Revista Brasileira de Ensino de Fısica, vol. 39, nº 4, e4311 (2017)www.scielo.br/rbefDOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2017-0119

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Electromagnetism as a world view: implications for theteaching of energy

Fabiana Botelho Kneubil∗1,2, Ricardo Karam2, Iva Gurgel3, Manoel Roberto Robilotta3

1Department of Physics, Federal University of Santa Catarina, Brazil2Department of Science Education, University of Copenhagen, Denmark

3Institute of Physics, University of Sao Paulo, Brazil

Received on April 10, 2017. Accepted on May 14, 2017.

There are considerable differences in the ways mechanics and electromagnetism conceptualize energy. In theformer, energy is defined in terms of force and work in a process-like reasoning that remits us to the ”past” ofa given body or system. Moreover, in the mechanical framework energy is ascribed to matter, i.e., one speaksof the energy of a system of bodies. The electromagnetic picture is quite different. There, energy assumes aform localized directly in the fields, i.e., if a region in space contains ~E and/or ~B fields, then there is energy.Furthermore, in electromagnetism energy is no longer a property of an object, but some kind of extensive quantityin space. We argue that traditional physics instruction in introductory courses fails to highlight these differencesand thus students do not become aware of important traits of the electromagnetic ”world view”. Analyzing thecase of electrostatic energy, we both review critically the standard mechanical discourse and introduce a didacticalapproach to motivate the electromagnetic way of thinking of energy without the need of an advanced mathematicalformalism. The latter is more akin to the practice of present day physics, which relies on lagrangians based onfield energy densities.Keywords: Electrostatic energy, electromagnetism, mechanics, physics, epistemology, electromagnetic world view

1. Introduction

In the historical development of physics theories it ispossible to identify the invention of new concepts, butmore often we find a resignification of old ones (see, forinstance, Max Jammer’s classical books on the evolutionof physics concepts). One clear example of this conceptualresignification is the history of energy. In the mechanicsframework, energy is closely associated with work andthe image it conveys is related to a process that refersto the “past” of a given body or system. For instance,the kinetic energy of a body moving with speed v isequivalent to the work required to bring the body fromrest to speed v. Similarly, the potential energy of a systemis the work necessary to bring the system to a particularconfiguration, against a conservative force. Furthermore,in the mechanics rationale energy is associated withmatter, i.e. one speaks of the energy of an object or asystem.

The electromagnetic theory introduced a radically dif-ferent way to conceive energy. This conceptual shift wasexplicitly promoted at the end of the 19th century bysome of Maxwell’s followers such as Poynting, Heavisideand Hertz [1]. The key turning point is traced back toPoynting’s seminal paper On the Transfer of Energy inthe Electromagnetic Field [2]. Departing from the usualmechanics framework - just as Maxwell did in his Trea-∗Correspondence email address: fkneubil@gmail.com.

tise [3] - Poynting introduces a new qualitative way tothink about energy and applies it in seven examples. Ineach one of them, Poynting guides our minds to reasonabout the flow of energy by focusing on the electric andmagnetic fields1. It is possible to say that this work marksa shift in the reasoning about energy from the mechanicallanguage of processes (force, work) to the electromag-netic language of fields in space. Max Jammer highlightssome important aspects of this change as follows:

In Poynting’s theory of energy flow the trans-port of energy is no longer confined to con-ductors. The surrounding medium or emptyspace is the arena where energy moves. En-ergy, thus disjoined from mater, raised itsontological status from a mere accident ofa mechanical or physics system to the au-tonomous rank of independent existence, andmatter ceased to be the indispensable vehiclefor its transport. As a result of this emancipa-tory change or reification of energy, the ideathat only differences in energy are of phys-ical significance had to be abandoned andan absolute magnitude had to be ascribed toenergy ([1],p.173).

1Nowadays we write ~S = 1µ0

~E × ~B where ~S - is Poynting vectorand represents the flux of energy in W/m2.

Copyright by Sociedade Brasileira de Fısica. Printed in Brazil.

e4311-2 Electromagnetism as a world view: implications for the teaching of energy

This new image of the world is one of the most impor-tant traits of field theories and conveys a very differentmindset when compared with the force and work frame-work. Despite its importance, the electromagnetic way ofconceptualising energy is rarely present in introductorycourses. We argue that focusing on the relation betweenenergy and field as soon as possible is essential for pro-viding a better understanding of how electromagnetism,as a field theory, works2.

This is important to the education of young scientists,because the presence of fields in contemporary physics isoverwhelming and theories, such as Quantum Electrody-namics, Quantum Chromodynamics, among others, areexpressed in terms of lagrangian densities. We feel thatthe earlier students are exposed to this way of lookingat the world, the better for their formation. In this pa-per, we choose the electrostatic energy as an exampleto highlight the differences between the force/work andfield approaches and propose a didactical strategy tointroduce the latter.

We start by reviewing the traditional derivation of theelectrostatic energy of two systems - two point chargesand capacitor - using the concepts of force and work, anddiscuss some of the difficulties of fitting a fully compre-hensive mechanical discourse into the time available forlectures3. Next, we introduce a didactical approach tomotivate the electromagnetic way of thinking of energywithout the need of an advanced mathematical formal-ism. The same expressions for the previous situations areobtained with the electromagnetic framework and thedifferences in their “world views”, as well as the differentquestions raised by the two approaches, are highlighted.We do not advocate in favour of the exclusion of theusual derivations via mechanics, let alone that the fieldapproach should precede the classical force/work one.Our main goal here is to stress the conceptual gainsassociated with the electromagnetic framework.

2. Electrostatic Energy: the UsualDiscourse with Force and Work

2.1. Point-like charges

The typical textbook discourse on electromagnetism be-gins with electrostatics, the part of the theory that dealswith charges at rest and, normally, with no mention toreference frames. As their state of rest suppresses theaction of time, one has the feeling that one is dealing withan eternal world. This is the scenario for the introductionof the Coulomb field ~E, due to a point-like charge q, at

2Of course one can also give mechanics a field formalism, for in-stance, in the theory of gravitation. However, it is interesting tonote that this formulation was proposed later by Heaviside [4], inanalogy with the electromagnetic field.3This might shed light on some of the reasons why students strugglewith the concept of electric field, as widely shown in the physicseducation literature (e.g. [5], [6] and [7]).

a point P in space, by means of the expression

~E = q

4πε0~r

r3 , (1)

where ~r is the vector which links the charge to P . Inorder to associate the field with interactions, one invokesa second point-like charge Q and derives the force itsuffers owing to the other one, given by

~F = Q ~E . (2)

In this kind of presentation, one normally omits theinformation that two charges cannot remain at rest bythemselves. This qualitative omission is important, forthe use of correct equations does not ensure that theapproach adopted is logical.

After electric field and force, one introduces the po-tential energy, usually already discussed in a previousmechanics course. The concept of work of a force ~F , alonga displacement d~,

dW = ~F · d~ (3)

is recalled and combined with eqs.(1) and (2). Whenwork is used in the derivation of the electrostatic po-tential energy of two point-like charges, displacementsof charges along a path between two arbitrary pointsin space are required. This apparently simple procedureinvolves an intellectual trick: the displacement requiredis not based on actual movements but is, rather, an intel-lectual construction. Even if this may seem too abstractto beginners, one must and does go on, by choosing apath, usually a radial straight line, which is to be mademore general later. This allows one to write d~= dr ~r asbeing parallel to ~F and the work to bring the charge Qfrom point A (~rA) to point B (~rB) is given by

WAB =∫ B

A

~F · d~=∫ rB

rA

drq Q

4πεo1r2

= − q Q

4πεo1rB

+ q Q

4πεo1rA

. (4)

Next, one must argue that the work WAB is related toenergy, and recalls the living forces theorem, which yields

WAB = mv2B

2 − mv2A

2 , (5)

where m and v are respectively the mass and velocity ofthe body. Of course, the teacher knows that these symbolsrefer to properties of a mechanical body. However, chargesand bodies are very different things. Moreover, the pre-viously static charges acquire kinetic energy. Studentsare therefore entitled to see this sudden transformationof static point-like charges into a moving bodies as beingboth ad hoc and arbitrary.

As the discourse has to go on, results (4) and (5)are used to induce the conclusion that the work of the

Revista Brasileira de Ensino de Fısica, vol. 39, nº 4, e4311, 2017 DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2017-0119

Kneubil et al. e4311-3

electrostatic force is also related with energy. With thispurpose, one writes

WAB = mv2B

2 − mv2A

2 = − q Q

4πεo1rB

+ q Q

4πεo1rA

→ mv2B

2 + q Q

4πεo1rB

= mv2A

2 + q Q

4πεo1rA

.(6)

As the value of the bottom line does not depend on thepoint considered, one may be tempted to identify it withthe total energy of the system and, subsequently, theterm proportional to q Q as the potential energy. However,strictly speaking, one should not use results based on theCoulomb field for moving charges. Eq.(6) correspondsthen to an approximation, valid for low velocities only, apoint not usually discussed in textbooks. Moreover, inmechanics, the potential energy is determined up to anunknown constant K, since it is based on the work neededto set the system in a given configuration, departing fromanother one, chosen arbitrarily as having no potentialenergy. This point, again, may seem too cumbersome tostudents.

For the sake of simplicity, one may choose K = 0in eq.(6), which amounts to setting the reference levelat infinity. Eventually, and in spite of possible gaps inthe logical discourse, one reaches the expression for thepotential energy of two point-like charges separated by adistance d, which is given by

U = q Q

4πε01d. (7)

This result has costs and benefits, that must be as-sessed. It does look simple, but its physical content stilldepends on a surrounding conceptual environment. Letus consider the symbol d, for instance. Taken at its facevalue, it represents the distance between the point-likecharges. However, it also corresponds to the end-pointof a virtual path, needed to bring the charges from aconfiguration in which they were infinitely separated toanother one, where their relative distance is d. The pointto be stressed here is that the derivation of eq.(7) in-volves a large number of decisions, which demand somematurity in physics. The problem is not mathematics,since all passages made are simple but, rather, the manyverbal provisos which deviate one from the mathematicaltrack. Both the quantity and the apparent artificiality ofintermediate explanatory remarks may prevent studentsfrom grasping the physics. At the end, one gives the fish,without teaching how to fish. This kind of situation mayconvey to students a feeling that they are in the midstof a process full of arbitrariness.

2.2. Capacitors

The case of capacitors is similar. Owing to their concep-tual simplicity, they are important in textbook discus-sions of potential energy. A pedagogical capacitor can

be made by two flat and parallel metal plates of areaS, separated by a small distance a. When the platesare connected to a battery, they get charges +Q and−Q, and an electric field ~E appears in the surroundingempty space, both between and outside the plates. Asit is difficult to evaluate this field using simple analyticmethods, one resorts to the infinite plane approximation,which amounts to a homogeneous field confined just tothe region between the plates. Then, Gauss’ law yields

| ~E| = σ

ε0= Q

S ε0. (8)

In order to evaluate the energy accumulated within thesystem, the usual procedure, based on mechanical work,consists in making a thought experiment, assuming thatthere is no initial charge in the capacitor and to carryhypothetical charge elements, from one plate to the other,until the full charge Q is restored. In this way, one getsthe electrostatic potential energy of the system, which isgiven by

U(a) = 12Q2

S ε0a , (9)

where we have stressed its dependence on the separationa between the its flat plates. The derivation of this resultlooks simple only if one does not think about it. Here,again, one must resort to virtual displacements withconstant velocity and students may want to ask why,now, U → ∞ when the two plates are set well apart,and not U → 0, as in the case of two point-like charges.Again, this is not a difficult problem, but there is noshort satisfactory answer to it.

This discussion could be extended, but it already suf-fices to show that the introduction of potential energy inbasic undergraduate courses on electromagnetism is nota particularly pleasant one. Courses begin by promot-ing a world view based on charges and fields, with theirontology of spatial extension, and it may intellectuallydisappointing for students to be conducted back to me-chanical modes of reasoning in the treatment of energy.Especially if this is accompanied by several technicalsubtleties, which demand lengthy and time consumingexplanations.

3. Restoring the Fields

The path followed to introduce electrostatic energy onlybecomes meaningful when looked at a distance, after itis completed. In principle, this is not a problem as, inphysics, it is usual that one does research employing re-sults which are just partially understood. People engagedin these activities are quite aware of this, indicating thatthe same is acceptable in physics teaching, provided thatone has enough maturity. The problem occurs when acumbersome discourse used to introduce a subject ispresented as conveying sharp logic.

DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2017-0119 Revista Brasileira de Ensino de Fısica, vol. 39, nº 4, e4311, 2017

e4311-4 Electromagnetism as a world view: implications for the teaching of energy

In this work, we argue that the world view associatedwith electrostatic potential energy does have a role toplay in education. The widespread notion that the po-tential energy of a spring is located within itself is quiteinstructive. However, the treatments introductory text-books give to potential energy, both gravitational andelectrostatic, normally skip the issue of their localizationin space, in spite of the close similarities with the case ofsprings. In undergraduate courses on electromagnetism,the localization of the electrostatic energy can be dis-cussed just after its introduction based on mechanicalwork. This has the advantage of bringing back the elec-tric field into the physical picture and being easier tounderstand. In the long run, it also paves the way forgoing beyond electrostatics, by allowing students to be-gin to learn the language to be fully explored later, inelectrodynamics.

The alternative procedure for treating the electrostaticenergy U of a system employs the idea that this energyis spread over space, with a volume density given by

dU

dV= ε0

2~E2 , (10)

where ~E is the electric field and d V is the volume elementat the point considered. This is the electrostatic version ofa quite general result from electrodynamics: if an electricfield ~E and a magnetic field ~B exist in a region withoutcharges and currents, the electromagnetic energy densitythere reads

dU

dV= ε0

2~E2 + 1

2µ0~B2 . (11)

This result can be derived rigorously from Maxwell’sequations and the interested reader is directed to tra-ditional electromagnetism textbooks. In electrostatics,the mechanisms which generate the magnetic field ~B areturned off and one recovers eq.(10). The dependence ofdU/dV on the square of the fields is, of course, compati-ble with the fact that it is a scalar quantity, whereas fieldsare vectors. The importance of these results is that theyare local and state that the electromagnetic energy of asystem is distributed in space, as the fields are. If thereare fields at a point, there is also a certain amount of en-ergy there. This holds for all sorts of situations, rangingfrom a simple capacitor to interfering electromagneticwaves.

The electrostatic energy density, given by eq.(10), isrelated to the total potential energy of a system by avolume integral, of the form

U =∫∫∫

d V[ε0

2~E2], (12)

over the whole volume occupied by the fields. This expres-sion sheds light on the ontological differences betweenpotential energy in mechanical and electromagnetic dis-courses. In the former, one evaluates a macroscopic globalquantity, whereas the latter is more microscopic and

therefore, gives rise to a picture of the world which isboth more comprehensive and more interesting.

The full electrodynamics treatment of the subject is notsuited to basic undergraduate courses. There, it is moreconvenient to introduce eq.(10) by means of plausibilityarguments. This path is followed, for instance, in thetextbook Electricity and Magnetism, by Purcell ( [8],p.31-33), who motivates the expression for the energydensity by using the case of a charged sphere. Here,instead, we discuss the case of a plane capacitor, as inRef. [9], which is somewhat simpler.

3.1. Potential energy of a capacitor

The energy contained in the capacitor described in theprevious section, given by eq.(9), is linear in the inter-plate separation a. If its charge is kept constant andthe separation between the plates is increased to a valueb > a, still small, as in fig.1, the modulus of the electricfield remains the same as that given by eq.(8), and thepotential energy becomes

U(b) = 12Q2

S ε0b > U(a) . (13)

The energy increase ∆U = U(b)−U(a) can be ascribedto the work done by an external force ~Fext, needed toseparate the plates. Considering the upper plate to bepositively charged and fixed, the ~Fext which displaces thenegative lower plate must be, at least, equal in modulusto the electric force ~F+ which attracts it to the other.

This electric force is obtained by considering that eachcharge in the lower plate feels the electric field ~E+ dueto the upper plate. In the infinite plane approximation,this field is uniform, with an intensity given by

| ~E+| =σ

2 ε0, (14)

as illustrated in figure 2(a). All the charges of the lowerplate, in figure 2(b), are immersed in the field ~E+ and

|~F+| = Q | ~E+| = Qσ

2 ε0= Q2

2 S ε0. (15)

Thus, the work done by the external agent for sepa-rating the plates is

W = |~Fext| (b− a) = Q2

2 S ε0(b− a) = U(b)− U(a) . (16)

Figure 1: Separation of plates

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Kneubil et al. e4311-5

As expected, this result shows that the work W accountsfor the difference in the potential energies of the system.This looks like an ordinary exercise, but it prepares oneto motivate the introduction of eq.(10).

At this point, it is interesting to ask students to con-sider the proposition: if the potential energies of the twosystems, shown in fig.1, are different, this difference mustbe directly associated with physical aspects of the systemwhich did change when one moved from one configurationto the other.

In order to guide the ensuing discussion, it is worthdistinguishing the features of the system which do notchange. Within the infinite plate approximation, theseinclude: (i) the absence of electric field inside the metallicplates; (ii) the way charges are distributed over the innersurfaces of the plates; (iii) the absence of electric fieldoutside the plates; (iv) the direction of the electric field~E in the region between the plates and (v) the modulus| ~E| = Q/(S ε0) of electric field in the region betweenthe plates. Concerning the features that do change, onehas: (i) the conspicuous increase in the inter-separationdistance, from a to b, which is geometrical and (ii) thefact that this geometrical change is accompanied by thefilling of the volume S(b− a) with electric field.

This last statement relates directly the increase ofpotential energy with the existence of electric field inspace. In order to explore this relationship, we recall thatthe volume occupied by the field when one passes fromone configuration to the other is ∆V = S(b− a) and, inthe case of the plane capacitor, the volume density ofpotential energy, is given by

dU

dV= ∆U

∆V = U(b)− U(a)S(b− a) = 1

2Q2

S2 ε0. (17)

Using eq.(8), we obtain

dU

dV= ε0

2~E2 ,

which is eq.(10). Once this result is reached, it is impor-tant to tell students that it is rather general.

In order to fix the idea that electrostatic energy isdirectly associated to the electric field, one could use

Figure 2: field and force over a charge

eq.(10) to calculate again the energy of the plane ca-pacitor, given by eq.(9). Now, this can be done just bymultiplying the energy density by the volume betweenthe plates:

U =[dU

dV

][S a ] =

[Q2

2 ε0 S2

][S a ] = 1

2Q2

S ε0a , (18)

3.2. Potential energy of a sphere

The confidence of students on result (10) can be rein-forced by means of other examples. An interesting oneconcerns the self-energy of a spherical surface of radiusR, bearing a total charge Q. In order to evaluate it byreasoning in terms of force and work, as usually found intextbooks, one assumes the surface to be initially with-out charge and then begins to bring charge elements dqfrom infinity. In an intermediate situation, in which thecharge of the surface is 0 < q < Q, the total work doneby an external force which compensates the electrostaticrepulsion reads

dU = dq q

4π ε01R

(19)

and the total potential energy is

U =∫ Q

0

dq q

4π ε01R

= 12

Q2

4π ε01R. (20)

In the approach based on the energy density, deals justwith the full charge Q and writes

dU

dV= ε0

2

[Q

4πε01r2

]2, (21)

together with

U =∫∫∫

d Vε02

[Q

4πε01r2

]2

=∫ ∞

R

4πr2 drε02

[Q

4πε01r2

]2

= 12

Q2

4π ε01R. (22)

This example is interesting because it involves integra-tions, respectively over dq and dV , which convey differentontological contents. The former represents a successionof virtual actions, developed along a kind of logical time,whereas the latter involves a “static” sum over space(from R to ∞!). As these crucial differences may go un-noticed to students, it is important that they are askedto produce their own assessments regarding possible im-plications of each approach. This may open the way forepistemological discussions, based on concrete instances.

3.3. Two point-like charges

Our last example concerns the energy of two point-likecharges, already evaluated through the mechanical work

DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2017-0119 Revista Brasileira de Ensino de Fısica, vol. 39, nº 4, e4311, 2017

e4311-6 Electromagnetism as a world view: implications for the teaching of energy

in eq.(7). Potential energy, whether electrostatic or grav-itational, is usually associated with a system and notwith charges or bodies. This is so, even if sometimes onefinds loose statements concerning the potential energy ofan electron in an accelerator or the potential energy ofa stone of mass m at Earth’s surface as being mgh. Inthe latter case, for example, one knows that the body-Earth interaction remains tacit in the gravitational fieldg. Electromagnetism, however, has taught us to movebeyond these limitations.

In the case of two point-like charges q and Q, locatedat points Pq = (0, 0,−d/2) and PQ = (0, 0,+d/2), thefield ~E at the generic point P in space, described incylindrical coordinates as P = (ρ cosθ, ρ sinθ, z), is givenby

~E = ~Eq + ~EQ , (23)

with

~Eq = q

4πε0~rq

r3q

, ~rq = (ρ cosθ, ρ sinθ, z + d/2) , (24)

~EQ = Q

4πε0~rQ

r3Q

, ~rQ = (ρ cosθ, ρ sinθ, z − d/2) . (25)

According to eq.(10), the electrostatic energy density atpoint P is given by

dU

dV= ε0

2~E2 = ε0

2

[~Eq + ~EQ

]2= ε0

2

[~E2

q + ~E2Q + 2 ~Eq · ~EQ

]. (26)

The interpretation of the three terms in the bottomline of this result is quite interesting and reveals the ex-istence of something new, not foreseen in the mechanicalapproach. The first term depends only on the chargeq and remains always the same, independently of theposition, or even the very existence, of the other charge.For this reason, this term is called density of self-energyof charge q, and describes the effect this charge producesover itself. The second term is totally analogous andcorresponds to the density of self-energy of charge Q. Inthe sequence, we will come back to these contributions.

Before, we discuss the third term, which involves thefields from charges q and Q in a symmetric combina-tion and represents the density of interaction energy.One notes that the fields ~Eq and ~EQ appear entangledtogether and the density of interaction energy is given as

dUint

dV= ε0 ~Eq · ~EQ. (27)

This expression cannot be written as a sum of moreelementary contributions and represents the spatial dis-tribution of potential energy of the system composed bycharges Q and q. The potential energy of this systemcorresponds to the total interaction energy spread over

space, given by the integral

Uint =∫∫∫

d V

[dUint

dV

]=∫∫∫

d V[ε0 ~Eq · ~EQ

]. (28)

This integral can be calculated by elementary methods [9],but the calculation is both too long and too involved tobe presented here. The important point for us is that theresult reads

Uint =∫∫∫

d V[ε0 ~Eq · ~EQ

]= q Q

4πε01d, (29)

which is identical to eq.(7). One notes, however, thatthe arbitrary integration constant K, present in the me-chanical approach, is now absent. This corresponds toan important difference between the “world views” un-derlying electromagnetism and mechanics, as describedin the passage by Jammer quoted in the introduction.The constant K in the force/work approach indicatesthat this technique allows one to access just potentialenergy differences whereas, in electromagnetism, one hasa direct measure of energy.

We now come back to the self-energy terms, whichare not explicitly present in the standard derivation andyield another indication of the power of electromagnetism.Conceptually the self-energy means that a single chargealready has energy associated with its field. If this chargeis at rest, this energy is purely electrostatic. If it is moving,one should also account for magnetic contributions, asin eq.(11). Considering the case of a single charge q atrest, its full self-energy is given by

Uself =∫∫∫

d V[ ε0

2~E2

q

]. (30)

As this energy refers to a particle at rest, its meaningbecomes clear only in the framework of the mass-energyrelation defined by relativity. Following Okun [10], wewrite it as E0 = mc2, where E0 is the energy in therest frame and m is the relativistic invariant mass. Themotivations for this form of the mass-energy relationswere discussed by Kneubil [11]. As the energy Uself ispart of E0, one concludes that the electrostatic energyof a charged particle contributes to its mass.

Using the explicit form of ~Eq in eq.(24), one has

Uself =∫ ∞

R

d r1

8π ε01r2 = 1

8π ε01R, (31)

where R is the radius of the charge. In the case of point-like charges, R → 0 and Uself diverges. The history ofthis problem is rather rich and a proper discussion iswell beyond the scope of this paper. And it still hauntscontemporary quantum electrodynamics.

Our purpose here is mainly to stress that the directassociation of the electrostatic energy with electric fieldscan and should be present in introductory undergrad-uate courses on electromagnetism. It is free from thecumbersome verbal provisos of the traditional force/work

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Kneubil et al. e4311-7

approach and yields a clear physical picture, by meansof eq.(10). Moreover, it gives rise to broader and morepowerful results. In association with the magnetic field,it accounts for many other aspects of the nature, whichgo beyond the potential energy, such as the self-energiesof charged particles and the energy of electromagneticwaves. It brings, therefore, the possibility of more pro-found interpretations about electromagnetic phenomenain nature.

4. Final Discussion

We have shown that a standard electromagnetic problemcan be solved by means of two alternative approaches,which we suggest to be taught in a close sequence. Thedidactical relevance of this procedure can be understoodby making explicit some epistemological assumptions em-ployed tacitly. The inclusion of elements of epistemologyin physics teaching should not be taken as a kind ofluxury associated with the wasting of “useful” time inthe classroom. Rather, when dressed in actual physicsproblems, as in the instance discussed here, it becomesa means of bringing to students a mode of dealing withphysics which is close to what many scientists do. Theclose relationship between science and epistemology wasstressed by Einstein, in 1949 ([12], p. 684):

The reciprocal relationship of epistemologyand science is of noteworthy kind. They aredependent upon each other. Epistemologywithout contact with science becomes an emptyscheme. Science without epistemology is - in-sofar as it is thinkable at all - primitive andmuddled.

The mechanical and electromagnetic approaches topotential energy involve rather different world views. Aworld view, in Dewitt’s definition, consists of an intercon-nected set of symbolic elements which allows one to givemeaning to an entity or phenomenon that one intendsto know [13]. As a consequence, the power of isolatedideas to produce meanings for the natural world is ratherlimited. Thus, the character of net or conceptual struc-ture which underlies a world view forces one to considerthat the process of its construction involves a weavingof meanings which transcends the simple relationshipbetween an entity and its measurement.

Our suggestion that both approaches to the electro-static energy should be presented in sequence is intendednot only to enrich the acquaintance with the specificsubject, but also to explore the possibilities open forreflection about science. This is aligned with ThomasKuhn’s [14] argument that scientific formation occursdominantly through exemplary problems, which are notjust means of applying knowledge learned somehow pre-viously. The construction of the solution to a problemand its implementation amount to the very elaborationof the scientific thought.

Mechanics, with the epistemological project establishedby Newton and followers, considers interactions betweenbodies. However, the very nature of these interactions isnot discussed and phenomena are reduced to individualactions. This leads to the construction of a world viewin which interactions are mere requirements needed tojustify the existence of forces, these being the actualcause of phenomena, such as changes in the state ofmotion of bodies. This version acquires strength withthe concept of work which relies on actions of forces onbodies. Accordingly, mechanics emphasizes the parts ofa system, which may be scrutinized as individual units.

The world view based on electromagnetism, the frame-work proposed at the end of the nineteenth century,involves another epistemology. In this case, the funda-mental interest is the very nature of interactions, whichbecome the protagonists. Charges are assumed to be thesources of fields, which acquire an autonomous “reality”and become the conveyors of physical qualities. The deal-ing with fields requires a perception of totalities because,in essence, one looks for a topology in space whereby thefields ascribe vectors to each of its points.

The very concept of energy ends up being resignifiedwithin this new framework. The systemic and “holistic”character of energy derives from the mode of thought em-bodied into electromagnetism, which does not reduce atotality to its parts. Going back to the instances discussedin the previous section, the procedure for obtaining theenergy by means of the volume integral of a densitybased on the field amounts to locate this energy all overspace. However, the most interesting feature suggested bythis approach consists in realizing that energy totalitiesinvolve terms referring both to individual charges only(self-energy) and others which encompass the whole sys-tem necessarily (interaction energy). Here, in summary,the whole transcends its parts.

The instances discussed in this paper show that thedifferent solutions convey their own world views. Bothscientific and educational practices do not require neces-sarily that physics should amount to a world view andlegitimate work can be produced renouncing to it. How-ever, if one sticks to the so called ‘cultural value” ofsciences, it becomes difficult to dismiss epistemologicaldiscussions which can highlight the world view underlyinga theory. Even within this more restricted perspective,the specific situation tackled in this work, the early in-troduction of students to the picture of potential energybased on fields is welcome, since it paves the way bothto more advanced courses on electromagnetism and tothe practice of contemporary theory which relies largelyon lagrangian densities. On the other hand, when onelooks at concepts from a more refined perspective, the-ories must be suitably taught in classroom. Therefore,the understanding of electromagnetic theory can only becomplete if the solutions to its exemplary problems coulddisplay its world view in full.

DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2017-0119 Revista Brasileira de Ensino de Fısica, vol. 39, nº 4, e4311, 2017

e4311-8 Electromagnetism as a world view: implications for the teaching of energy

Acknowledgements

F.B.K. thanks the financial support by CAPES (BEX0816/15-9) and M.R.R. thanks the hospitality of the De-partment of Science Education of the University of Copen-hagen and the financial support by FAPESP (BrazilianAgencies).

References

[1] M. Jammer,. Concepts of Mass in Classical and ModernPhysics (Harvard University Press, New York, 1961).

[2] J.H. Poynting, Phil. Trans. R. Soc. Lond. 175, 343(1884).

[3] J.C. Maxwell, A Treatise on Electricity and Mag-netism (Cambridge University Press, Cambridge,1873), available from http://dx.doi.org/10.1017/CBO9780511709340, accessed May 26, 2016.

[4] O. Heaviside, The Electrician 31, 281 (1893).[5] L. Viennot and S. Rainson, International Journal of

Science Education 14, 475 (1992).[6] I. Galili, International Journal of Science Education, 17,

371 (1995).[7] C. Furio and J. Guisasola, Science Education 82, 511

(1998).[8] E. Purcell and D. Morin, Electricity and Magnetism

(Cambridge University Press, Cambridge, 2013), 3rd ed.[9] M.J. Bechara, J.L.M. Duarte, M.R. Robilotta and S.

Salem, Fısica 4, Lecture Notes University of Sao Paulo,Sao Paulo, 2014).

[10] L.B. Okun, Physics Today 42, 31 (1989).[11] F.B. Kneubil, The Epistemological Path of Knowledge

and the Mass-Energy Equivalence. Ph.D Thesis (text inPortuguese), University of Sao Paulo, 2014.

[12] P.A. Schilpp (ed.) Albert Einstein: Philosopher-Scientist(The Library of Living Philosophers, Evanston, 1949).

[13] R. DeWitt, Worldviews: An Introduction to the His-tory and Philosophy of Science (Wiley-Blackwell, Oxford,2003).

[14] T.S. Kuhn, The Structure of Scientific Revolutions (Uni-versity of Chicago Press, Chicago, 1962).

Revista Brasileira de Ensino de Fısica, vol. 39, nº 4, e4311, 2017 DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2017-0119