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The London Equation Università degli Studi di Palermo Tesi di Dottorato SUPERCONDUCTIVITY EXPLAINED WITH THE TOOLS OF THE CLASSICAL ELECTROMAGNETISM Educational path for the secondary school and its experimentation Dottorando Sara Roberta Barbieri Tutor Coordinatore Prof. Claudio Fazio Prof. Aldo Brigaglia Co-Tutor Marco Giliberti SSD: Fis08 Università degli Studi di Palermo - Dipartimento di Fisica e Chimica Corso di Dottorato di Ricerca in Storia e Didattica delle Matematiche, della Fisica e della Chimica (XXIV Ciclo) – 2013 Università degli Studi di Palermo Tesi di Dottorato SUPERCONDUCTIVITY EXPLAINED WITH THE TOOLS OF THE CLASSICAL ELECTROMAGNETISM Educational path for the secondary school and its experimentation Dottorando Sara Roberta Barbieri Tutor Coordinatore Prof. Claudio Fazio Prof. Aldo Brigaglia Co-Tutor Marco Giliberti SSD: Fis08 Università degli Studi di Palermo - Dipartimento di Fisica e Chimica Corso di Dottorato di Ricerca in Storia e Didattica delle Matematiche, della Fisica e della Chimica (XXIV Ciclo) – 2013 Physics is such stuff as dreams are made on Contents Abstract 1 1. Introduction 3 2. The Higgs boson and the superconductivity 7 2.1. The Lagrangian formalism for the description of a system . .7 2.1.1. An example: the conservation of the linear momentum . .8 2.1.2. The importance of the theoretical framework . .9 2.2. The characterization of the unknown interactions . 10 2.2.1. Experimental hints for the description of the strong interaction 10 2.2.2. Theoretical hints for the description of the strong interaction . 11 2.3. The problem of the particle masses in the Standard Model . 14 2.3.1. Short range interactions are mediated by massive quanta . 15 2.3.2. The screening currents and the generation of the photon mass 16 2.4. The Higgs mechanism . 18 2.4.1. The chiral symmetry and the fermion masses . 18 2.4.2. The spontaneous breaking of the symmetry . 19 2.4.3. The Higgs idea . 21 3. Some basic superconductivity for teachers 23 3.1. Resistivity: few words . 23 3.2. Superconductivity . 26 3.3. Magnetic field expulsion (Meisser effect) . 30 3.4. Few elements of thermodynamic of superconductors . 32 3.5. A first phenomenological description: the London equation . 34 3.6. Something more on magnetic fields inside type I and type II super- conductors . 37 3.7. Flux quantization . 39 4. Superconductivity at the secondary school, a proposal 41 4.1. Overview . 41 4.2. The Model of the Educational Reconstruction of the Contents . 41 4.3. Schema of the basic educational path on superconductivity . 43 4.4. Schema of the complete educational path on superconductivity . 44 i Contents Contents 5. Electrical conduction in secondary school 47 5.1. Overview . 47 5.2. A Design-Based Research for the development of the educational path 48 5.3. The initial path proposed to high school students . 49 5.3.1. Overview of the phenomenological description in terms of local quantities E and J ........................ 49 5.3.2. Microscopic picture of the electrical conduction . 51 5.4. The Conceptual Change Method as a framework in which we analyze results . 59 5.5. The first step of the experimentation . 62 5.5.1. Quantitative results of the first step . 63 5.5.2. Qualitative results of the first step . 71 5.5.3. Conclusions about the first step . 73 5.6. The second step of the experimentation . 73 5.6.1. Quantitative results of the second step . 74 5.6.2. Qualitative results of the first step . 79 5.6.3. Conclusions about the second step . 86 6. A path on electrical conduction 89 6.1. Overview . 89 6.2. Description of the electrical current by the electric field . 90 6.2.1. The experimental evidence of dissipation in electrical circuits . 91 6.2.2. A relation between the electric field and the current density . 92 6.2.3. The electric field in electrical circuits . 92 6.2.4. The e.m.f. of a circuit . 97 6.3. Resistors in series and in parallel . 100 6.3.1. A useful representation for resistors . 100 6.3.2. Resistors in series . 103 6.3.3. Resistors in parallel . 105 6.3.4. An example of circuit with resistors in series and parallel . 109 6.4. Phenomenology of the electrical current: syntesis of the path . 113 7. Essential lines of the magnetic vector potential 117 7.1. Overview . 117 7.2. The concepts of field and potential . 117 7.2.1. The concept of field . 117 7.2.2. The Maxwell’s equations in terms of potentials . 119 7.2.3. A new approach for the introduction of the vector potential . 123 7.2.4. The physical meaning of the magnetic vector potential . 126 7.2.5. A stimulating hint . 128 8. The magnetic vector potential in secondary school 129 8.1. Overview . 129 ii Contents 8.2. The concept of potential . 129 8.2.1. The definition of potential . 130 8.2.2. The gravitational potential . 131 8.2.3. The general scheme . 133 8.2.4. Examples of potential . 134 8.3. The magnetic vector potential for secondary school students . 137 8.3.1. Motivations to face the magnetic vector potential . 137 8.3.2. A definition of the vector potential in analogy with the scalar potential . 138 8.3.3. A definition of the vector potential from the Maxwell’s treatise 141 8.3.4. A new formulation for the electromagnetic induction using the vector potential A ........................ 143 8.4. Physical meaning of the vector potential . 144 8.5. Examples of determination of the magnetic vector potential . 145 8.5.1. Vector potential of a solenoid carrying a steady current . 145 8.5.2. Vector potential of two parallel planes carrying opposite currents147 8.5.3. Vector potential of a wire carrying steady current . 151 8.6. Vector potentials: starting from current distributions or from mag- netic fields . 154 8.7. The vector potential: synthesis of the path . 154 9. Experimentation of the vector potential in secondary school 157 9.1. Overview . 157 9.2. The electric and the magnetic fields for a current-carrying wire . 157 9.3. The flux of a vector field . 160 9.4. The circulation of a vector field . 161 10.An experimental lab on superconductivity for secondary school 165 10.1. Overview . 165 10.2. Electromagnetic induction lab . 165 10.3. Measure of the critical temperature . 167 10.4. Observations of superconductors in interaction with magnets . 168 11.An educational path on superconductivity for secondary school 171 11.1. Overview . 171 11.2. Main guidelines of our proposal . 171 11.3. The two fluid model to explain superconductivity . 173 11.3.1. A description of the superconductive fluid . 174 11.3.2. The Meissner effect for secondary school students . 176 11.4. The validity of the London equation . 181 11.4.1. Experimental evidences of the penetration of the magnetic field in the bulk of the superconductor . 181 11.4.2. Hints for the theoretical explanation of the fluxoids in a su- perconductor . 188 iii Contents Contents Bibliography 193 Acknowledgments 199 A. Test on Electrical Conduction 201 B. Pre-test on Electrical Conduction 205 C. Post-test on Electrical Conduction 209 D. Test on the basic electromagnetism 211 E. Lab sheet: the Meissner effect 215 F. Homework on superconductivity 219 iv Abstract The work of this thesis describes an educational path for the presentation of su- perconductivity at the secondary school, together with the experimentation of some particular parts of the path with high school students. The educational path that we have developed is mainly centred on the phenomeno- logical aspects of superconductivity and has been inspired by the two fluid theory of Gorter and Casimir (1934) where a superconductor is seen as a material in which two fluids are present, the normal fluid described by the Ohm’s laws, and the su- perconductive fluid descibed by the London equation (1935). Both the Ohm’s laws and the London equation give respectively the phenomenological descriptions of the conductive and the superconductive fluids. There is no reference to the microscopic structure of the conductor or of the superconductor. For this reason we have chosen these phenomenological models as the framework in which we contextualized our work and the reconstruction of the contents of super- conductivity. In order to develop these phenomenological models in secondary school, we have reconstructed the contents of the electrical conduction and we have subsequently developed a formal analogy between conduction and superconduction from a math- ematical point of view, by means of the introduction of the London equation. There- fore a very conspicuous part of this work pertains the presentation of the London equation, for whose writing it is necessary the introduction of the magnetic vector potential. Being the magnetic vector potential never treated in secondary schools, its intro- duction has required a lot of reconstruction work, mainly because the mathematics generally used in the literature for the vector potential is based on differential cal- culus, with operators as divergence and curl, that are really beyond the knowledge and the possibilities of most of the secondary school students. We have described the magnetic vector potential in analogy with the electric scalar potential, always present in the secondary school curriculum, we have used a mathe- matical formalism suitable for the level of the students, and we have realized how the introduction of this new, and somewhat complicated object at first glance, can be not only a suitable way for presenting superconductivity at secondary school level, but a stimulus for a deeper understanding of the electromagnetism as well. Once that the needed prerequisite are introduced and diffusely discussed, we could afford the actual description of the phenomenological characteristic of superconduc- 1 tivity.
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