High school students face the magnetic vector potential: some relapses in their learning and tips for teachers dealing with electromagnetism

Transcription

1 High school students face the magnetic vector potential: some relapses in their learning and tips for teachers dealing with electromagnetism Sara R. Barbieri and Marco Giliberti University of Milan, Italy. Abstract Although the magnetic vector potential is currently treated at university level in electromagnetism and modern physics courses, in secondary school it is never introduced. On the contrary, a quite important place is given to the electric scalar potential, especially in connection with the description of electrical circuits, thus creating a certain asymmetry in the didactical presentation of the electric and magnetic fields. Here we present the results of an experimentation carried out with 25 secondary school students to whom we have introduced the magnetic vector potential using the integral tools of circulation and flux, in close analogy with the usual presentation of the electrostatic scalar potential in secondary school. Since these mathematical tools are the same that students use for writing Maxwell s equations, our strategy for the introduction of the vector potential is also useful to strengthen and improving students learning of more common and basic concepts in electromagnetism. Besides a description of what can be done in classroom work, here we also present some considerations for teachers to further motivate the introduction of vector potential in secondary school. In particular we deal with some opportunities provided by a re-writing of the Maxwell s equations in terms of potentials. Therefore, we outline connections between electromagnetism and special relativity, so to highlight the importance of vector potential in addition to the traditional scalar potential in a modern presentation of electromagnetism at high school level. Keywords Electromagnetism, vector potential, secondary education. Introduction The idea of treating the vector potential A in secondary school has come out when we decided to deal with superconductivity. In order to keep mathematical consistence between the physical experiments performed in lab-room on superconductivity and their theoretical explanation, we found that the vector potential could be a very important tool if one does not want to give students only popular explanations. Although secondary students lack of the standard mathematical background of differential operators, generally needed to face the definition of magnetic vector potential, we found it possible to introduce the vector potential by means of integral notions, thus remaining in the standard secondary school mathematical formalism, already used in electromagnetism. Besides its application to give a phenomenological explanation of superconductivity, we found that the importance of A is great in itself, not only for advanced physics university courses, but

2 also for a deeper understanding of secondary school physics. In fact, magnetic vector potential, together with the electric scalar potential, can shed light on some interesting, though simple, problems of electromagnetism, open many didactical opportunities towards relativity and quantum physics and give many hints for discussing the meaning of physical quantities. Our aim in this work is double: from one hand, we would like to show that it is worth trying to include vector potential into secondary school curriculum, and on the other hand we will delineate some reflections for teachers about some fundamental concepts pertaining electromagnetism and symmetry. In fact, despite its shortness, an important leitmotif of this paper is the concept of symmetry, that should be recognized as one of the most useful guides, not only in physics itself, but also in physics education. Experimentation at high school on the vector potential We present here some of the result of an experimentation that we carried out with 25 high school students attending their last year (13 th grade) of a scientific high school. The curricular teacher gladly accepted to enrol their students in the proposed experimentation, even though he thought they had a very poor disciplinary preparation: students were used to a basically mnemonic study and unable of solving very simple text-book problems (they even found difficulties in substituting numbers into a given mathematical formula). Nevertheless, students were all aware of their poor disciplinary preparation of which, even, often nicely apologized with the experimenter throughout the lectures of the path. We monitored students learning process with oral interviews before and during the path and with two written tests. The two tests, that were given to students before and after the path on vector potential, were identical and did not contain any explicit question regarding the vector potential itself, but investigated students competencies on basic concepts of electromagnetism and its mathematical formulation. Students competencies about vector potential have been investigated later with other tests and interviews, that for brevity we cannot report here. Students had already treated the Maxwell s equations with their curricular teacher, for this reason we could give a pre-test with questions about the electric and the magnetic fields and the concepts of flux and circulation of a vector field. In the following, we present some of our results: for each topic, besides the results of the written tests, we also show some excerpts from oral interviews that have been made before the sequence on the vector potential. Electric and magnetic fields Excerpts from oral interviews. (T stands for teacher, S stands for student) T: What generates a magnetic field? S 1 : A charge? T: Is there a magnetic field in this room? S 1 : I don t know T: What could you do to answer? S 1 : I don t know T: If you consider a circuit with a LED connected to a battery, which are the fields involved in this case?

3 S 2 : If there is a circuit, then there will be a magnetic field, but if the wire of the circuit is insulated by some rubber, the magnetic field doesn t go outside otherwise our houses would be filled with magnetic fields. Question 1 proposed in the written tests: You have a very long wire carrying a direct current. Describe the fields present inside and outside the wire Results are summarized in Tab.1 for what concerns the electric field and in Tab.2 for what concerns the magnetic one. Table 1. Categorization of students conception about the electric field (from the answers to Question 1) with the percentage of students belonging to each category, pre-test and post-test. Categorization about the electric field E PRE POST An electric field is present inside the wire 28% 43% No electric field is present outside the wire 0% 6% Percentage of students answering the question 64% 91% Table 2. Categorization of students conception about the magnetic field (from the answers to Question 1) with the percentage of students belonging to each category, pre-test and post-test. Categorization about the magnetic field B PRE POST There are circular magnetic field lines around the wire 38% 93% A magnetic field is present inside the wire 0% 0% Percentage of students answering the question 64% 91% Tab.1 points out a very common misconception in the pre-test: all students believed that no electric field is present outside a current carrying wire [Jefimenko, 1962]. Moreover only 72% of them retained it is not present even inside the wire. These ideas are probably due to the fact that, in teaching, electric fields are presented almost in connection with electrostatic, while in dealing with circuits, the most important concept involved is that of the potential difference. From Tab.2 a similar (but reversed) situation emerges about the magnetic field: in the pre-test no one think it is present inside the wire and only 38% knows that it encloses the wire itself. Oral interviews also show great difficulties in describing electric and magnetic field in common situations. Results coming from the post-test show that the experimentation on the vector potential has somewhat sorted out students ideas only about the electric field inside the wire and the magnetic field outside the wire. A possible reason could be the brevity of the sequence on the vector potential that lasted only 5 hours while, probably, students needed much more time. Flux of a vector field Excerpts from oral interviews. T: What surface can I refer to when I want to determine the flux of a river? S: To the surface of the river that is to its higher part T: So, is that surface a part of the river?

4 S: Yes, it is! T: And what do you imagine when I say the water flows THROUGH a surface? S: I imagine some water moving on the surface, in a lot of different directions. Question 2 proposed in the written tests: You have a uniform vector field v and two different open circular surfaces of area S, as represented in the following figure (a). In (b) the surface is inclined by 30 respect to the horizontal. Find the flux of the vector field v through the surfaces. Results are summarized in Tab.3 for what concerns the flux through the surface (a), and in Tab.4 for what concerns the flux through the surface (b). Table 3. Categorization of students ability in calculating fluxes (from the answers to Question 2(a)) with the percentage of students belonging to each category, pre-test and post-test. Categorization about the flux in case (a) PRE POST Φ(v) = BS 57% 90% Φ(v) = BS cos(90 ) 36% 5% Table 4. Categorization of students ability in calculating fluxes (from the answers to Question 2(b)) with the percentage of students belonging to each category, pre-test and post-test. Categorization about the flux in case (b) PRE POST Φ(v) = BS cos(30 ) 71% 71% Φ(v) = BS cos(60 ) 24% 16% It might be interesting to observe that, as reported in Tab.3 and Tab.4, in answering the questions all students always wrote B instead of v for the vector field, even though in the text and in the picture they can always read v. This is likely due to the fact that they were used to calculate only fluxes of the magnetic field We can also observe that, after the sequence, students took more care in dealing with the flux of a uniform field. The circulation of a vector field Excerpts from oral interviews. T: For homework you were asked to recall the definition of circulation of a vector field? Did you do it, did look for the definition of circulation of a vector field? S 1 : The Maxwell s equation? T: I think that maybe you are confusing a definition with a theorem involving the circulation

5 T: What is the meaning of performing the circulation integral? S 2 : the evaluation of an area S 3 : Please, would you explain again what the circulation is? Question 3 proposed in the written tests: You have a uniform vector field v and two different closed square paths of side L. In (b) the square is rotated of 45 as represented in the picture. Find the circulation of the vector field v along the two paths. Results are summarized in Tab.5 for what concerns the circulation along the path (a) and in Tab.6 for what concerns the circulation along the path (b). Table 5. Categorization of students ability in calculating circulations (from the answers to Question 3(a)) with the percentage of students belonging to each category, pre-test and post-test. Categorization about the circulation in case (a) PRE POST C(v) = 0 0% 76% Percentage of students answering the question 19% 81% Table 6. Categorization of students ability in calculating fluxes (from the answers to Question 3(b)) with the percentage of students belonging to each category, pre-test and post-test. Categorization about the circulation in case (b) PRE POST C(v) = 0 0% 76% Percentage of students answering the question 33% 76% Although Maxwell s equations had been presented to students in their integral form, therefore making abundant use the concept of circulation, students were not at all comfortable with this concept. At first, the use of circulation caused some discouragement in students but, as they realized that it is strictly related to conservative forces and work, they were stimulated enough to participate in class reasoning. For example, students found it very interesting reviewing the old and nearly forgotten physical quantity work as a line integral (they had never before defined work in such a complicated mathematical and way). Therefore, the percentage of students answering the questions in the post-test was incredibly enhanced. Besides previous ones, we proposed also another question on circulation, as reported below. Question 4 proposed in the written tests: You have a situation very similar to that of Question 2, except for the different vector field: in this case the field changes verse in correspondence of the dashed line as, reported in figure.

6 Results are summarized in Tab.7 for what concerns the calculation of circulation along the path (a) and in Tab.8 for what concerns its calculation along the path (b). Table 7. Categorization of students ability in calculating circulations with the percentage of students belonging to each category, pre-test and post-test. Categorization about the circulation in case (a) PRE POST C(v) = 2Lv 0% 38% Students that answer 14% 57% Table 8. Categorization of students ability in calculating circulations (from the answers to Question 4(b)) with the percentage of students belonging to each category, pre-test and post-test.. Categorization about the circulation in case (b) PRE POST C(v) = 2 2Lv 0% 38% Students that answers 24% 57% From Tab.7 and Tab.8 we can notice an improvement in the number of answering students, in their capabilities in facing the new concept of circulation, and in doing calculations. Reflections for teachers: the importance of the vector potential Since this section is devoted to teachers, we feel free to use differential operators (curl and divergence) instead of only integral operators (flux and circulation) as we did in the previous sections. Our aim is to highlight some features of electromagnetism and add reasons to the claimed importance of vector potential in secondary school physics teaching [see also Barbieri et al., 2013]. Maxwell equations in terms of potentials With standard symbols, Maxwell s equations in terms of the fields can be written as: ρ E = ε 0 E B = µ 0J + ε 0µ 0 t B = 0 B E = -. t (1)

7 Now: B = 0 A B = A, (2) we call A the magnetic vector potential. We observe that it is implicitly defined by a simply vector property. Substituting eq.(2) into the last of eq.(1) we get: A A E = - ( A) E + = 0 φ E + = φ. (3) t t t We call ϕ the electric scalar potential (also ϕ is then implicitly defined by a simple vector property). In terms of the scalar and the vector potentials the first two of eq.(1) (those containing the sources) after some simple calculations become: 2 1 φ 2 ρ φ = 2 2 c t ε 0 (4) 2 1 A 2 A = µ 0J, 2 2 c t where the Lorenz gauge condition: φ A + ε 0 µ 0 = 0 (5) t has been imposed. So, when written in terms of the potentials A and φ with the condition (5), the Maxwell s equations are decoupled and have exactly the same mathematical form. Therefore, also their solutions will have the same form. In particular: 1 ρ ( ) ( r', t) dv ' φ r, t =, (6) 4πε r r' 0 V ' that can also be seen as a definition of the electric scalar potential (especially at high school where it is generally given in a slightly more simplified version) and: (, t) A r µ 0 = 4π V ' ( r', t) J dv '. r r' Equations (4), (5), (6) and (7) put clearly in evidence that the charge density ρ and current density J are the sources both of the fields and of the potentials. Magnetic vector potential for high school students Looking at the Maxwell s equations in terms of potentials, it clearly appears that the two potentials have the same importance and play a similar role in electromagnetism; therefore we can safely say that the asymmetry in the didactical consideration of A and φ has no physical reasons. We could also, jokingly, say that disregarding the vector potential is equivalent to choose to present only one half of the electromagnetism, a priori. Evidently, differential operators are not suited for secondary school students, but this mathematical problem can be overcome [Barbieri et al., 2013]. In this work we can t linger on this point but, during our experimentations with secondary school students, we have effectively introduced the vector potential in close formal analogy with the scalar potential, by means of a simplified version of eq.(7), in the case of slowly varying vector fields [Barbieri et al., 2014]. (7)

8 Nonetheless, in the following will give some hints for the introduction of the vector potential by means of mathematical tools suited for upper secondary school students. The two key points of the presentation for students are (a) a definition of the magnetic vector potential and (b) a property to allow students the determination of the vector potential in simple physical situations (similar to those for the determination of the magnetic field). For what concerns the point (a) we give a definition of the vector potential in analogy with the definition of the scalar potential usually known to students. This approach is useful also for a better understanding of the definition of scalar potential. In general secondary school students define the electric scalar potential for a discrete (and finite) distribution of charges. Therefore, indicating the scalar potential by φ, and the N charges considered by Q 1, Q 2 and Q N, with obvious meaning of the symbols, students write:. (8) With the aim of introducing the vector potential, first students will generalize the definition (8) in the case of continuous distributions of electric charges, and then, they will write an expression for the vector potential in formal analogy with the expression of the scalar potential. The conceptual difficulty is that students have to recognize that the electric currents are the sources of the vector potential as well as the electric charges are the sources of the scalar potential. This is an important point, that should be deeply discussed with students. Once this part is carried out, it is possible to write the definition of vector potential:. (9) Actually, recalling eq. (7) we immediately see that the formal mathematical analogy has a precise physical justification, and it can be encouraging for teachers that would deal with this topic with their students. By means of eq. (9) students may have an explicit definition of vector potential (that is a definition in the form A=, that appears very comfortable compared with B = A, in which the vector potential is implicitly defined). Moreover, eq. (9) gives an empirical referent that allows students to have a picture of the vector potential, in fact they can guess the behaviour of the vector potential by the behaviour of the currents, at least for the easier current distributions. The definition (9) of vector potential is instead useless in order to determine the mathematical expression of the vector potential, given the expression of the current distributions. It is for this reason that we developed the second key point (b) of the students presentation of the vector potential. Let us imagine to have a magnetic field B, in a certain region of space, a closed path γ, and two surfaces and that have g as a boundary. and are such that the surface is a closed surface. From the solenoidality of the magnetic field, we have that the flux of B through the closed surface is zero, that is:. (10)

9 If then, we choose the normal vectors to the surfaces in such a way to give the same orientation for and with respect to the orientation of γ, we can rewrite eq.(10) as:, (11) But and are two surfaces arbitrarily chosen. Therefore, following an argumentation used by Maxwell in his treatise, it must be possible to determine the flux of the magnetic field through an open surface, that has γ as a boundary, by a process that involves only the closed path γ and that does not involve the surface itself (in fact, the flux of the magnetic field through a surface bounded by a closed line cannot depend on the surface, that can be varied in infinite ways, but can depend only on its boundary). At this point Maxwell introduces a vector A (the vector potential), a vector whose circulation along the closed path γ could provide the flux of the magnetic field through the open surface that has γ as a boundary. In formulas:, (12) with the oriented element of the closed line γ and the oriented element of the surface. It is possible to rewrite eq.(12) in a more concise formula:. (13) Eq.(13) can be seen as another definition for the vector potential. It is possible to demonstrate that if the fields involved are slowly varying on time, the two definitions are equivalent. The theorem is beyond the level of a secondary school, but it can be stated to students without demonstration, thus allowing them to have a framework coherent and complete of the problem. For students, instead, eq.(13) can be very useful to determine the vector potential in some simple cases of current distributions or for some simple magnetic fields. The property of eq.(13) gives a relation between the vector potential A and the magnetic field B, therefore it follows that if the problem gives a distribution of currents, the solution will pass from an initial determination of the magnetic field generated by the particular current distribution given. Many examples can be found in [Barbieri et al., 2014]. We would also like to stress that one of the most common obstacles to the introduction of the vector potential is the claim that it has no physical meaning. But this claim is false. In fact, it is worth noticing that the usual physical meaning given to the scalar potential (that is the energy needed to transport a charge from the infinity to a certain point, divided by the charge itself) holds only for a quasi-static electric field, and not in general. The same is true for the magnetic vector potential that, for a slow varying magnetic field, can be given the meaning of momentum per unit charge [Giuliani, 2010] and [Barbieri et al., 2013]. Relativity as the natural context for electromagnetism Let us to return again to eq.(4). The identical mathematical structure of the two equations invite us to define two new four-vectors, the four-potential Aµ and the four-current Jµ as follows: A = φ, A, A, A µ ( ) ρ J,,, µ = J x J y J z. ε 0 The two equations (2) can now be combined into just one covariant equation: x y z (14)

10 2 1 Aµ 2 A 2 2 µ = J µ c dt This kind of formalism is certainly not suitable for students. But it is just an example of the deep connections between electromagnetism and special relativity. Our opinion is that teachers should more deeply link electromagnetism with relativity and we believe that the magnetic vector potential could be used to trigger off students interest and help them in founding out those connections. Here below, another example (immediately suitable for students) to show that in electromagnetism Galilean relativity is not enough, even at low velocities. Let us consider a direct current carrying wire (A) and a beam of negative charges (B) having the same velocity of the negative current carriers in the wire, as represented in Fig.1.. (A) (B) (15) Figure 1. Hypothetical experimental situation: a current carrying wire (A) and a beam of charges (B). In the rest frame of the wire (A), we have two parallel currents with the same verse and therefore, due to magnetic interaction, an attracting force will appear between the wire and the charges. Instead, in the rest frame of the beam of charges (B) we see only one current, the one given by the motion of the positive charges of the crystal lattice of the wire that is moving with the velocity v D (A) (see Fig.1). In this frame of reference there won t appear forces: no electric forces, because the wire is neutral, and no magnetic forces, because there are no charges moving in the magnetic field generated by the moving positive charges of the lattice (A). We thus have two inertial frames describing a completely different physics. This is a contradiction that cannot be overcome in the Galilean relativity context [Chinnici, 2013]. We suggest teachers to present many other examples of this kind, in order to describe to their students why something beyond Galilean relativity is needed to treat electromagnetism, even avoiding the formalism of special relativity that may appear too complicated in some cases. The problem of the gauge invariance A very interesting and important question is given by the gauge fixing. In fact, for obtaining eq.(4) we made a particular gauge choice: the Lorenz gauge, given by eq.(5) [Barbieri et al., 2013]. Other choices are obviously possible; they give rise to different A and ϕ fields, for given E and B, and to more complicated equations, different from eq.(4). For the existence of these possibilities, we found that students can be easily induced to think that the vector potential is not a well-defined physical quantity, or that it is quite a weird mathematical tool lacking of a real physical meaning. The problems that come from the gauge invariance of electromagnetism and its connections with the physical meaning of potentials are really delicate, and particular care must be taken not to confuse secondary school students. But this is an opportunity, since some reflections on the physical meaning of the vector potential can also stimulate important reflections on the concept of physical quantities in general.

11 Our efforts are markedly focused in giving students the opportunity of reflecting on these aspects starting from a reversed point of view. For this reason, in our secondary school path (in the slowly varying field approximation) the vector potential is defined by eq.(7), thus it is univocally given by the current distribution. Instead, the integral property, stated for the first time by Maxwell [Maxwell, 1873] in order to define the magnetic vector potential, that links the vector potential A to the magnetic field B: A dl = B n da, (16) L where L is the boundary of the open surface S, is just a theorem, useful to calculate vector potentials from a given magnetic field. Therefore, it is very instructive to calculate the vector potential using eq.(16) starting from simple given magnetic fields, but it is precisely from the use of eq.(16) that students may experience the problem of the gauge fixing. An example can clarify what we are saying. Let us consider a uniform magnetic field [Barbieri et al., 2013]. If students are asked to find out the field A, they can easily see that the particular choice of the closed path L breaks the spatial symmetry of the uniform magnetic field and determines the particular shape of the vector potential. Since the symmetry may be broken in infinite ways, one can get infinite different vector potentials from a given magnetic field B. But, at this point, students can return to the definition of A given by eq.(7): in fact, they know that the vector potential is univocally determined by the currents. This is a very important step for the comprehension since it is possible to understand that the different As (given by the different ways in which the symmetry can be broken, and arising from the different chosen Ls) are related to the different current distributions that can generate the same given magnetic field B (in this case a uniform field). The gauge freedom of the magnetic vector potential is only a particularly evident indication of an aspect that is common to many other physical quantities. Whilst only gauge-invariant quantities have physical meaning, nonetheless in our physical description we make use of many non-invariant fundamental quantities. For instance: position is univocally defined only relatively to a fixed reference frame; kinetic energy is defined only once the velocity is defined, that again depends on the reference frame; and the potential energy is univocally defined when not only a reference frame has been chosen, but when also the zero point of the potential has been fixed. These simple and well-known examples show that many physical quantities, although without possessing an absolute existence, as they strongly depend on some choices of ours, nonetheless maintain an important relative physical meaning; and no one could think to get rid of them. Conclusions Secondary school teachers could consider the vector potential too difficult an issue to be faced by their students. That idea usually comes out from the fact that the textbooks do not contain this topic and, in general, teachers themselves are not familiar with it. On the contrary, we believe that the vector potential is of great help in secondary school physics teaching. In fact, the outcomes of our (only 5-hours lasting) experimentation on vector potential show that some deep difficulties with basic concepts of electromagnetism, such as those of circulation and flux and some understanding of the notions of electric and magnetic fields have, at least partially, been overcome. Moreover we believe that magnetic vector potential can be a great stimulus in order to reflect, solve problems and shed light on electromagnetism. Also S

12 many different branches of modern physics, i.e. superconductivity [Barbieri et al., 2014], may be more simply faced using the vector potential. Moreover, the vector potential provides a direct way to connect electromagnetism with special relativity and to reflect about the meaning both of the physical quantities and physical laws, even at secondary school level. References Barbieri, S., Cavinato, M. and Giliberti M. (2013). An educational path for the magnetic vector potential and its physical implications, European Journal of Physics. 34, Barbieri, S. (2014). Superconductivity explained with the tools of the classical electromagnetism, PHD thesis, Corso di dottorato in Storia e Didattica delle Matematiche della Fisica e della Chimica, XXIV. Barbieri, S., Cavinato, M. and Giliberti M. (2013). Riscoprire il potenziale vettore nella scuola superiore, Il Giornale di Fisica. LIV(2), Chinnici, G. (2013). Horror absoluti, Ti Pubblica, Vignate (Milano ), 143, 144. Giuliani, G. (2010). Vector potential, electromagnetic induction and physical meaning, European Journal of Physics. 31, Jefimenko, O. D. (1962). Demonstration of the electric fields of current-carrying conductors. American Journal of Physics. 30, Maxwell, J. C. (1873). A Treatise on Electricity and Magnetism, Oxford Clarendon Press. Vol.II, 405. Sara R. Barbieri Department of Physics University of Milano Via Celoria, Milano, Italy sararoberta.barbieri@unimi.it