-matically, as a result of the inversesquare law, the total outward flux ofH is

E fire d§ o 417k1q,

s closed

where qm is the total pole strengthinside the volume, as in Fig. 1.9. (Werecall that in the electrical case,owing to the inverse square of the dis-tance in the quantity to be summed, it

6 MAGNETOSTATICS

Fig. 1.9

does not matter where the charge islocated within the volume, and theprinciple of superposition then insuresthat the total flux is independent ofthe distribution of charge within thevolume.) But the surface involved inthis theorem of Gauss's it mathemati-cal; there is no more physics in thenew statement than in the original lawof Michell and Coulomb. The power ofthe Gauss form of Coulomb's law inelectrostatics is that net electricalcharges can be isolated with emptyspace completely surrounding them, thatfor any electric charge q there is nouniquely associated -q. As a result,Gaussian surfaces can be chosen withthe same symmetry as that of thecharge, so as to yield an expressionfor the electric feld intensity. Theanalogous theorem for the flux of H isnot so useful, since isolated spheresand lines of magnetic pole strengthcannot be constructed.

And yet the concept of magneticflux suggests a way of stating the in-separability of poles and the inversesquare law at the same time. Let us as-

Fig. 1,10

sums a quantity B welch is indistin-guishable from the magnetic field in-tensity in empty space outside themagnet hu;, defined inside the magnetby the condition that its total outwardflux from every closed surface is zerowhether there are magnets or not. Thuswe may write

E 5- . YS 0,S closed

for all possible surfaces. This condi-tion is satisfied by IL itself for sur-faces which do not cut through magnets,but not in general _for surfaces whichdo. The demand on B is equivalent todemanding a physical cut through themagnet, which would create a polestrength to cancel that already insidethe volume, rather than the mere math-ematical surface of Gauss's theorem forthe field intensity. The behavior oflines of B through a magnet is shownin Fig. 1.10.

We shall see in the followingchapters that B can be given an opera-tional definition in connection withanother aspect of magnetism. The onlyvirtue in introducing it here is tostate in mathematical form the insep-arability of poles as we find them innature. In the mks system of units, Band H are expressed in different units:

110 HH=

in empty space, where M0 is 4v x 10-7.Again in empty space outside the mag-net, a single pole of strength qm web-ers gives rise to B,

B = (qm/4rr2)T,

so that B is measured in webers persquare meter. We should note the occur-rence of the geometrical factor 4r,just as in electrostatics. One placeor another this factor is sure to en-ter the description.

If this were all there were tomagnetism, there would be no connectionwith electricity, and the two would beconsidered as separate subjects. It

MAGNETS AND MAGNETISM 7

was, in fact, one of Gilbert's achieve-ments that he distinguished clearlybetween magnetism, as produced by mag-netite and magnetized iron, and staticelectricity PS produced by rubbingglass with silk. But there is current

electricity which consists of a netflow of charge, whether as free chargeor in electrical conductors, and mag-netism is connected with the motion ofcharge, as we shall see in the nextchapter.

PROBLEMS

1.1 Suppose you were confronted withtwo iron bars that look identicalin every respect, but one has been"permanently" and strongly magnet-ized with its poles well localizedat the two ends, and the other not.Without any additional equipmentwhatsoever, how could you deter-mine which is a permanent magnetand which has no residual magnetismof its own? Describe in detail theoperations you could perform andthe conclusions you would draw ateach step.

1,2 When one end of a magnet is broughtclose Lo one end of an initiallynnmagneLlved nail, the nail itselfbecomes a magnet and will attractother nails. Tho effect is evenmore pronounced if the end of the

magnet is actually brought intocontact with the end of the nail.(If you have not observed suchphenomena you can easily do so withtoy magnets obtained at a varietystore.) What electrical phenomenonis this analogous to? Is the analo-gy complete, i.e., in what way dothe phenomena differ? (flint: Whatwould happen in the electricalcase after the two objects came incontact with each other?)

1.3 Make a detailed list of ways inwhich electrostatic and magneto-static phenomena clearly differfrom each ()Lacy. (Note such aspectsas the ellAire of materlals that ex-hibit relevant properties, phenom-ena of conduction, phenomena ofpolarity, etc.)

2 THE MAGNETIC INTERACTIONS OF STEADYCURRENTS

The electrical effects first studiedwere those of static charges producedby rubbing amber with cloth, but noquantitative results were obtained un-til the properties of conductors weredistinguished from those of insulators.In electrostatics a conductor is anobject whose surface has an equilib-rium distribution of charge: There isno difference of potential between anytwo points on the surface of a conduc-tor. This follows from the definitionof a conductor as something in whichcharge is free to move.

But if a difference of potentialcan be maintained between two pointsof a conductor, there will be a flowof charge from one point to the other.We no longer have a static situation,but we may have a steady flow. Let usconsider a linear conductor such as astraight wire, and assume that someexternal device can maintain a constantdifference of potential between theends (see Fig. 2.1). This device willneed to supply charge, but the chargedues not build up anywhere - it leavesthe wire at the same rate that it en-ters, and the conductor need have nonet charge. The result is a steady flowof charge in the wire. The amount ofcharge per unit time which passes anyposition P is the current:

I -Act coulombs

At second '

and one coulomb/second is called anampere. The direction of the currentis that of positive charge flow. A flowof negative charge is equivalent to acurrent whose direction is opposite tothe motion of the charge.

Many practical uses of conductorsinvolving the transfer of charge aretreated in the laboratory monograph ofthis series. Here we shall not be con-cerned primarily with any particularrelation between the magnitude of cur-

8

BATTERY

Fig. 2.1

rent in a conductor and the appliedpotential difference, but demonstrationof the effect under immediate consider-ation does involve setting up electri-cal circuits, for which it usually suf-fices to know Ohm's law: For many con-ductors the current I of Fig. 2.1 isdirectly proportional to the appliedpotential difference denoted by V inthe figure. Mathematically,

V RI,

where R, called the resistance, de-pends on the material of which the con-ductor is made - whether copper oraluminum, for example. For a uniformwire the resistance is directly pro-portional to the length and inverselyproportional to the cross-sectionalarea of the wire. Ohm's law is impor-tant as describing the behavior of manyconductors, but it is not a fundamentallaw of electricity and magnetism. Themagnetic effects of electric currentsdo not depend on the applicability ofOhm's law.

The first currents observed werethose obtained by discharging conduc-tors which had been charged electro-statically, but such currents are usu-ally small and sporadic. Production offairly large steady currents became

TNE MAGNETIC INTERACTION OF STEADY CURRENTS

possible only after Volta's inventionof the chemical battery at the begin-ning of the nineteenth century. Devel-opment of the battery as a practicaldevice facilitated many kinds of elec-trical experiments including tho ef-fects of current electricity.

Even earlier (1752), BenjnminFranklin had demonstrated that light-ning is an electrical discharge. Obser-vation of occasional erratic behaviorof compass needles daring a thunder-storm suggested to Hans Christian Oer-sted of Denmark some connection be-tween electricity and magnetism, andled to his remarkable discovery in1819 that current electricity is ac-companied by magnetic effects. Oer-sted's experiment consisted of settinga long straight portion of an electriccircuit above and parallel to a com-pass needle, and finding that the nee-dle is deflected from its originalnorth-south orientation when the cir-cuit is closed (see Fig. 2.2). This isnot a temporary effect: The deflectionis maintained so long as the current ismaintained. With a strong current theneedle is very nearly at right anglesto the line of the current, and thedeflection of the needle is reversedwhen the current is reversed. If thecompass is held above the wire insteadof below, and the direction of the cur-rent is unchanged, the deflection ofthe needle is again reversed.

We have seen that a compass alignsitself along the direction of the mag-netic field intensity, and that fieldlines can be traced out with a smallcompass. The "sense" or direction ofthe arrow on a field line is that ofthe dipole moment of the compass: fromnegative to positive, or from S pole toN pole. The field lines so traced for along straight wire carrying a currentare circles, directed in accord with aright-hand rule: If the wire is graspedwith the right hand, the thumb point-ing in the direction of positive chargeflow, the direction of the curved fin-gers is that of the field lines. (Checkthis rule with Fig. 2.3.)

A quantitative study of the mag-

9

Fig. 2.2

netic field intensity accompanying along straight linear current was under-taken by Biot and Savart in Paris, im-mediately after hearing of Oersted'sdiscovery. They found that the magni-tude of the magnetic field intensity Hat any point is directly proportionalto the strength of the current, and in-versely proportional to the shortestdistance from the point to the wire.Quantitatively,

I/2vr (2.1)

for a long straight wire carrying cur-rent of magnitude 1, in mks units. Thecurrent is measured in amperes and rin meters. There is no explicit arbi-trary constant (for a change!), because

Fig. 2.3By pwinistion PSSC letoD.C. Heath and Company

10 MAGNETOSTATICS

Fig. 2.4

the ampere and the weber, which we havealready encountered in Chapter 1 as aunit of magnetic pole strength, are sodefined that the only factor in thisformula is 2r, the ratio of the cir-cumference of a circle to its radius._Clearly the magnetic field intensity Hmay be expressed in amperes per meterinstead of newtons/weber, and it isusually so expressed in mks units. Infact, H can be defined by this equation,with the magnitude of H at a distanceof one meter from a long straight wirecarrying a current of one ampere beingequal to (1/2n) amperes/meter. Theequation as it stands, however, doesnot give the direction of the magneticfield intensity, and the right-handrule must be kept in mind as well asthe relation of magnitudes.

The form of the Biot-Savart lawgiven above, together with the rirht-hand rule, suggests another way of put-ting the relation between I and the

Fig. 2.5

Fig. 2.6

magnitude of H. Let us define what iscalled the circulation of it Considera closed path, s, in the field; forevery part of the path multiply theelement of length Ls by the componentof H parallel to Ls', and sum the prod-uct over the entire path. The resultis called the circulation of g aboutthe path chosen. For a circular pathin a plane perpendicular to the currentwhose line passes through the center ofthe circle (Fig. 2,4), .his process iseasily carried out. The field intensityis everywhere in the same direction asthe path, the entire path length iss 2nr, and the circulation is simply

E ff. a 2ur 2rr 1/2,7r I,

h CIONVd(2.2)

just the current in the wire. But letus evaluate the circulation of H overa somewhat more complicated path con-sistiag of concentric circular arcsconnected with radial lines as in Fig.2.5, The length of each arc is propor-tional to both its radius and the angleit subtends, but the field intensity isinversely proportional to the radius.The radial portions of the path con-tribute nothing, since they are per-pendicular to ff, so that the circula-tion about this pat'a is I, just as be-fore. Even a slant element of pathcontributes to the circulation only0/2u, where 8 is the angle it subtendsas shown in Fig. 2.6. Thus the t I,

THE MAGNETIC INTERACTION OF STEADY CURRENTS 11

lation of fl is equal to I if the areabounded by the closed path has I pass-ing through it, and equals zero if thecurrent circuit does not link throughthe loop. Otherwise, the shape of theclosed path does not matter, nor doesthe exact position of the current; theonly consideration is whether the cur-rent "threads" the loop, that is,whether therc, is flow of charge throughthe area bounded by the oop (Fig.2,7).

That this result is perfectly gen-eral follows from its independence ofthe shape of the loop for a lino cur-rent, and from the principle of super-position for a. We may write

circulation of IT E II 'As ..- I,

s closed(2.3)

where I is the total (net) currentti.reading the path of the circulation.The circulation of H about two equaland opposite currents is zero, eventhough the currents are displaced from,each other and the magnetic field in-tensity itself may have quite appreci-able values at various points along the

The circul ;,in law for H is ex-treme4 nsefui t, finding the fieldintensity associat d g.r,h all currentconfigurations which hive cylindricalsymmetry. An important example is thesolenoid, a coil of insulated wirewound in a close helix or spiral on ahollow cylinder, or having the shape ofof a cylinder. Let us consider a verylong Coil of this kind, of n turns perunit length, eacy carrying current I.We may investigate the field intensitywell away from both ends by taking acirculation path partly inside andpartly outside the coil, as in Fig,2.8, barely including the current car-rying wires. For each turn of wire themagnetic field in the plane of the turnis at right angles to the plane, and ifthe contributions of the loops aboveand below are considered in pairs itcan be seen that the whole field isparallel to the axis of the solenoid,so long as we stay far from the ends.

rig. 2.7

Thus the short sides of the circula-tion rectangle contribute nothing, and

IL Inside noutbidoi of I.

where f is the long dimension of thecirculation rectangle. But this sameequation holds if the circulation pathis changed to the dotted line insidethe coil, or for any other position ofthe inside leg of the rectangle. We cantherefore conclude that the field in-

Fig. 2.8

12 gAGNETOSTATICs

Fig. 2.9

7

tensity inside the cylinder is uniform,having the same strength and directionover the entire cross section. Exactlythe same argtmont can be made for theleg of the rectangle outside the cylin-der; the sides which are short in Fig.2.8 may be made as long as we please.The field intensity outside the sole-noid, in a plane that cuts the cylin-der far from the ends, is also uniform.But the field intensity outside must bevery small indeed; full justificationof this statement is left to a problem.Therefore:

Inside lioutside nI ° Hinside (2.4)

to a very good approximation.Except in current configurations

of cylindrical symmetry, the circula-tion law is not so very useful in find-ing the magn.tic field intensity accom-panying a current. To express the fieldintEnsity at a point in terms of thecurrent in a circuit of arbitrary geom-etry we shall need the vector productof two other vector quantities.

The prototype vector is almostliterally the directed line segment inthree-dimensional space by which othervector quantities such as force andvelocity are represented. We are famil-iar with the scalar product of two vec-torstors as it occurs in F s work, orR s - electrical potential differ-ence, giving a scalar quantity whichcan be expressed as a single number.The prototype cross product of twovectors is the area of the parallelo-

gram defined by two directed line seg-ments, represented in a direction per-pendicular to that area. That twolines specify a definite parallelogramin a plane is shown in Fig. 2.9, Bydefinition

d -Xxff- AB sin&

where C is a unit vector at right an-gles to the plane of A and B. The senseor sign of X x B is determined by aright-hand rule: With the plane of yourhand at right angles to the plane of Aand B, let the fingers of your openright hand point in the direction ofthe first named vector (A) with thehand oriented so that the parallelo-gram is in front of your palm - partialclosing of the fingers would bring themparallel to the second vector (B); thedirection of your extended thumb isthat of A X B.

Vector cross multiplication isnot commutative: It is readily seenthat

x A - -(A x ff).

The cross product of A and B vanishesif A and B are parallel, and has itsmaximum magnitude if they are at rightangles to each other. We note that anarea is represented as normal to itsplane, but that the sign of the normalis chosen by an arbitrary rule. Manyphysical quantities share this charac-teristic. Cross products may be repre-sented by directed line segments, and,for most purposes, such as additionand multiplication, they behave likeordinary vectors. Actually a vectorproduct is not exactly the same kindof quantity as at least one of its vec-tor factors - that a directed area isnot quite like a directed line segmentis shown in one of the problems. In theproblems the cross product is also ex-pressed in terms of the components ofA and irt in Cartesian coordinates.

From other experiments of Riotand Savart and of Ampere (to whose fur-ther work we shall turn our attentionshortly) on circular circuits and those

THE MAGNETIC INTERACTION OF STEADY CURRENTS 13

Fig. 2.10

which combine circular arcs and radialcircuit elements, it became evidentthat magnetic effects are always di-rectly proportional to the magnitudeof the current, and that the effect ofeach element of current at a particu-lar point P depends not only on thedistance of the point, but also on theorientation of the current with respectto the line between it and the point atwhich H is to be determined. With ref-erence to Fig. 2.10, the contributionof a length of circuit As to the fieldintensity at a point whose distance isr is given by

AH 'As sin Oblurl, (2.5)

where 0 is the angle between IAW (takenpositive in the direction of the cur-rent) and the line between the currentelement and the point. The direction of

is at right angles to both la and r,and in this instance into the page.All this information is conveyed moresimply by means of the cross product

IAs x T417r2 '

(2.6)

where r is a unit vector along r, di-rected from the current element to thepoint in space. This formula has to beinferred from experiments with completecircuits, for which the field intensityis correctly given by

E IAs X f.

TiTi--*e closed

(2.7)

Let us apply this last formula tofind the field intensity at the centerof a circular loop of radius r, carry-

Fig. 2.11

ing current I. The lead wires from thebattery produce no effect, (Why?) SinceAs is perpendicular to the radius ofthe loop, and the distance r is thesame for all elements of current, thesum over all parts of the circuit canbe evaluated at once:

H -I 2wr I

0:1'2 2r(2.8)

at the center of a circular loot. Thedirection of H is that of As x 2., outof the page for the current indicatedin Fig. 2.11.

Ampere, who learned of Oersted'sdiscovery at the same time as did Blotand Savart, reasoned that there shouldbe forces between two current circuitsif both produce magnetic effects, sincetwo magnets interact with each otner.Within a week ho had shown that twoparallel wires carrying currents in thesame direction (see Fig. 2.12) attracteach other, and repel each other ifthe currents are in opposite direc-tions. The magnitude of the force be-tween the wires is directly propor-tional to both currents. As the resultof a remarkable series of experimentsperformed during the next three years,

F

Fig. 2.12

14 HAGNETOSTAT ICS

Fig. 2.13

Ampere was able to infer that the forcebetween two parallel current elementsIlbs, and 13622 is given by

k' 1112 sin 0 tis, 6s2/r2

La. 1112 tisibs2 sin 0

4n r 2

where 110/47r is the constant of propor-tionality in mks units (see Fig. 2.13).The value of k' is taken arbitrarily as10-7 newton/ampere2, so that p, 4vX 10-7 newton/ampere2. The size of theampere (and thus also the coulomb) isin fact determined by taking the con-stant of exactly this magnitude.

Again the presence of sin 0 sug-gests a cross product. Still anotherangle must be taken into account if thetwo current elements are not parallel,and the formula begins to look evenmare complicated: s,

po 1,46;71 x (12a2 X F)tor(on A.;', )

477 r2

po 124g2 X T )(2.9)14.;1 X (

sir2

where r is a_unit vector directed fromAs2 toward As,. For the total force oncurrent element Jibs, exerted by cir-cuit 2, we must sum over all I2A32:

F(on Ill ) Ill

2]122`.322< F. 1

r 2 jsa closed

But this way of writing tho force sug-gests a simplification, since the termin parentheses is, apart from the con-stant po, the magnetic field intensityfound by Diot and Savart as a force per

ieFig. 2.14

unit pole. Moreover, it is found thata current element experiences a forcewhen placed in the vicinity of an or-dinary magnet, as indeed we might ex-pect from the fact that a current ex-erts forces on a magnetic compass nee-dle. The force on our current elementIlbsi may be written as

x Aoff x B, (2.10)

where

- 1: a I24g22

x F(2.11)

ry rsa closed

if if is produced by current 12. Ingeneral throughout empty space

- (2.12)

whatever the sources of magnetic fieldintensity (see Fig. 2.14).

In Chapter 1 the field if was de-fined so that its net flux through thesurface enclosing any volume of spacevanishes, although the net flux of Hmight be different from zero. The linesof both B and H as produced by currentsare without beginning or end, and thenet flux of both quantities through thesurface enclosing any volume is zero.To see this we need consider only anincrement of field arising from a sin-gle current element Ms, and rememberthat the entire field at a point is thesum of such increments. Let us examine

10 A; x F4613

N°1-1" r

The direction of di is perpendicularto both a and 1, and its magnitude de-pends on r. If we choose any point andmove along the direction of 615, wo

THE MAGNETIC INTERACTION OF STEADY CURRENTS 15

shall tracc out a circle in a plane atright ningles to the direction of As,which would yield no net flux from Lhosurfnce of n volume such as shown inFig. 2.15. Increments of [3 arisingfrom other current elements have thesame property: The circles of di theycontribute may lie in different planes,but the lines of each are continuous,Thus for B as a whole, the summationof the flux over any closed surface,

E ag 0 E HAASS close.' S closed

if II is produced by currents.The equation V lbs x B is often

taken as the definition of the vectorfield B, which is called the magneticinduction field. The units of ff arcnewtons/ampere-meter. It is left to theproblems to show that these units areconsistent with those given in Chap-ter 1. In empty space, where our equa-tions hold, the field quantities B andare really indistinguishable, al-

though their units are arbitrarily dif-ferent in the mks system. We shall in-vestigate further the equivalence ofcurrents and magnets, and how thisequivalence depends on the absence ofseparable magnetic poles, but let usfirst look again at the role of if inthe interaction of two currents.

We have investigated the magnitudeand direction of B (although we some-times called it a) In relation to itssources in some simple cases. If weknow B ai; any point we can immediatelyfind the force on a current elementIAs placed at that point by computinglAs x B. In terms of Ampere's experi-ments, the interaction between two cur-rents is thus for convenience consid-ered in two steps, the production of afield B by one circuit and the actionof the field B on the other circuit.In view of the complicated dependenceof the forces on the angles involved,this procedure has great advantages,since we need now consider only oneangle at a time. Furthermore, the con-tributions to if frc.la different sources,

be they current? or magnets, are addi-

Fig. 2.15

LINE OF AT

Live, so that simultaneous interactionsmay be considered in a relatively sim-ple way. Even so, it should be remarkedthat the greatest advantages of thefield concept become apparent only whenthe sources, and therefore the fields,are permitted to vary in time. The sub-ject of time varying fields is reservedfor another monograph in this series,but we should note that light and otherelectromagnetic radiation can be simplyunderstood only on the basis of elec-tric and magnetic field quantities.

If we consider only forces on cur-rent elements, we need only one mag-netic field quantity, that defined togive the force per unit current atright angles to the direction of thefield, namely, B. The necessity forconsidering a second field quantity,the magnetic field intensity ff, doesnot then arise until we consider mag-netic materials, within which B and iTare different. But can we confine ourattention exclusively to currents? Itwas Ampere's hypothesis that all mag-netic interactions can in fact betraced to currents. whether they occurin macroscopic circuits or are assumedto exist in the most elementary formof matter. To establish the basis forthis hypothesis we must consider theforces on a loop of current in afield B.

Let us take a rectangular loop ofwire cdef, as shown in Fig. 2.16, hav-ing dimensions a and b, placed ini-tially so that its plane is parallel toa field 6 which is uniform in space ar,dconstant in time. The current in theloop is 1. We nay compute the force oneach straight section of the loop fromthe formula F 16; X B. With the cur-rent as indicated, we see that there

16 MAGNETo3TATIC8

Fig. 2,16

is a force IbB directed out of theplane on the wire cd, a force IbB di-rected into the plane on the wire ofand no forces on fc and de since thesewires are in the same direction as thelines of B. The net force on the loopis zero, but there is a torque of mag-nitude ablB tending to turn it out ofthe plane, into such a position thatwhat is now the front of the loop facesdown, with the plane of the loop per-pendicular to the lines of B. This isexactly what would happen to a rectan-gle of magnetic material whose frontface is a negative (south-seeking) poleand whose back face is positive (north-seeking). The loop is thus equivalentto what Ampere called a magnetic shell,a flat magnet of magnetic moment pro-portional to the product of itu areaand the current on its boundary,

Fig. 1.17

S

Fig, 2.18

We may recall that the torque ona magnot of magnetic moment m in a re-gion of field intensity II is mil whenthe direction of the moment is atright angles to H. The torque on ourloop is IAB, where A - a x b is thearea of the loop. If we want to keepthe same units as before for magneticmoment, we may ascribe to the loop amoment of magnitude 'OA, directed per-pendicular to the plane of the loop,and positive toward a right-hand thumbwhose curved fingers point along thecurrent. For other orientations of theloop the torque is IAB s;r1 0, where 8is the angle between the magnetic mo-ment and the field lines, in agreementwith the torque mH sin 0 on a magnetin a field intensity n.

The magnetic moment of a currentloop does not depend on the shape ofthe loop. A circular loop is equivalentto a magnetic disk whose faces are ofequal and opposite polarity, and whosemagnetic moment is again 'OA, with Athe area of either face. Ampere wentfurther; according to his hypothesis,all magnets are current configurations,which exist on a submicroscopic scale.A "magnetic shell" would consist of anindefinite number of tiny currentwhirls, all oriented in the sane sense,so that the net current is zero exceptat the boundary of the shell, or loop;internally the currents of contiguouswhirls cancel each other as is evidentin Fig. 2.17. A helix of wire with ad-jacent turns of current is thus equiv-alent to a stack of magnetic disks, asin Fig. 2.18; the net effect as a nag-net is a positive (north-seeking) poleat one end of the helix aid a negative

THE MAGNETIC INTERACTION OF STEADY CURRENTS 17

pole at the other. The impossibilityof separating poles is then just theimpossibility of separating the facesof a disk.

The modern view of magnetism isnot very different from this simplepicture, and Ampere's hypothesis hasbeen accepted in principle. Because ofthe absence of separable magneticpoles, all magnetism is traced to cur-rents, even when the physical currentsare not accessible to measurement. Theneutron, for example, is an unchargedparticle, but it does have a magneticmoment; in this sense it behaves likea circulating negative charge. Recentexperiments have shown that the neu-tron does behave more like an infini-tesimal current whirl than like an in-finitesimal linear magnet, although itscurrent is quite inaccessible for de-tailed investigation.

Search for the isolated magnetic,)ole continues, but all the evidencein hand supports the view that ff, a

field which acts on currents and canbe traced to moving charge, is a morebasic concept than that of the magneticfield intensity fl, simply because elec-tric charges and currents do exist.Nevertheless we continue to exploremagnetic fields with iron filings andcompass needles, and the concept ofmagnetic field intensity IT can hardlybe avoided in the description of wig-nntiv materials.

Einstein has pointed out in his"scientific autobiography" that con-cepts used to describe the physicalworld are in truth intellectual crea-tions, devised by scientific imagina-tion but not freely: The disciplineimposed by observation and experimentis very strict. The concepts of elec-tromagnetic theory are very intimatelyrelated to one another, reflecting anenormous body of diverse but relatedexperimental results. More than one ofthe concepts plays a dual role, andexperiments can often be described inmore than one way. Yet we shall see inMonograph III that there are only fourfundamental empirical laws of electric-ity and magnetism, which can be de.

scribed in a most elegant and simpleway in terms of field quantities. Wehave already considered three of theselaws: Coulomb's law for the interac-tion of two static charges, the law ofMichell and Coulomb for the interactionof two magnetic poles plus a statementof the ineeparability of poles, and thelaw describing the magnetic effect ofcurrents or the interaction of two cur-rents, usually called Ampere's law. Theremainder of this booklet will be de-voted to further development of thisthird law. Before undertaking a moremathematical and thus more powerfulformulation of the law, however, letus examine the effect of magneticfields on individual charges.

The currents we have consideredthus far were assumed to exist in un-charged conductors, but a current isdefined as a flow of charge. The ques-tion of whether the mechanical motionof a charged body produces magneticeffects was first tested experimentallyby Henry Rowland of Johns Hopkins Uni-versity in 1878. Rowland electrostatic-ally charged the rim of an insulatingdisk, rotated the disk, and found thatmagnetic effects were indeed produced.The experiment is very difficult toimrform because the currents producedin this way are small, but the resultis unambiguous. That moving chargesexperience a force in n magnetic fieldis much easier to demonstrate: Streamsof electrons in a vathotle ray tube pro-duce a visible glow on the glass enve-lope of the tube which is shifted veryreadily by even a small magnet. Infact, it was concluded that cathoderays are charged particles as a resultof their deflection by both electricfields and magnetic fields.

The correct formula for the forceon a charge moving in a magnetic fieldcan be obtained from that for the forceon an element of current in a conduc-tor. Let us assume that a conductor hasN movable charges per unit volume, eachcharge of magnitude q. These movablecharges may undergo very complicatedmotions, but if there is a net flow ofcharge in one direction, we may ascribe

18 MAGNETOSTAT ICS

As

Fig. 2.19

to them an average "drift" velocity v.A linear conductor of unit cross sec-tion, a segment. Of .8WA.C.114 I al

in Fig. 2.19, would then carry a cur-rent Nqv - this is the nmount of chargecrossing the face shown per unit time.The amount of charge per unit timeflowing through a portion of the facewhose area is A is then NqAv, and thegeneral formula is I - NqAv. For acurrent element of length As,

Ids NqAAsv.

But Ads is just the volume of the cur-rent element and NAM is the total num-ber of charges involved. Thus the forceper charge q is

lk; X? N Abs

qv x B. (2.13)

This derivation of F q; x if in-volves too many assumptions to be rig-

orous, but the result is entirely cor-rect. For a charge q moving with ve-locity v, qv is equivalent to a currentelement. The force q; x ii is called theLoreiltz force, first. derived rigorouslyby the famous Dutch physicist H. A. Lo-rentz in 1892. It is often taken as aFundamental equation, and the force ona current element derived from it. It

can be taken as the defining equationfor the magnetic field quantity IS isthat field which gives a velocity de-pendent force on a charge q, in accordwith the equation for the Lorentzforce, as distinguished from the elec-tric field intensity E, which producesa force which is independent of the ve-locity. We note that both E and B aredefined in a particular frame of ref-erence, that in which the velocity ofthe charge is v.

One of the most interesting prop-erties of the Lorentz force is that itis incapable of changing the speed orkinetic energy of a moving charge,since the force (and therefore the ac-celeration) is at right angles to thevelocity. In a uniform magnetic fieldwhich is itself perpendicular to thevelocity, the motion of a charged par-ticle is circular.

PROBLEMS

2.1 Show that if I, 9, and E are unitvectora in the direction of in-creasint x, y, and s in a right-handed Cartesian coordinate system,

;x; 0 iiitxfokixi

ixf. I;xi. ix I

Show also that in terms of Carte-sian components,. the vector productof A Axit Ar9 Ali and B Bx*

4. By; + Bar may be written in theform of a determinant:

A x Ii Ax Ay As

It., By 1311,

2.2 Suppose you have a long cylindri-cal conductor of rodius r0, carry-ing total current I0. The currentis distributed uniformly over across section of the conductor, sothat 10 tr0 5 j, where J is the

current density in amperes/meter'.

THE MAGNETIC INTERACTION OF STEADY CURRENTS 19

(a) The field lines of II outsidethe cylinder are circles centeredat the axis. Would this also betrue of the magnetic field lineswithin the cylinder? Give yourreasons.

(b) What is the magnetic field in-tensity on the axis of the cylin-der?

(0 What is the magnitudsl of II atr - Arft?

(d) Plot the magnitude of II againstthe distance r from the axis of thecylinder, both inside and outsidethe cylinder. At what distance fromthe axis is this magnitude great-est?

2.3 Let P be a point outside the longsolenoid of Fig. 2.8, whose dis-tance from the axis is large com-pared with the radius of the sole-noid. According to Eq. (2.6), eachelement of current contributes tothe field intensity a om la XPirt.

(a) Consider qualitatively the con-tributions to the magnetic field

intensity from current elementslAs parallel to P and those per-pendicular for from a single turnof the coil in the plane of P per-pendicular to the axis. Do theytend to reinforce or to cancel eachother?

(b) Make the same qualitative esti-mate of contributions &I for apoint inside the solenoid.

(c) Consider, again qualitatively,the contributions dit from twoturns, one above and one below theplane of P perpendicular to theaxis.

(d) What :e your conclusions con-cerning the validity of Eq. (2.4)7

2.4 If a_charged body moves with veloc-ity v at right angles to a uniformfield D, its path is circular,since F cl; x 6, and thus it ac-celeration is perpendicular to v.What sort of path would :exult ifv were in the same direction as thelines of 0? What sort of path wouldbe produced if the angle between vand 13 were neither 0° nor 904?

3 MAONETOSTATICS REFORMULATED

Ampere's law is all there is to mag-netostatics if we confine our attentionto the magnetic effects of steady cur-rents, but some Aspects of the subjectbecome apparent only if more elegantmathematical methods are available.Mathematics is more than a powerfultool for the solution of problems Re-lations between physical quantities areoften revealed by mathematical analy-sis, so that more physics emergesclearly. The danger lies in a tendencyto substitute mathematical formaliamfor physical thought, to overlook orneglect the physical content of a math-emntien1 equation or a line of mathe-matical reason, instead of being guidedby it. Let us try to keep the physics,or sometimes only geometry, firmly inmind in our further analysis of sag-netostatics. For this chapter we shallonly assume knowltdgt of basic calcu-lus, that branch of mathematics in-vented by Newton not only to make hardproblems easier but also to sharpen andclarify his ideas concerning the physi-cal world.

Let us begin by rewriting the re-lations considered earlier in terms ofdifferentials and integrals. Ampere'slaw has appeared in essentially twodifferent forms. The direct expressionof the force on a current element Iola,is

non Ildi.1) I1c1;2 x e, (3.1)

where

B- Et f Isds, x I

4w-----T--- ' (3.2)

where t is a unit vector from Ilds2 tothe point at which if is computed. Thesubstitution of ds for As is routine,but tho circle at the middle of the in-tegral sign is shorthand notation forthe 'm closed" below the summation signof Chapter 2. Nero it should bo road

20

"the integral over the entire circuits" of the integrand as written. Thisintegral is, of course, a vector, everyelcaent contributing to B in a direc-tion at right angles to both ds2 andto the lino from d'Si2 to tho positionof lids, where B is to be determinedto give the correct force.

We have also .toted that B so de-fined yields a flux such that the totaloutward flux of D through any closedsurface is zero. If dg is an elementof surface, directed ot,t from the vol-ume inclosed by the surface,

f 6 dg 0 (3.3)cloned

Unfortunately one must write in wordsthat this surface integral is closed.A little sphere at the middle of theintegral sign could be a convenientshorthand notation except for the factthat a sphere is indistinguishable froma circle in two dimensions. In thebooklet on electrostatics we have con-sidered a function corresponding tcsuch a closed surfce integral whichdescribes a property of the integrandvector at each point in the inclosedspace. We shall consider this propertyof if at a later stage of the dis-cussion.

We have seen that the physical con-tent of Aapere'c law may be written al-ternatively as a circulation law for 6,which may also be written as an inte-gral. The field II is a vector quantitydefined at all points (x,y,t) in theregion of interest, The line inte6ralof II on a path C from pointP_to pointP' is a scalar, written es f:B .ds,where ds is an element of length inthe direction of the local tangent tothe path, and the integrand is theproduct of ds and the component ofparallel to ds (see Fig. 3.1). (interms of magnetic poles this integralwould represent 06 times the work doneby the field in moving a unit pole

MON ETOSTAT ICS REFORMULATED 21

from P to P'.) The circulation of avector implies a closed path, not nec-essarily in a plane, and the integralsign can again be written with a circleat the middle to indicate that the ini-tial and final points are coincident:

B d; (3.4)

is simply a neater way of writing thecirculation law for B.

We have thus far considered almostexclusively linear currents such asthose carried by thin wires, but thetotal current through a surface boundedby the circulation loop may be distrib-uted over the surface and may vary fromone part of the surface to the next.We may readily take account of suchvariations if we express the total cur-rent I In terms of the current density3. For a conductor of cross-sectionalarea A in which the density of currentis uniform, I I. JA. As a vector quan-tity, j is the current per unit areaof a plane normal to the direction ofcharge flow. Since an element of sur-face can be represented by a vector dSnormal to its surface,

I f dg (3.5)

is the total current through a surfaceover which the integral is evaluated(see Fig. 3.2). (In this monograph weconfine our Attention to steady cur-rents, for which the integral of 3over a closed surface would net aero.)Therefore the circulation law for gmay be written

lB.dB -po S dg, (3.6)

where the integral on the right is tobe carried out over a surface, in factany surface, which is bounded by thecurve C. We should recall that the pos-itive direction for dS is chosen by aright-hand rule; for a curve traversedcounterclockwise in the plane, dS ispositive out of the page; if clockwise,dg is positive into the page. Thus farwe have hardly changed the form of thiequations relating magnetic fields to

Fig. 3.1

their sources: g is written as an in-tegral over all portions of a linearcurrent, instead of the correspondingsum, and the line integral B is re-lated to current through a surface, aswas the sum of B As in Chapter 2. isit possible to relate the magneticfield to its source strength at eachpoint, much as we found in the Electro-statics monograph that we could relatethe electric field intensity to thecharge density at each point? If therewere only magnets with poles, and nomagnetic effects of currents, the an-swer would be completely analogous tothe electrostatic relations. It is in-deed true that the net outward flux ofthe magnetic field intensity g (but notthat of g) from a closed volume Is thetotal pole strength within the volume,and that H is related to density of

Fig. 3.2

22 MAGNETOSTATICS

Fig. 3.3

pole strength in the same way that theelectric field intensity is related tothe charge density. But we have seenthat the net flux of both g and g froma closed volume is zero if these fieldsare produced by currents. This is avery interesting and important propertyof magnetic fields, but it is of no im-mediate help in relating fields to thecurrents which produce them.

The answer comes from the circula-tion law for D (or for 11, since the twofields are the same in empty space, ex-cept for an arbitrary constant factorno). It will be necessary to examinecirculation more closely and to developfurther the mathematics associated withit. This mathematics applies to allvector fields, but we shall call thefield D, and feel free to apply themathematical consequences to other vec-tor fields as the need arises.

In electrostatics we were guiledto the appropriate mathematical theo-rem by Gauss's law: Beginning with arelation between the surface integralof the electric intensity E over aclosed surface and the charge withinthe enclosed volume, we found that thefunction of E which can be identifiedwith the charge density at each pointin space is the divergence of I (divE). The circuital form of Ampere's lawrelates a closed line integral of B tothe Integral of the current densityover a surface bounded by the liner Byanalogy we shoule expect to find afunction of B at each point of apace

such that its integral over any surfacebounded by a_line is equal to the cir-culation of Ii about the perimeter ofthe surface. If it is to be identifiedwith the current density, it must be avector quantity. In other words, weseek a vector C which is a mathematicalfunction of ff such that

fff . d' fe dg .1' j dg.

(3.7)

If the second equality is to be truefor any surface bounded by the curve ofof the line integral, it follows thatthe integrands must be the same, andC

Consider any simple closed curvein a region where there is a field B.An area bounded by a closed curve,whether plane or not, can be dividedinto two or several areas by lines be-tween two points on the boundary. InFig. 3.3 there are three areas, aroundeach of which we may take the circula-tion of d in the counterclockwisesense indicated. In the sum of thesecirculations all the interior bound-aries are traversed twice, in oppositedirections. Thus

fgdW-f if al f LI dW,1

+ f g dg3. (3.8)

The sum of counterclockwise circula-tions of B about all the subdivisionsis just the circulation about thewhole, and this result is independentof whether the surface, or the curvebounding it, is plane. The same ift trueof clockwise circulations, of course,but the sense of all the circulationsin the sum must be the same for can-cellation of adjacent interior bound-aries. The rule is equally justifiedfor ten subdivisions, or a hundred. Ingeneral

f 8. d; E d;,. (3.9)311 i

(We are here reminded of Ampere's hy-pothesis on the equivalence of a closed

MAGNETOSTATICS REFORMULATED 23

Fig. 3.4

current and a shell of magnetic di-poles, which involves submicroscopiccurrent whirls; there is indeed a sim-ilarity, which we shall pursue later.)

Circulation itself is a scalarquantity, but any plane surface has anorientation in space which can bespecified by a vector normal to theplane, and any well behaved surfacecan be broken up into elements suffi-ciently small to be considered plane.(We shall exclude surfaces with infi-nite peaks.) Let us see what the ori-entation of a surfacc has to do withthe circulation of a vector about itsboundary.

Consider a small triangular planeboundary abc, Fig. 3.4, and two sur-faces bounded by it, one the plane abc,the other a surface made up of segmentsof three pieties chosen at right anglesto each other. According to the sumrule,

f 6 de f ff di7, + f asabe *de bcal

fB des .abd

(3.10)

It is most unlikely that the threeterms on the right contribute equallyto the total circulation, for severalreasons. Each small circulation speci-fies a different area and a differentlength of perimeter, and of course Bitself Pnd its variation in space isquite independent of the surfaces we .

happened to choose. There is a relationbetween the surfaces themselves, which

Fig. 3.5

follows from the theorem that the vec-tor sum of all the outward surfaces ofa polyhedron equals zero. (The proofof this theorem is left to a problem.)We note that the four triangles bounda tetrahedron (Figs. 3.4 and 3.5), forwhich the right-hand rule for circula-tion leads to positive inward direc-tions for surfaces 1, 2, and 3, and apositive outward surface for the tri-angle abc. If AS AS3, ag3. and dSrepresent these four surfaces, then

af - aSc af + as, . (3.11)

But does this vector relation betweensurfaces have anything to uo with thescalar relation of the circulations?If there is a vector related to thecirculation about each surface suchthat its scalar product with the sur-face itself would net the circulationthe answer would certainly be affirma-tive.

A vector related to the circula-tion about an elementary plane surfacecan be constructed by multiplying thecirculation of B about its boundary bya unit normal to the surface, and di-viding by the magnitude of the surface:Let us write (f B dsa/ASI)A, as avector in the direction of the unitnormal AI to AS1 The scalar productof this vector with SS is just f 6del, since 61 AS AS,. In fact,

If we take the vector

C16 dal i 5

A +AS

.as,

DSa

(3.12)

24 lIAGNET0STATIC6

Fig, 3.6

then C dg is the original sum of cir-culations. We note that e is not in gen-eral the same as (fri dir/AS);, where; is the unit vector normal to the sur-face abc, although a*. J. if ' ds.

The quantity f D d;/AS is the compo-nent of e normal to the surface AB.

The components of e are thus fardefined in relation to plane surfaces,which may be oriented in three mutu-ally perpendicular and thus independentdirections. Our surfaces at, a52, andAS3 are themselves components of Ag.Now the surface a may be taken assmall as we please, and in the limitof small AS we define a vector whichis called the curl of g, writtencurl B. For the component. of curl ifnormal to a surface element AS whoseorientation is

a(curl g)/ 1;:

/ If(3.13)

with the line integral taken aroundthe boundary of AS. The integral formof the circulation sun is then

a curl if a (3.14)

where the surface integral extendsover the surface (any surface) boundedby the closed curve of the line inte-gral, the positive direction of 4g be-ing determii.ed in relation to the cir-culation path by the right-hand rule.

The vector curl 13- is then a functionof B at every point in space which ismathematically related to the circula-tion of B by this formula.

The mathematical relation betweenthe line integral of a vector 5 abouta closed path and the surface Integralof curl B was derived by George GabrielStokes, and is known as Stokes' theo-rem. There is no physics in it. But wehave seen that if B represents themagnetic field,

f Q d; f j dg, (3.15)

whore y is the current density, There-fore, for any surface f curl dgmust equal f pj dg, a demand whichis impossible to satisfy unless

curl I3 - poi (3.16)

at every point, This is the physicalrelation we have sought between 11 andthe current at any point in space.

The name curl suggests goingaround, and we have arrived at theidea of curl by considering circula-tion, but of course curl is not iden-tical with circulation. Consider, asa simple example, a long conductingcircular cylinder in which there is auniform current density j, as indicatedin the cross-section diagram (Fig.3.6). Curl B poj has nonvanishingvalue only within the cylinder, but

fl3 ds a.. I, the total current thread-ing the circulation loop, even if everypoint on the loop is outside the cylin-der. There is, of course, a magneticfield B both outside and inside thecylinder, and it is B itself of whichone takes the circulation.

The magnetic field intensity it-self can be mapped out by a compassneedle, a single small sagnetic dipolefree to orient itself along the fieldlines; the curl of the field intensitycan be demonstrated with a magnetic"quadrupole' - two small permanentmagnets with like poles cemented to-gether - but an "octopole" is more sta-ble and convenient. The negative(south- seeking) poles of four small

MAGNETO3TATICE REFORMULATED 25

permanent magnets can be cemented to awire as indicated in Fig. 3. ?, so thattheir positive or N poles are the fourtips of a cross at right angles to thewire. A cork attached to the wire willmake the whole contrivance float in asolution of NaC1, for example. If theelectrodes are arranged so that thecurrent flows vertically in a cylinderof electrolyte, the "curl -}l- meter" willrotate continuously, the sense of rota-tion depending on the direction of thecurrent. Outside the cylinder of cur-rent a dipole will show the presenceof a magnetic field, but the "curl -7-meter" does not rotate in the absenceof current in the solution.

We have noted in electrostaticsthat the line integral of the electricfield intensity I between any twopoints is independent of the path con-necting the points and thus fg d;0 around any closed curve. It is now

seen from the definition of the curlof a vector that this absence of cir-culation corresponds to the statementthat curl r . 0 at every point in anelectrostatic field. We also found thatthe point by point relation satisfiedbyS so as to express the physicalcontent of Coulomb's law is divp/(0, where p is the electric charge

density. From the definition of thedivergence of a vector and the factthat the net flux of the magneticfield g from any closed volume is zero,it follows that div 6 . O. All theserelations can be summarized:

Electrostatics Magnetostatics

div I - p/cc di*? g . 0

curl g 0 curl B - piT

These are the basic equations of elec-trostatics and magnetostatics. Theirphysical content is Coolonl's law andAmpere's law.

In electrostatics we went further,and found that the determination of thefield g corresponding to any conf1f,ura-lion of static charges was much facili-tated by the introduction of a scalarpotential function 0. The propertiesof the static field included the con.

Fig, 3.7

dition that the line integral of tfrom one point to another is given by

, E a 0, - 01,1

(3.17)

the difference of potential between thetwo points. The possibility of relyingon 0 to obtain g depends on the condi-tion that curl g 0. It is left tothe problems to show that the curl ofthe gradient of any scalar function ofposition vanishes identically.

It is clear that we cannot dependon a scalar potential to obtain 5 ifthe magnetic field owes its existenceto currents, since the circulation ofg does not in general vanish. But thedivergence of 6 does vanish - the linesof 6 never begin or end. These condi-tioas suggest that the magnetic fieldmay be written as the curl of anothervector, for it can be shown that thedivergence of any vector which is it-self a curl is identically zero. Toshow this let us again consider afinite volume (Fig. 3.8), on the sur-face on which there is a closed curvethat divides the surface into SI andSI. For anE vector field A the circu-lation of A about the closed curve is

f X d; f curl X dgi

- I curl X dg,

by the right-hand rule, since wo havetaken dg positive outward from the en

28 MAGNSTOSTATICS

Fig. 3.8

Si

closed volume for both surfaces. Butthe total flux of curl A out of thevolume is

f curl X . dEc + j curl 7, '

f8 closed

curl X dg

f X d; -f X 0.

This is true for any volume, and forany closed curve on the surface of thevolume, and must therefore be true inthe limit:

div curl X h.

i curl X dgI'm s closed

ar-s AV

property of d as produced by magnetsto include the inseparability of mag-netic poles along with Coulomb's lawfor magnets.

In view of the limiting processby which the curl was defined it isnot surprising that it resembles thegradient and the divergence in being adifferential operator with respect tocoordinate, in three-dimensional space.In order to make quantitative use ofthe concept we must write it in termsof coordinates, although the physicalquantity it represents is quite inde-pendent of the particular coordinatesystem chosen. For our purposes, thefamiliar Cartesian coordinates willsuffice, particularly if we rememberto choose the origin of coordinatesand the orientation of the axes so asto make the description of the physi-cal problem as simply as possible.

Let us consider the x componentof curl A in a right-handed Cartesiancoordinate system at the point (x,y,x).In Fig. 3.9, dy and dz are shown asfinite increments in the direction ofincreasing y and i; eventually we shalllet dy and di become as small as weplease. Cy definition,

a 0. (curl X)* limit 1

Thus writing g as the curl of anothervector insures that its divergence iszero, a condition ilposed physicallyon the magnetic field If if it is pro-duced by currents, and defined as a

fig. 3.4

A dS 11,did).

(3.19)

with the line integral taken aroundthe boundary of the small rectangleshown. The vector A(x,y,c) must varywith changing y or t (or both) if theline integral is to be different fromzero, and we must allow for this vari-ation to first order in dy and da. Forthe legs of the rectangle adjacent tothe point mo,y0,10 A Is A(xo.YO.Ite).but all of the leg dt on the right isat yo dy, and the leg dy at the topis at a1 + dz. The line integral isthen

Ay(xotym,s0) dy + At(xm,y0 + dy,so) da

- Ay(xmasore + di) dy

As(Noye,se) di,

MAGNETOSTAT ICS REFORMULATED 27

the last two terms being negative be-cause the path is traversed in the di-rection of decreasing y and z, respec-tively. To take into account the factthat in one term Az is evaluated atyo + dy, we write

Az(x0,y0 + dy, zo) Az(xopYoPzo)

(aAz(x0,y,z0))+ dy

ayye

where (DA lay)yo is the slope of Azplotted against y, evaluated at thepoint yo, the other two coordinatesremaining unchanged. This does not im-ply that As is a linear function of y.If dy were to rennin a finite lengthwe should have to worry in more detailabout the dependence of Az on y. Simi-larly,

Ay(x0,y0,zo + dz) a Ay(x0,y0,zo)

(aAy(xo,y0,z))dz.

azzo

When these expressions are substitutedin the closed line integal all termswhich do not involve derivatives can-cel, and we are left with

aA, aAi(curl X) - _

x py Etz(3.20)

where the coordinates need not bewritten explicitly.

The other components of curl Xcan be derived in the same way, but itis equally valid to invoke the sym-metry of a right-handed Cartesian co-ordinate system and obtain the y and zcomponents by cyclic permutation ofx,y,z:

(curl X) aAx8z ex

aAz

(curl X) . aAz ax ay (3.21)

The result is reminiscent of the formof the cross product of two vectors.If x, y, and z are unit vectors in the

direction of increasing x, y, z, we re-call that

a x (Apz AzBy)I

+ (A213 - AzBz)T + (A,By AyBOZ.

In writing the gradient and the diver-gence in Cartesian coordinates we havealready made use of _a vector differen-tial operator V E (Wax) + (ia/ay)+ (ia/a,), rnd have found it convenientto write grad 0 - and div E - VV E.Here we may write curl X 0 x A, andthe determinantal form o the crossproduct of two vectors s again a help-ful mnemonic device:

curl X - x X - d a

ax 8y 8z

A, Ay Az

(3.22)

The symbol V (del) is useful in themanipulation of mathematical relationsbetween physical concepts, since itcan be treated as an ordinary vectorso long as its role as a differentialoperator is kept firmly in mind, butthe meaning is more apparent if we say"curl" instead of "del cross" in read-ing a formula.

We have seen that the magneticfield B can be written as the curl ofsome other vector which is also definedfor all points of space. The relationbetween a vector and its curl is clar-ified by consideration of some simpleexamples. Let us take

B =, curl X (3.23)

where A - B0(9x gy)._It is easilyfound that curl A - Boi, a uniformfield parallel to the z axis of coor-dinates, but what about the lines ofA? It is a simple exercise to show thatthey are concentric circles, lying inplanes perpendicular to the z axis.

As a second example considerX (p0,) /9) (x' + y2)2 parallel to theAxis but depending symmetrically on

28 MAGNETOSTATICEI

F1g. 3.10

x and y. (The first factor imconstantand a scalar.) Find II B., curl A. Nowfind curl B. The nnswor has boon antic-ipated in writing the constant factorin the expression for A, but the rela-tion between the succession of vectorsfound by taking the curl is interest-ing.

The vector X of which the mag-netic field B is the curl is called the"vector potential." The scalar poten-tial in electrostatics was defined aswork per unit charge, measured involts, and could be traced to itssources by summing the effects of allcharges giving rise to the field in-tensity E. We saw that its gradient isjust the electric field intensity (ex-cept that we change the sign), that is,the physically observable force perunit charge is derived from the scalarpotential by means of the operator"del." The relation between work andforce is then quite apparent, and themeasurement of E in volts per meterreinforces the connection. The role ofthe vector potential in magnetic fieldsis more complicated, largely becausethe force per unit current element isnot in the direction of the field Bbut at right angles to it. The sim-plest justification for using the wordpotential here is that A represents aquantity whose derivative (its curl,this time) is a.physically measurablefield, namely, B.

Just as the scalar potential 0can be traced to electric charges weshould expet that A can be traced tocurrents. We shall write down the cor-rect relation between the vector po-tential and linear current sources,then show that this relation is com-patible with the dependence of B onthese same curronts, as known from

Ampere's law. Before doing so, however,we should note that X is not completelydetermined by the demand that Its curlgive the correct magnell., field; anyvector whose curl Is zero could beadded to A without affecting a at all.We have noted in the description ofthe fields P. and ri that it is neces-sary to know both the curl and the di-vergence to specify a vector, Since Ahas been introduced only so that itscurl represents a, nothing has beensaid about its divergence, which maybe anything. It is customary in magne-tostatics to require that div A - 0,but this restriction Is arbitrary. Thefact that the vector potential is notcompletely defined by requiring thatits curl give the right magnetic fieldis reminiscent of the ambiguity of thesr dr potential of electrostatics, towhich any arbitrary constant could beadded.

It is possible to show that theexpression for B in terms of a current,Eq. (3.2), may be written as the curlof some other vector quantity which canthen be identified as the vector poten-tial. It is somewhat simpler, mathe-matically, to write down a formula /orthe vector potential and show that itscurl gives Eq. (3.2). Let us put

A1.10 Ids

4v r(3.24)

where Ids is an element of current, asusual, and r is the distance from Idsto the point where A (and hence is

to be computed, which we may call thefield point (see Fig. 3.10). The curlof A is to be taken at the field point,and depends on the coordinates of thatpoint, not those of the source. (Afterall, the same field B at some pointcould be produced equally well by avariety of source configurations.)Moreover, owing to the principle ofsuperposition, it makes no differencewhether we take the curl at each pointof the integrand and then sum, orfirst sum over all parts of the cir-cuit Ids and then take the curl at thefield point. In other words,

MAGNETOSTAT/C5 REFORMULATED 29

A- Po f la Po r (Ia )V x

40 r 4w

f x Ida"

where in the las. term we have madeexplicit use of the fact that the deloperator does not act on the coordinatesor the current element. The field pointis involved in the integrand onlythrough the factor l/r, where r is thedistance from Ids to the field point,and thus depends both on the variableof integration ds and those at whichthe vector derivative is taken.

In writing curl K in tho finalform above wo have taken advantage ofthe fact that the operator del behaveslike a vector as well as a differentialoperator, but the integrand now readsdifferently: V(1/0 grad (l /r), andwe have the cross product of a gradientof a scalar and Ids. The gradient oflir is very familiar from electrost-t-ics, since the electrostatic poteW'alof a point charge is properti.onal tol/r, where r is the distance from thepoint charge to the point at which wetake the gradient to find E, the elec-tric field intensity. (Of course, wecould also simply compute it again.)If we take the source point, the posi-tion of Ids, as the origin of coordi-nates, grad (1/0 -T/r2, where F isa unit vector directed from Ids to thefield point. With this substitution,and a change of order in the crossproduct which changes its sign,

po r Ids Xcurl A ,

411 r 2 (3.25)

which is identical with Eq. (3.2).Thus our expression for A is justified.

The vector potential, like thescalar potential, is defined at thefield point, but we see that it is veryclosely related to the current. Infact, for each element of current

-Id;

4w r(3.26)

and thus each increment of A is in thesame direction as the current elementwhich produces it. From the definitionof the curl we see again that thefield B is at right angles to A, aswell as to Ids.

Tho vector potential is sometimesusoft'l in solving problems in magneto-statics, but it does not play nearlyso practical a role in determining Bfrom steady currents as does the sca-lar potential in electrostatics prob-lems. On the other hand, it is almostindispensible in relating fields tononsteady currents, the fields producedby time varying currents. We shall re-turn to this point in another chapter.But before- leaving the '4hject, lot usnote an interesting relation betweenthe vector potential and the flux ofthe magnetic field B through a surface.Thus far we have considered the circu-lation of B in relation to the currentthrough a surface bounded by the cir-culation path, and have noted that theflux of B through a closed surface al-ways vanishes, but we can now derive anew circulation law. Consider the fluxof ET through a surface bounded by aclosed path. By definition this fluxis the integral of B d6 over the sur-face. But g - curl I. Therefore theflux through the surface is

0 f fVx X' dg f X 67,

(3.27)

just the line integral of I round theboundary of the surface. In the earlychapters of Monograph III you will havelearned that a changing flux of Bthrough a surface is accompanied by acirculation of the electric field in-tensity Z. Thus

Llta. aBdt f TT . dg - Jr a fat ds,

(3.28)

and therefore an electric field whichhas a circulation is related to thevector potential: Z -69X/60.

The reformulation of magnetostat-ics in terms of vector calculus has in

30 MAONETOSTATICS

itself added nothing to the physicalcontent of Ampere's law; in fact, thephysics of curl if A T may be lesstransparent than fir . ds I, exceptfor the "curl-II-meter" which works onlyin fluid conductors, As for the vectorpotential, A is sometimes, but not al-ways, useful for solving problems, butit is not oven directly observable bymeans of classical currents or magnets,The power of the differential formula-tion of the laws of electricity andmagnetism is fully realized only whenvariations of the fields in time aretaken into account. If 2. and B (or ff)are permitted to vary in time, as theyare bound to do, a whole new set of

electromagnetic consequences are ob-served, The differential forms of Am-pere's and Faraday's laws helped Max-well to conclude that "light itself(including radiant heat, and otherradiations if any) is an electromag-netic disturbance in the form of wavespropagated through the electromagneticfield according to electromagneticlaws." Even Maxwell did not succeed intracing electromagnetic radiation toits sources; to accomplish this in anunambiguous way requires the vectorpotential, or something equivalent toit, a single quantity to which boththe electric field and the magneticfield are related,