Abstract
Due to its fundamental role for the consolidation of Maxwell’s equations, the displacement current is one of the most important topics of any introductory course on electromagnetism. Moreover, this episode is widely used by historians and philosophers of science as a case study to investigate several issues (e.g. the theory–experiment relationship). Despite the consensus among physics educators concerning the relevance of the topic, there are many possible ways to interpret and justify the need for the displacement current term. With the goal of understanding the didactical transposition of this topic more deeply, we investigate three of its domains: (1) The historical development of Maxwell’s reasoning; (2) Different approaches to justify the term insertion in physics textbooks; and (3) Four lectures devoted to introduce the topic in undergraduate level given by four different professors. By reflecting on the differences between these three domains, significant evidence for the knowledge transformation caused by the didactization of this episode is provided. The main purpose of this comparative analysis is to assist physics educators in developing an epistemological surveillance regarding the teaching and learning of the displacement current.
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Notes
All the citations from these papers (I, II and III) refer to The Scientific Papers of James Clerk Maxwell, edited by W. D. Niven (1890).
Where a, b and c are the Cartesian components of the electric current (today we think in terms of the current density vector) and α, β, γ the Cartesian components of the magnetic force (magnetic field in actual terms). When compared with the actual representation of the components of the curl of a vector field, the different signs are due to a different orientation of the axes (x–west, z–south and y–upwards).
This is equivalent to our current notion of the divergence of a vector field.
Note that now the axes are oriented according to the actual convention.
The term field is not to be found in this work. In II Maxwell is talking about forces.
The page numbers for IV refer to Maxwell, J. C. (1878).
According to Chevallard (2007, p. 11), the admission of this very transformation of knowledge can be a traumatic process for the ones involved in teaching. In his words: “Knowledge is not a given […] it is built up, and transformed, and—such was the keyword—transposed. The wound was twofold. For some people, especially for teachers, the statement was a threat to the unconscious belief that the world of knowledge was homogeneous, isotropic and indefinitely unblemished—therefore unquestionable. To others, […] who regarded themselves as the true masters of knowledge [scholars], to them, the transposition principle came as a repudiation of their as yet unchallenged authority.
See also the term pseudo-history in Allchin (2004).
The sample of the textbooks consists mainly of traditional and well established collections (e.g. Halliday et al. 2011; Tipler 1982), the ones used as references in some of the lectures analyzed (Giancoli 2000; Serway and Jerwey 2008) as well as other famous “unorthodox” approaches such as Feynman et al. (1964) and Purcell (1985). The articles mentioned are based on a literature review we conducted (mainly in the AJP) of papers where alternative ways to justify the displacement current insertion are presented.
This expression was used by Holton (1969) to justify the didactic use of the Micheslon-Morley experiment in the teaching of Special Relativity, despite its dispensable role for Einstein’s theory.
It is important to mention that Feynman clearly states that Maxwell did not reason this way. The sequence of this passage shows Feynman’s careful position concerning assertions about Maxwell’s thought: “It was not yet customary in Maxwell’s time to think in terms of abstract fields. Maxwell discussed his ideas in terms of a model in which the vacuum was like an elastic solid. He also tried to explain the meaning of his new theory in terms of the mechanical model”.
Gauthier (1983) proposes a method that exploits the general implications of electric charge conservation and is formulated in the integral representation of Maxwell’s equations. However, we were not able to find textbooks using this method.
\( \vec{r} \) represents the distance between the wire element and the point at which the field is calculated (given that \( \vec{r}^{\prime} = 0 \)) and d l ′ is a vector whose magnitude is the length of the differential element of the wire and whose direction is equal to the conventional current.
A detailed presentation can be found in Greiner (1998).
The differential equation describing a propagation of the vector potential was derived by Maxwell, who defined A as electrokinetic momentum and his choice was \( \nabla \cdot \varvec{A} = 0, \) called afterwards Coulomb’s gauge (Buchwald 1988, p. 103).
Time reversal symmetry is considered when the variable t is substituted by –t and the quantity remains invariant or not (symmetric or anti-symmetric, respectively).
See Diener et al. (2013) for the process of obtaining these coefficients.
Here symmetry is seen as invariance of an object or system to a set of changes/transformations (Lederman and Hill 2004).
In his book, Purcell focuses on relativistic discussions from the very beginning. When analyzing the way the fields (electric and magnetic) transform from one frame of reference to another, a deep symmetry between these fields is highlighted. See Purcell (1985, pp. 235-240) for a detailed approach.
\( \nabla_{{{\mathbf{r}}^{ '} }} \times \varvec{B}^{\prime} = \frac{1}{c}\left[ {{\mathbf{v}}^{\prime} \left( {\nabla_{{{\mathbf{r}}^{ '} }} \cdot \varvec{E}} \right) - \left( {{\mathbf{v}}^{\prime} \cdot \nabla_{{{\mathbf{r}}^{ '} }} } \right)\varvec{E}^{\prime} } \right] = \frac{1}{c}\left[ {{\mathbf{v}}^{\prime} \left( {4\pi \rho } \right) - \frac{\partial }{\partial t}\varvec{E}^{\prime} } \right] = \frac{1}{c}\left[ {4\pi \varvec{J} - \frac{{\partial \varvec{E}^{\varvec{\prime}} }}{{\partial \varvec{t}}}} \right] \). Different constants are due to unit systems choice.
When asking for permission for using his lectures with research purposes, Prof. Walter Lewin explicitly demanded that his name should be mentioned. The names of the other lecturers will not be mentioned to preserve confidentiality according to ethic standards in qualitative educational research.
References
Allchin, D. (2004). Pseudohistory and pseudoscience. Science & Education, 13, 179–195.
Bork, A. M. (1963). Maxwell, displacement current, and symmetry. American Journal of Physics, 31, 854–859.
Bromberg, J. (1968). Maxwell’s electrostatics. American Journal of Physics, 36, 142–151.
Buchwald, J. (1988). From Maxwell to microphysics: Aspects of electromagnetic theory in the last quarter of the nineteenth century. Chicago: The University of Chicago Press.
Buchwald, J. (1996). Mathematics in 19th century physics. Journal of the Center for Applied Computing, 9, 41–50.
Carver, T. R., & Rajhel, J. (1974). Direct “literal” demonstration of the effect of a displacement current. American Journal of Physics, 42, 246–249.
Chalmers, A. F. (1975). Maxwell and the displacement current. Physics Education, 10, 45–49.
Chaves, A. S. (2001). Física: Eletromagnetismo. Rio de Janeiro: Reichmann & Affonso Ed.
Chevallard, Y. (1991). La transposition didactique—Du savoir savant au savoir enseigné. Grenoble: La Pensée sauvage.
Chevallard, Y. (2007). Readjusting didactics to a changing epistemology. European Educational Research Journal, 6, 9–27.
Darrigol, O. (2000). Electrodynamics from Ampère to Einstein. New York: Oxford University Press Inc.
Demtröder, W. (2009). Experimentalphysik 2: Elektrizität und Optik. Berlin: Springer.
Diener, G., Weissbarth, J., Grossmann, F., & Schmidt, R. (2013). Obtaining Maxwell’s equations heuristically. American Journal of Physics, 81, 120–123.
Duhem, P. (1902). Les Théories Électriques de J. Clerk Maxwell: Etude Historique et Critique. Paris: A. Hermann.
Duhem, P. (1991). The aim and structure of physical theory. Princeton: Princeton University Press.
Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman lectures on physics (Vol. 2). Reading, MA: Addison-Wesley.
French, A. P. (2000). Is Maxwell’s displacement current a current? The Physics Teacher, 38, 274–276.
French, A. P., & Tessman, J. R. (1963). Displacement currents and magnetic fields. American Journal of Physics, 31, 201–204.
Gauthier, N. (1983). Displacement current, transport current, and charge conservation. American Journal of Physics, 51, 168–170.
Giancoli, D. C. (2000). Physics for scientists and engineers (Vol. 2). Upper Saddle River, New Jersey: Prentice Hall.
Greiner, W. (1998). Classical electrodynamics. New York: Springer.
Halliday, D., Resnick, R., & Walker, J. (2011). Fundamentals of physics. London: Wiley.
Holton, G. (1969). Einstein, Michelson, and the ‘crucial’ experiment. Isis, 60, 133–197.
Kuhn, T. S. (1963). The function of dogma in scientific research. In A. C. Crombie (Ed.), Scientific change (pp. 347–369). New York and London: Basic Books.
Lederman, L., & Hill, C. (2004). Symmetry and the beautiful universe. Amherst: Prometheus Books.
Maxwell, J. C. (1878). A treatise on electricity and magnetism (Vol. 2). Oxford: Clarendon Press.
Meissner, H. (1964). Demonstration of the displacement current. American Journal of Physics, 32, 916–918.
Mello, P. A. (1972). A remark on Maxwell’s displacement current. American Journal of Physics, 40, 1010–1013.
Niven, W. D. (1890). The scientific papers of James clerk Maxwell. (Vol. 1). Cambridge: Cambridge University Press.
Poincaré, H. (1958). The value of science. New York: Dover publications.
Purcell, E. M. (1985). Electricity and magnetism (Berkeley Physics Course, Vol. 2). New York: McGraw-Hill.
Rizotto, M. G. (1999). Visualizing displacement current—A classroom experiment. Physics Teacher, 37, 398.
Serway, R. A., & Jerwey, J. W, Jr. (2008). Physics for scientists and engineers with modern physics (7th ed.). Belmont: Thomson.
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15, 4–14.
Siegel, D. M. (1991). Innovation in Maxwell’s electromagnetic theory: Molecular vortices, displacement current, and light. Cambridge: Cambridge University Press.
Steiner, M. (1998). The applicability of mathematics as a philosophical problem. Cambridge: Harvard University Press.
Tipler, P. A. (1982). Physics (2nd ed.). Belmont: Worth.
Weber, T. A., & Macomb, D. J. (1989). On the equivalence of the laws of Biot-Savart and Ampère. American Journal of Physics, 57, 57–59.
Whitaker, M. A. B. (1979). History and quasi history in physics education part 1. Physics Education, 14, 108–112.
Whittaker, E. T. (1910). A history of theories of Aether and electricity. Dublin: Longman, Green and Co.
Yano, A. F. (1968). Is charge conservation needed to derive the displacement current? American Journal of Physics, 36, 599–601.
Acknowledgments
The financial support provided by the Alexander von Humboldt Foundation to Ricardo Karam (postdoctoral fellowship) is greatly acknowledged. The authors are very grateful to the reviewers for their constructive suggestions and to the four professors who kindly allowed us to use their lectures for research purposes.
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Karam, R., Coimbra, D. & Pietrocola, M. Comparing Teaching Approaches About Maxwell’s Displacement Current. Sci & Educ 23, 1637–1661 (2014). https://doi.org/10.1007/s11191-013-9624-3
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DOI: https://doi.org/10.1007/s11191-013-9624-3