field line motion in classical electromagnetism - mit

Field line motion in classical electromagnetismJohn W. Belchera) and Stanislaw OlbertDepartment of Physics and Center for Educational Computing Initiatives,Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

~Received 15 June 2001; accepted 30 October 2002!

We consider the concept of field line motion in classical electromagnetism for crossedelectromagnetic fields and suggest definitions for this motion that are physically meaningful but notunique. Our choice has the attractive feature that the local motion of the field lines is in the directionof the Poynting vector. The animation of the field line motion using our approach reinforcesFaraday’s insights into the connection between the shape of the electromagnetic field lines and theirdynamical effects. We give examples of these animations, which are available on the Web. ©2003

American Association of Physics Teachers.

@DOI: 10.1119/1.1531577#

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I. INTRODUCTION

Classical electromagnetism is a difficult subject for begning students. This difficulty is due in part to the complexof the underlying mathematics which obscures the physi1

The standard introductory approach also does little to cnect the dynamics of electromagnetism to the everydayperience of students. Because much of our learning is dby analogy,2 students have a difficult time constructing coceptual models of the ways in which electromagnetic fiemediate the interactions of the charged objects that genethem. However, there is a way to make this connectionmany situations in electromagnetism.

This approach has been known since the time of Farawho originated the concept of fields. He was also the firsunderstand that the geometry of electromagnetic field linea guide to their dynamical effects. By trial and error, Faraddeduced that the electromagnetic field lines transmit asion along the field lines and a pressure perpendicular tofield lines. From his empirical knowledge of the shape offield lines, he was able to understand the dynamical effeof those fields based on simple analogies to stringsropes.3–5

As we demonstrate in this paper, the animation of the mtion of field lines in dynamical situations reinforces Farday’s insight into the connection between shape adynamics.6 Animation allows the student to gain insight inthe way in which fields transmit stresses by watching hthe motion of material objects evolve in response to thstresses.7 Such animations enable the student to better mthe intuitive connection between the stresses transmittedelectromagnetic fields and the forces transmitted by mprosaic means, for example, by rubber bands and string

We emphasize that we consider here only the mathemaof how to animate field line motion and not the pedagogiresults of using these animations in instruction. We are crently using these animations in an introductory studio phics course at MIT that most closely resemblesSCALE-UP course at NCSU.8 The instruction includes preand post-tests in electromagnetism, with comparisons to ctrol groups taught in the traditional lecture/recitation formAlthough some of the results of this study have appearepreliminary form,9 the full study is still in progress, and thresults will be reported elsewhere.

We note that the idea of moving field lines has long beconsidered suspect because of a perceived lack of phy

220 Am. J. Phys.71 ~3!, March 2003 http://ojps.aip.org/a

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meaning.10,11 This skepticism is in part due to the fact ththere is no unique way to define the motion of field lineNevertheless, the concept has become an accepted and uone in space plasma physics.12–14In this paper we focus on aparticular definition of the velocity of electric and magnefield lines that is useful in quasi-static situations in which tE and B fields are mutually perpendicular. Although thdefinition can be identified with physically meaningful quatities in appropriate limits, it is not unique. We choose herparticular subset of the infinite range of possible field limotions.

Previous work in the animation of electromagnetic fielincludes film loops of the electric field lines of acceleraticharges15 and of electric dipole radiation.11 Computers havebeen used to illustrate the time evolution of quasi-staelectro- and magneto-static fields,16,17 although pessimismhas been expressed about their educational utility for theerage student.18 Electric and Magnetic Interactionsis a col-lection of three-dimensional movies of electric and magnefields.19 The Mechanical Universeuses a number of threedimensional animations of the electromagnetic field.20 Max-well World is a real-time virtual reality interface that allowusers to interact with three-dimensional electromagnfields.21 References 11 and 15 use essentially the methoddefine here for the animation, although they do not explicdefine the velocity field that we will introduce. The othcitations on the animation of field lines are not explicit abothe method they use. None of these treatments focuserelating the dynamical effects of fields to the shape offield lines. Our emphasis on dynamics and shape is similaother pedagogical approaches for understanding forceelectromagnetism.4,5

II. NONUNIQUENESS OF FIELD LINE MOTION

To demonstrate explicitly the ambiguities inherent in dfining the evolution of field lines, consider a point charwith chargeq and massm, initially moving upward along thenegativez-axis in a constant background fieldE52E0z.The position of the charge as a function of time is given

Xcharge~ t !5~ 12 t223t1 9

2!z ~1!

and its velocity is given by

Vcharge~ t !5~ t23!z. ~2!

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The charge comes to rest at the origin att53 time units, andthen moves back down the negativez-axis.

Figure 1 shows the charge at the origin when it comesrest. The strength of the background field is such thattotal field is zero at a distance of one unit above the chaFigure 1 also shows a set of electric field lines at this timethe total field~that is, the field of the charge plus that of thconstant field!. Note that in Fig. 1 and in our other figurewe make no attempt to have the density of the field linrepresent the strength of the field~there are difficulties associated with such a representation in any case22!. We discussother features of Fig. 1 below.

Now suppose we want to make a movie of the scenarioFig. 1, showing both the moving charge and the timchanging electric field lines. How would we do this? Ocourse, there is no problem in animating the charge motbut there are an infinite number of ways of animatingfield line motion. We show one of these ways in Fig.which gives two snapshots of the evolution of six differefield lines. The animation method that we use here isfollows ~this method isnot the method we use later!. At anytime ~frame!, we start drawing each of our six field linessix different points, with each of the six initial points fixedtime and space for a given field line. These six pointsindicated by the spheres in Fig. 2. Figure 2~a! shows the fieldlines computed in this way when the charge is still outsight on the negativez-axis att50, moving upward. Figure2~b! shows these ‘‘same’’ field lines when the charge harrived at the origin att53.

Although we have a perfectly well-defined animationthe field line motion using this method, the method has mundesirable features. To name two, the innermost four filines in Fig. 2, which were originally connected to thcharges producing the constant field in Fig. 2~a!, are con-nected to the moving charge at the time shown in Fig. 2~b!.Moreover, the motion of all of the field lines in the horizontdirection is opposite to the direction of the flow of enerwhenever the charge is in motion. A much more satisfyway to animate the field lines for this problem is outlinedSec. IV.

Fig. 1. Electric field lines for a point charge in a constant electric field.also show the Maxwell stress vectorsdF5T"ndA on a spherical surfacesurrounding the charge. Because the magnitude of the stress vectorgreatly around the surface, the length of the displayed vectors is protional to the square root of the length of the stress vector, rather thalength of the stress vector itself.

221 Am. J. Phys., Vol. 71, No. 3, March 2003

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We again emphasize that there are an infinite numbeways to animate electric field line motions for the problemhand. We have shown one way in Fig. 2. Another way wobe to start out with the same six points shown in Fig. 2~a!,but now let their positions oscillate sinusoidally in the hozontal direction with some amplitude and frequency. At ainstant of time, we could then construct the field lines thpass through those six points. This construction would agive us an animation of the field line motion for these sfield lines which would be continuous and well-behaveHowever, other than the fact that the field lines wouldvalid field lines at that instant, the manner in which thevolve has little to do with the physics of the problem.

III. DRIFT VELOCITIES OF ELECTRIC ANDMAGNETIC MONOPOLES

Before we present our preferred method of animationfield lines, we first discuss the drifts of electric and magne

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Fig. 2. Two frames of a method for animating field lines for a charmoving in a constant field. At any instant of time, we begin drawing the filines from the six spheres, which are fixed in space. In~a!, the particle is stillout of sight on the negativez-axis, but the influence of its field can be seeIn ~b!, the particle is at the origin and has come to rest before beginninmove back down the negativez-axis.

221J. W. Belcher and S. Olbert

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monopoles in crossed electric and magnetic fields. Thiscussion will help motivate our definitions of field line motioin the following sections. For an electric charge with velocv, massm, and electric chargeq, the nonrelativistic equationof motion in constantE andB fields is

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If we define theEÃB drift velocity for electric monopoles tobe

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and make the substitution

v5v81Vd,E , ~5!

then Eq.~3! becomes~assumingE andB are perpendicularand constant!:

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The motion of the electric charge thus reduces to a gyraabout the magnetic field line superimposed on the stedrift velocity given by Eq.~4!. This expression for the drifvelocity is only physically meaningful if the right-hand sidis less than the speed of light. This assumption is equivato the requirement that the energy density in the electric fibe less than that in the magnetic field.

For a hypothetical magnetic monopole of velocityv, massm, and magnetic chargeqm , the nonrelativistic equation omotion is23

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If we define theEÃB drift velocity for magnetic monopolesto be

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and make the substitution analogous to Eq.~5!, then we re-cover Eq.~6! with B replaced by2E/c2. That is, the motionof the hypothetical magnetic monopole reduces to a gyraabout the electric field line superimposed on a steady dvelocity given by Eq.~8!. This expression for the drift velocity is only physically meaningful if it is less than the speedlight. This assumption is equivalent to the requirement tthe energy density in the magnetic field be less than thathe electric field. Note that these drift velocities are indepdent of both the charge and the mass of the monopoles.

In situations whereE andB are not independent of spacand time, the drift velocities given above are still appromate solutions to the full motion of the monopoles as longthe radius and period of gyration are small compared tocharacteristic length and time scales of the variation inE andB. There are other drift velocities that depend on bothsign of the charge and the magnitude of its gyroradius,these can be made arbitrarily small if the gyroradius ofmonopole is made arbitrarily small.24 The gyroradius de-pends on the kinetic energy of the charge as seen in a frmoving with the drift velocities. When we say that we aconsidering ‘‘low energy’’ test monopoles in what follow


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we mean that we take the kinetic energy~and thus the gyro-radii! of the monopoles in a frame moving with the drivelocity to be as small as we desire.

The definition we will use to construct our electric fieline motions in Sec. IV is equivalent to taking the local vlocity of an electric field line in electro-quasi-statics to be tdrift velocity of low energy test magnetic monopoles sprealong that field line. Similarly, the definition we will use tconstruct our magnetic field line motions in Sec. V is equivlent to taking the local velocity of a magnetic field linemagneto-quasi-statics to be the drift velocity of low enertest electric charges spread along that field line. Thchoices are thus physically based in terms of test partmotion and have the advantage that the local motion offield lines is in the direction of the Poynting vector.

IV. A PHYSICALLY-BASED EXAMPLE OFELECTRIC FIELD LINE MOTION

We now introduce our preferred way of defining the eletric field line motion in the problem described in Sec. II.this section and in Secs. V–VI we concentrate on explainour calculational technique. In Sec. VII, we return to tquestion of the physical motivation for our choice of animtion algorithms.

First, we need to calculate the time-dependent electricmagnetic fields for this problem in the electro-quasi-staapproximation, assuming nonrelativistic motion. By electquasi-statics, we imply that there is unbalanced charge,also that the system is constrained to a region of sizeD suchthat if T is the characteristic time scale for variations in tsources, thenD!cT. By using Ampere’s law including thedisplacement current, we can argue on dimensional grouthat cB'DE/cT5(V/c)E, whereV5D/T. With this esti-mate ofB, it is straightforward to show that if we neglecterms of order (V/c)2, then the curl of the electric field iszero. In this situation, our solution forE(x,t) as a function oftime is just a series of electrostatic solutions appropriatethe source strength and location at any particular time. Tour time-dependent electric field due to the charge in tproblem is given by

Echarge~x,t !5q

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x2xcharge~ t !

ux2xcharge~ t !u3 . ~9!

The time-dependent magnetic field in the same approxition can be found by using the Lorentz transformation proerties of the electromagnetic fields, and is given by

B~x,t !51

c2 Vcharge~ t !3Echarge~x,t !. ~10!

To verify directly that Eq.~10! is the appropriate solutionsimply take the curl of this magnetic field. The total electfield is then the sum of the expression in Eq.~9! and thebackground electric field, and the total magnetic field is jgiven by Eq.~10!. Note that the electric and magnetic fieldare everywhere perpendicular to one another.

Given these explicit solutions for the electric and magnefield, we can calculate at every point in space and timemagnetic monopole drift velocityVd,B(x,t) given in Eq.~8!.We call this velocity field the velocity of the time-dependeelectric field lines in electro-quasi-statics. Note that this vlocity is parallel to the Poynting vector, so that an observmotion of a field line in our animations indicates the dire


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tion of energy flow at that point. This definition is valid onwhen we have a set of discrete sources~point charges, pointdipoles, etc.!. In other situations, for example, a continuocharge density, it is not appropriate~see Sec. VII C!.

We show in Fig. 3 two frames of an animation using thdefinition of the velocity of electric field lines to animatheir motion. In addition to the field line representation, walso show in Fig. 3 a representation of the electric fieldwhich the direction of the field at any point is indicatedthe orientation of texture correlations near that point. Tlatler representation of the field structure is the line integconvolution technique of Cabral and Leedom.25 Sundquist26

has devised a novel way to animate such textures usingdefinitions of field motion contained in this paper. Thatthe texture pattern in Fig. 3 evolves in time according to E~8!.27

This animation is constructed in the following waChoose any point in spacex0 at a given timet, and draw anelectric field line through that point in the usual fashion~forexample, construct the line that passes throughx0 and is

Fig. 3. Two frames of a physically-based method for animating field lifor a charge moving in a constant field. We show both a field line repretation and a method of representing the electric field in which the direcof the field at any point is indicated by the orientation of the correlationsthe texture near that point. The texture is color-coded so that black resents low electric field magnitudes and white represents high electricmagnitudes.


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everywhere tangent to the electric field at that time!. Thisprocedure defines a particular field line at that instant of tit. Now increment the time by a small amountDt. Move thestarting pointx0 to a new point at timet1Dt given by x0

1Dt Vd,B(x0 ,t). Draw the field line that passes through thnew point at timet1Dt. That field line is now the time-evolved field line at the timet1Dt. Using this method, thetwo outer field lines in Fig. 3~a! evolve into the two outerfield lines in Fig. 3~b!. A more elegant way to follow theevolution of the field lines is to follow contours on which thvalue of an electric flux function is constant, and we explathat method elsewhere.28

An immediate question arises as to why this methodconstruction yields a valid set of field lines at every instanttime. What if we had considered at timet a different startingpoint x1 on the same field line asx0? For our method tomake sense, the pointsx01DtVd,B(x0 ,t) and x1

1DtVd,B(x1 ,t) must both lie on the same field line at timt1Dt. This is in fact the case, but it is by no means obvioWe postpone the explanation as to why this is true to SVII C.

What are the advantages to the student in viewingkind of animation? To answer this question, let us firstview how Faraday would have described the downward foon the charge in Figs. 1 and 3. First, surround the chargean imaginary sphere, as in Fig. 1. The field lines piercinglower half of the sphere transmit a tension that is parallethe field. This is a stress pulling downward on the chafrom below. The field lines draped over the top of the imanary sphere transmit a pressure perpendicular to themseThis is a stress pushing down on the charge from above.total effect of these stresses is a net downward force oncharge. Over the course of the animation the displayedometry simply translates upward or downward, so that ialso obvious from the animation that the force on the chais constant in time as well.

Figure 1 also demonstrates how Maxwell would haveplained this same situation quantitatively using his stress

sor. We show the Maxwell stress vectorsdF5TI"ndA on theimaginary spherical surface centered on the charge. Becthe stresses vary so greatly in magnitude on the surfacthe sphere, we show in Fig. 1 the proper direction ofstress vectors, but the stress vectors have a length thproportional to the square root of their magnitude, to reduthe variation in the length of the vectors. The downwaforce on the charge is due both to a pressure transmperpendicularly to the electric field over the upper hemsphere in Fig. 1, and a tension transmitted along the elecfield over the lower hemisphere in Fig. 1, as we~and Fara-day! would expect given the overall field configuration.

We now argue that the animation of this scene greaenhances Faraday’s interpretation of the static image in1. First of all, as the charge moves upward, it is apparenthe animation that the electric field lines are generally copressed above the charge and stretched below the char29

This changing field configuration makes it intuitively plasible that the field enables the transmission of a downwforce to the moving charge we see as well as an upward foto the charges that produce the constant field, which we cnot see. That is, it makes plausible the stress analysis of1 that we carried out at one instant of time. Moreover,know physically that as the particle moves upward, therecontinual transfer of energy from the kinetic energy of t

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particle to the electrostatic energy of the total field. This idifficult point to argue at an introductory level. However,is easy to argue in the context of the animation, as follow

The overall appearance of the upward motion ofcharge through the electric field is that of a point beiforced into a resisting medium, with stresses arising in tmedium as a result of that encroachment. Thus it is plausto argue based on the animation that the energy of thewardly moving charge is decreasing as more and moreergy is stored in the compressed electrostatic field, and cversely when the charge is moving downward. Moreovbecause the field line motion in the animation is in the dirtion of the Poynting vector, we can explicitly see the electmagnetic energy flow away from the immediate vicinitythez-axis into the surrounding field when the charge is sloing. Conversely, we see the electromagnetic energy flback toward the immediate vicinity of thez-axis from thesurrounding field when the charge is accelerating back dothe z-axis. All of these features make viewing the animatia much more informative experience than viewing a sinstatic image.

V. AN EXAMPLE OF MAGNETIC FIELD LINEMOTION

Let us turn from electro-quasi-statics to a magneto-qustatic example. By magneto-quasi-statics, we assumethere is no unbalanced electric charge, and that the systeconstrained to a region of sizeD such that ifT is the char-acteristic time scale for variations in the sources, thenD!cT. Then using Faraday’s law, we can argue on dimsional grounds thatE'DB/T5VB, whereV5D/T. By us-ing Ampere’s law including the displacement current, itstraightforward to show that if we neglect terms of ord(V/c)2, then we can neglect the displacement current,the B field is determined by

¹3B~X,t !5m0J~X,t !. ~11!

That is, as time increases our solution forB(X,t) as a func-tion of time is just a series of magneto-static solutionspropriate to the source strength at any particular time. Telectric field is then derived from this magnetic field usiFaraday’s law.

Now consider a particular situation. A permanent magis fixed at the origin with its dipole moment pointing upwarOn thez-axis above the magnet, we have a co-axial, conding, nonmagnetic ring with radiusa, inductanceL, and resis-tanceR @see Fig. 4~a!#. The center of the conducting ring iconstrained to move along the vertical axis. The ring isleased from rest att50 and falls under gravity toward thstationary magnet. Eddy currents arise in the ring becausthe changing magnetic flux as the ring falls toward the mnet, and the sense of these currents is to repel the ring.

This physical situation can be formulated mathematicain terms of three coupled ordinary differential equationsthe positionXring(t) of the ring, its velocityVring(t), and thecurrent I (t) in the ring.28 We consider here the particulasituation where the resistance of the ring~which in ourmodel can have any value! is identically zero, and the masof the ring is small enough~or the field of the magnet is largenough! so that the ring levitates above the magnet. Wethe ring begin at rest a distance 2a above the magnet. Thring begins to fall under gravity. When the ring reaches


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distance of abouta above the ring, its acceleration slowbecause of the increasing current in the ring. As the currincreases, energy is stored in the magnetic field, and wthe ring comes to rest, all of the initial gravitational potentof the ring is stored in the magnetic field. That magneenergy is then returned to the ring as it ‘‘bounces’’ andturns to its original position a distance 2a above the magnetBecause there is no dissipation in the system for our partlar choice ofR in this example, this motion repeats indenitely.

How do we compute the time evolution of the magnefield lines in this situation? As before, we first find the totelectric and magnetic fields for this problem. In the reframe of the ring, the magnetic field of the ring can be drived from the vector potentialA(x,t).30 The electric field ofthe ring in its rest frame is given by the negative of tpartial time derivative ofA(x,t). We then calculate the electric field in the rest frame of the laboratory using the Lorentransformation properties of the electric field@for example,we use the equation analogous to Eq.~10!#. The magneticfield in the laboratory is just the sum of the magnetic fieldthe magnet and the magnet field of the ring in its rest fra~assuming nonrelativistic motion!. Given these explicit solu-

Fig. 4. Two frames from an example of a physically-based method ofmating magnetic field lines for a conducting ring falling in the magnefield of a permanent magnet. The spheres in the ring in the animationthe sense and the relative strength of the eddy current in the ring as itand rises.


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tions for the total electric and magnetic field, we can ncalculate for all space and time the electric monopole dvelocity Vd,E(x,t) given in Eq.~4!. This is the velocity fieldwe call the velocity of the time-dependent magnetic fielines in magneto-quasi-statics. Note again that this velocitparallel to the Poynting vector, so that the observed moin our animations indicates the direction of energy flowthat point.

We show in Fig. 4 two frames of an animation using thdefinition of the velocity of magnetic field lines to animathe motion. The animation is constructed in the same manas that described for Fig. 3, except that we use Eq.~4! for thevelocity field. There are two zeroes in the total magnetic fiwhen the ring is in motion@see Fig. 4~b!#. These zeroesappear just above the ring at the two places where theconfiguration looks locally like an ‘‘X’’ tilted at about a 45degree angle. Very near these two zeroes, our assumthatE!cB is violated, which means that our magneto-quastatic equation approximation is no longer valid. Howevercan be shown28 that the region where the drift velocity ivery large, and thus where our magneto-quasi-static apprmation is invalid, is a region that for all practical purposesvanishingly small. Similar considerations apply to electquasi-statics.

What are the advantages to the student of presentingkind of animation? One advantage is again the reinforcemof Faraday’s insight into the connection between shapedynamics. As the ring moves downward, it is apparent inanimation that the magnetic field lines are generally copressed below the ring. This makes it intuitively plausibthat the compressed field enables the transmission of anward force to the moving ring as well as a downward forcethe magnet. Moreover, we know physically that as the rmoves downward, there is a continual transfer of enefrom the kinetic energy of the ring to the magnetostaticergy of the total field. This point is difficult to argue at aintroductory level. However, it is easy to argue in the contof the animation, as follows.

The overall appearance of the downward motion ofring through the magnetic field is that of a ring being forcdownward into a resisting physical medium, with stresthat develop due to this encroachment. Thus it is plausiblargue based on the animation that the energy of the dowardly moving ring is decreasing as more and more eneis stored in the magnetostatic field, and conversely whenring is rising. Moreover, because the field line motion isthe direction of the Poynting vector, we can explicitly selectromagnetic energy flowing away from the immedivicinity of the ring into the surrounding field when the ringfalling and flowing back out of the surrounding field towathe immediate vicinity of the ring when it is rising. All othese features make watching the animation a much minformative experience than viewing any single image of tsituation.

VI. AN EXAMPLE OF MOVING FIELD LINES INDIPOLE RADIATION

Before turning to a physical justification of our choices fdefining motion of the electromagnetic field, we consider ofinal example of field line animation, this time in a situatiothat is not quasi-static. Some of what we discuss here


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been treated elsewhere,11,31 but without explicit reference tothe connection between field line motion and the directionthe Poynting vector.

Consider an electric dipole whose dipole moment variestime according to

p~ t !5p0„110.1 cos~2pt/T!…z. ~12!

The solutions for the electric and magnetic fields in this siation in the near, intermediate, and far zone are commoquoted, and will not be given here. We take those fieldsuse Eq.~8! to calculate the electric field line velocity in thisituation. Figure 5 shows one frame of an animation of thfields.

Close to the dipole, the field line motion and thus tPoynting vector is first outward and then inward, correspoing to energy flow outward as the quasi-static dipolar elecfield energy is being built up, and energy flow inward as tquasi-static dipole electric field energy is being destroyThis behavior is related to the ‘‘casual surface’’ discusselsewhere.31 Even though the energy flow direction changsign in these regions, there is still a small time-averagenergy flow outward. This small energy flow outward repsents the small amount of energy radiated away to infinThe outer field lines detach from the dipole and move offinfinity. Outside of the point at which they neck off, thvelocity of the field lines, and thus the direction of the eletromagnetic energy flow, is always outward. This is thegion dominated by radiation fields, which consistently caenergy outward to infinity.

VII. THE PHYSICAL MOTIVATION FOR OURDEFINITIONS

A. Thought experiments in magneto-quasi-statics

To understand more fully what our animation procedumean physically, consider a few simple situations. If we ha magnet in our hand and move it at constant speed, dofield lines move with the magnet? Most readers would agthat the familiar dipole field line pattern should move withe magnet, and it is easy to see how to calculatemotion—we simple translate the familiar static dipolar fielines along with the magnet. But consider a more compcated situation. Suppose we have a stationary solenoid

Fig. 5. One frame of an animation of the electric field lines around a timvarying electric dipole. The frame shows the near zone, the intermedzone, and the far zone.


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carries a current that is increasing with time. At any instawe can easily draw the magnetic field lines of the solenoHowever, in contrast to the moving magnet case, thernow no intuitively obvious way to follow the motion ofgiven field line from one instant of time to the next.

The use of the motion of low-energy test electric charghowever, offers a plausible way to define this motion.‘‘test’’ we mean electric charges whose charge is vanishinsmall, and that therefore do not modify the original curredistribution or affect each other. To see how to approachconcept of field line motion using test electric charges,first return to the case of translational motion, but considemoving solenoid instead of a moving magnet. Considerfollowing thought experiment. We have a solenoid carryinconstant current provided by a battery. The axis of the snoid is vertical. We place the entire apparatus on a cart,move the cart horizontally at a constant velocityV as seen inthe laboratory frame (V!c).

As above, our intuition is that the magnetic field linassociated with the currents in the solenoid should mwith their source, for example, with the solenoid on the caHow do we make this intuition quantitative in this casFirst, we realize that in the laboratory frame there will be‘‘motional’’ electric field E52VÃB, even though there iszero electric field in the rest frame of the solenoid. Nogiven the existence of this laboratory electric field, imagplacing a low-energy test electric charge in the center ofmoving solenoid. The charge will gyrate about the magnefield and the center of gyration will move in the laboratoframe because it drifts in the2VÃB electric field with adrift velocity is given by Eq.~4!. An EÃB electric driftvelocity in the2VÃB electric field is justV. That is, the testelectric charge ‘‘hugs’’ the ‘‘moving’’ field line, moving atthe velocity our intuition expects. The drift motion of thlow energy test charge constitutes a convenient way to dethe motion of the field line. This definition has the advantathat this motion is in the direction of the local Poyntinvector. In the more general case~for example, a stationarysolenoid with current increasing with time, or two sourcesmagnetic field moving at two different velocities!, we simplyextend this concept. That is, we define field line motion tothe motion that we would observe for hypothetical loenergy test electric charges initially spread along the gimagnetic field line, drifting in the electric field that is assciated with the time changing magnetic field. This is tdefinition of motion that we have used above.

Let us now return to the original case of the stationasolenoid with a current varying in time. We can find a sotion for B that is the curl of a time-varying vector potentiA. Then our electric fieldE is simply given by the timederivative of this vector potential, and we can calculatevelocity of a given field line directly by using Eq.~4!. Thusas desired, we have found a~nonunique! way to calculatefield line motion in this case.

B. Thought experiments in electro-quasi-statics

Now we turn to electro-quasi-statics. Consider the folloing thought experiment. We have a charged capacitor csisting of two circular, coaxial conducting metal plates. Tcommon axis of the plates is vertical. We place the capacon a cart, and move the cart horizontally at a constant veity V as seen in the laboratory frame (V!c). Our intuition isthat the electric field lines associated with the charges on


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capacitor plates should move with their source, for examwith the capacitor on the cart. How do we make this intuitiquantitative? We realize that in the laboratory frame thwill be a ‘‘motional’’ magnetic fieldB5VÃE/c2. We thenimagine placing a hypothetical low energy test magnemonopole in the electric field of the capacitor. The monopwill gyrate about the electric field and the center of gyratiwill move in the laboratory frame because it drifts in thVÃE/c2 magnetic field. TheEÃB magnetic monopole driftvelocity is given by Eq.~8!. An EÃB magnetic drift velocityin this motional magnetic field is justV. That is, the testmagnetic monopole ‘‘hugs’’ the ‘‘moving’’ electric field linemoving at the velocity our intuition expects. The drift motioof this low energy test magnetic monopole constitutes oway to define the motion of the electric field line. This mtion is not unique, but is physically motivated. As above,also has the advantage that this motion is in the directionthe local Poynting vector.

In the more general case~for example, two sources oelectric field moving at two different velocities, or sourcwhich vary in time!, we choose the motion of an electrfield line to have the same physical basis as above. Thathe electric field line motion is the motion we would obserfor hypothetical low energy test magnetic monopoles itially spread along the given electric field line, drifting in thmagnetic field that is associated with the electric field athe moving charges.

Finally, consider the animation of the electric field lineselectric dipole radiation that we presented in Sec. VI. In tsituation our definition of the motion of field lines using E~8! no longer permits a physical interpretation in termsmonopole drifts, as our ‘‘drift’’ speeds so defined can excethe speed of light in regions that are not vanishingly smEven though this velocity is no longer physical in suchgions~that is, corresponding to monopole drift motions!, it isstill in the direction of the local Poynting vector. Thus thanimation of the motion using Eq.~8! shows both the fieldline configuration at any instant of time, and indicates tlocal direction of electromagnetic energy flow in the systeas time progresses. For this reason, even in non-quasi-ssituations, animation of the field is useful, even if no longphysically interpretable in terms of the drift motions of temonopoles.

C. Why are monopole drift velocities so special inelectromagnetism?

We now return to the question of why the method of costruction of moving field lines described in Sec. IV yieldsvalid set of field lines at every instant of time. For any geeral vector fieldW(X,t), the time rate of change of the fluof that field through an open surfaceSbounded by a contouC that moves with velocityv(X,t) is given by

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Now we specialize to those configurations of electrmagnetic, and velocity fields that preserve the electric flthat is for which the left-hand side of Eq.~14! is zero, andthus for which the right-hand side of Eq.~14! is also zero.Because the surfaceS and its contour are arbitrary~otherthan the requirement that they not sweep over sourceelectric field!, we must have the integrand on the right-haside be zero as well in those regions. Note that becauseintegrand is zero, we are implicitly assuming that the elecfield is perpendicular to the magnetic field at all pointsspace. This requirement is satisfied for all of the exampwe considered above. We further assume that the velofield is everywhere perpendicular toE. This restriction isreasonable because there is no meaning to the motionfield line parallel to itself. Then the requirement that tintegrand be zero leads to our equation for magnetic mopole drift, Eq.~8!.

Thus the velocity field defined by our magnetic monopdrifts preserves electric flux in source free regions. In pticular, if we define the open surfaceS at time t to be suchthat the electric fieldE is everywhere perpendicular to itsurface normal, then the electric flux through that surfaczero at timet. If we let that surface evolve according to thvelocity field in Eq. ~8!, then Eq.~14! guarantees that thelectric flux continues to remain zero. Note that the surfdoes not have to be infinitesimally small for this argumenhold, because the velocityv(x,t) varies at every point arounthe contour according the local values ofE and B on thecontour, in such a way that the integrand on the right-haside of Eq.~14! is zero. Thus theE field continues to belocally tangent to the surface as it evolves.

We can now argue that our method of constructing elecfield lines in Sec. IV using Eq.~8! yields a valid set of fieldlines at every instant of time, as follows. For our surfacetime t, take the field line joining the pointsx0 and x1 dis-cussed in Sec. IV, extended an infinitesimal amount perpdicular to itself in the local direction ofB. The electric fluxthrough this surface is zero by construction. Let this surfevolve to timet1Dt using the velocity field described bEq. ~8!. Equation~14! guarantees that the flux through thevolved surface at timet1Dt remains zero, and thus thelectric field is still everywhere tangent to that surface. Tmeans that the points x01Dt Vd,B(x0 ,t) and x1

1Dt Vd,B(x1 ,t) discussed in Sec. IV still lie on the samfield line at timet1Dt, as was to be shown. Similar consierations apply to our construction in magneto-quasi-statexcept that we are not limited here to point sources, becathe divergence ofB is zero everywhere.

VIII. SUMMARY AND DISCUSSION

It is worth noting that the idea of moving field lines halong been considered suspect because of a perceived laphysical meaning.10,11This is in part due to the fact that theris no unique way to define the motion of field lines. Nevtheless, the concept has become an accepted and usefuin space plasma physics. In this paper we have focusedparticular definition for the velocity of electric and magnefield lines that is useful in quasi-static situations in which t


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E and B fields are mutually perpendicular, and that canidentified with physically meaningful quantities in appropate limits.

In situations that are not quasi-static, for example, dipradiation, the velocities of field lines calculated accordingthe above prescriptions are not physical. That is, they docorrespond to the drift motions of low-energy test monpoles. For example, in some regions of nontrivial extethese velocities exceed the speed of light. Even so, themation of field line motion using these definitions is infomative. Such animations provide both the field line configration at any instant of time, and the local directionelectromagnetic energy flow in the system as timprogresses. Such a visual representation is useful in unstanding the energy flows during electric dipole radiation,in the creation and destruction of electric fields.

In conclusion, we return to the question of the pedagogusefulness of such animations. Emphasizing the shapfield lines and their dynamical effects in instruction undescores the fact that electromagnetic forces between chaand currents arise due to stresses transmitted by the fieldthe space surrounding them, and not magically by ‘‘actiona distance’’ with no need for an intervening mechanism. Fthis reason, our approach is advocated as a useful pedacal approach to teaching electromagnetism.4 This approachallows the student to construct conceptual models of helectromagnetic fields mediate the interactions of the charobjects that generate them, by basing that conceptual unstanding on simple analogies to strings and ropes.

Our further contention is that animation greatly enhanthe student’s ability to perceive this connection betweshape and dynamical effects. This is because animationlows the student to gain additional insight into the waywhich fields transmit stress, by observing how the motionsmaterial objects evolve in time in response to the stresassociated with those fields. Moreover, watching the evotion of field structures~for example, their compression anstretching! provides insight into how fields transmit thforces that they do. Finally, because our field line motioare in the direction of the Poynting vector, animations pvide insight into the energy exchanges between materialjects and the fields they produce. The actual efficacy of thanimations when used in instruction is the subject of angoing investigation,9 which will be reported elsewhere.

ACKNOWLEDGMENTS

We thank an anonymous referee for the amount of efhe or she expended in the review of this paper. This efhas substantially increased the accessibility of the paperbroader audience as compared to the original version. Weindebted to Mark Bessette and Andrew McKinney of tMIT Center for Educational Computing Initiatives for technical support. This work was supported by NSF Grant N9950380, an MIT Class of 1960 Fellowship, The HeleFoundation, the MIT Classes of 51 and 55 Funds for Edutional Excellence, the MIT School of Science EducationInitiative Awards, and MIT Academic Computing. It is paof a larger effort at MIT funded by the d’Arbeloff Fund FoExcellence in MIT Education, the MIT/Microsoft iCampuAlliance, and the MIT School of Science and DepartmentPhysics, to reinvigorate the teaching of introductory physusing advanced technology and innovative pedagogy.


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a!Electronic mail: [email protected] one of his first papers on the subject, Maxwell remarked that ‘‘.order therefore to appreciate the requirements of the science@of electro-magnetism#, the student must make himself familiar with a considerabody of most intricate mathematics, the mere retention of which inmemory materially interferes with further progress...,’’ J. C. Maxwell, ‘‘OFaraday’s lines of force,’’ as quoted by T. K. Simpson,Maxwell on theElectromagnetic Field~Rutgers U.P., New Brunswick, NJ, 1997!, p. 55.

2E. F. Redish, ‘‘New models of physics instruction based on physics ecation research,’’ invited talk presented at the 60th meeting of the Dschen Physikalischen Gesellschaft, Jena, 14 March 1996, online at hwww.physics.umd.edu/perg/papers/redish/jena/jena.html

3‘‘...It appears therefore that the stress in the axis of a line of magnetic fis a tension, like that of a rope...,’’ J. C. Maxwell, ‘‘On physical linesforce,’’ as quoted by T. K. Simpson,Maxwell on the Electromagnetic Field~Rutgers U.P., New Brunswick, NJ, 1997!, p. 146. Also ‘‘...The greatestmagnetic tension produced by Joule by means of electromagnetsabout 140 pounds weight on the square inch...,’’ J. C. Maxwell,A Treatiseon Electricity and Magnetism~Dover, New York, 1954!, Vol. II, p. 283.

4F. Hermann, ‘‘Energy density and stress: A new approach to teacelectromagnetism,’’ Am. J. Phys.57, 707–714~1989!.

5R. C. Cross, ‘‘Magnetic lines of force and rubber bands,’’ Am. J. Phys.57,722–725~1989!.

6J. W. Belcher and R. M. Bessette, ‘‘Using 3D animation in teaching intductory electromagnetism,’’ Comput. Graphics35, 18–21~2001!.

7See EPAPS Document No. E-AJPIAS-71-005302 for animations ofexamples given in this paper. A direct link to this document may be foin the online article’s HTML reference section. The document may alsoreached via the EPAPS homepage~http://www.aip.org/pubservs/epaps.html! or from ftp.aip.org in the directory /epaps. See the EPAhomepage for more information.

8A description of the SCALE-UP program at North Carolina State Univsity is available athttp://www.ncsu.edu/per/&.

9Y. J. Dori and J. W. Belcher, ‘‘Technology enabled active learninDevelopment and assessment,’’ paper presented at the 2001 annualing of the National Association for Research in Science Teaching Conence, St. Louis, 25–28 March 2001.

10‘‘...Not only is it not possible to say whether the field lines move or do nmove with charges—they may disappear completely in certain coordiframes...,’’ R. P. Feynman, R. B. Leighton, and M. Sands,The FeynmanLectures on Physics~Addison-Wesley, Reading, MA, 1963!, Vol. II, pp.1–9.

11‘‘...Also the idea of moving field lines isa priori suspect because the fielshould be regarded not as moving relative to the observer, but rathchanging in magnitude and direction at fixed points, as is proper...,’’ RGood, ‘‘Dipole radiation: A film,’’ Am. J. Phys.49, 185–187~1981!.

12B. Rossi and S. Olbert,Introduction to the Physics of Space~McGraw–Hill, New York, 1970!, p. 382.

13V. M. Vasyliunas, ‘‘Non-uniqueness of magnetic field line motion,’’Geophys. Res.77, 6271–6274~1972!.

14J. W. Belcher, ‘‘The Jupiter-Io connection: An Alfven engine in spaceScience238, 170–176~1987!.

15J. C. Hamilton and J. L. Schwartz, ‘‘Electric fields of moving chargesseries of four film loops,’’ Am. J. Phys.39, 1540–1542~1971!.


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16J. F. Hoburg, ‘‘Dynamic electric and magnetic field evolutions with coputer graphics display,’’ IEEE Trans. Educ.E-23, 16–22~1980!.

17J. F. Hoburg, ‘‘Personal computer based educational tools for visualizaof applied electroquasi-static and magnetoquasi-static phenomenaElectrost.19, 165–179~1987!.

18J. F. Hoburg, ‘‘Can computers really help students understand electromnetics?’’ IEEE Trans. Educ.E-36, 119–122~1993!.

19R. W. Chabay and B. A. Sherwood, ‘‘3-D visualization of fields,’’ inTheChanging Role of Physics Departments In Modern Universities, Proceed-ings of the International Conference on Undergraduate Physics Educaedited by E. F. Redish and J. S. Rigden, AIP Conf. Proc.399, 763 1996.

20S. C. Frautschi, R. P. Olenick, T. M. Apostol, and D. L. Goodstein,TheMechanical Universe~Cambridge U.P., Cambridge, 1986!.

21M. C. Salzman, M. C. Dede, and B. Loftin, ‘‘ScienceSpace: Virtual reaties for learning complex and abstract scientific concepts,’’ ProceedingIEEE Virtual Reality Annual International Symposium~1996!, pp. 246–253.

22A. Wolf, S. J. Van Hook, and E. R. Weeks, ‘‘Electric field line diagramdon’t work,’’ Am. J. Phys.64, 714–724~1996!.

23This equation can be derived by considering the four dot product offour momentum of the monopole with the dual field-strength tensor, JJackson,Classical Electrodynamics~Wiley, New York, 1975!, 2nd ed., p.551.

24N. K. Krall and A. W. Trivelpiece, Principles of Plasma Physics~McGraw–Hill, New York, 1973!.

25B. Cabral and C. Leedom, ‘‘Imaging vector fields using line integral covolution,’’ Proc. SIGGRAPH ’93~1993!, pp. 263–270.

26A. Sundquist, ‘‘Dynamic line integral convolution for visualizing streamline evolution’’ IEEE Trans. Visual. Comput. Graph.~in press, 2003!.

27The only motion that is physical in this representation is the motion ppendicular to the instantaneous electric field~in electro-quasi-statics! ormagnetic field~in magneto-quasi-statics!. However, because of statisticacorrelations in the random texture used as a basis for the field represtion, the Sundquist method for the calculation of the animated textusometimes leads to apparent patterns in the texture which appear atto move along the local field direction, as well as perpendicular to it. Thapparent field aligned flows in the evolving texture patterns are artifactthe construction process, and have no physical meaning.

28Many of the mathematical and physical concepts in this paper arecussed at a much more detailed and sophisticated level in an expaversion of this paper available at the URL given in Ref. 7.

29This statement is true on a global scale, even though the local endensity in the electric field just above the moving charge is decreasrather than increasing. If we calculate the total electric energy insidcube fixed in space whose dimensions are large compared to the transstructure shown in Fig. 3, and compare the value of this field energy wthe charge is at the center of the cube to its value when the chargedistanceDz above the center, we find that the total electrostatic energthe fixed cube increases by an amountq E0 Dz as the particle moves fromthe center of the cube to a distanceDz above the center. This is the amouof decrease in the kinetic energy of the particle as it moves upward othis same distance, as it must be.

30Ref. 23, p. 177.31H. G. Schantz, ‘‘The flow of electromagnetic energy in the decay of

electric dipole,’’ Am. J. Phys.63, 513–520~1995!.


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