In J.D. Jackson's Classical Electrodynamics textbook, the analysis of Dirac's charge quantization condition in the presence of a magnetic monopole has a mathematical omission and an all-too-brief physical argument that might mislead some students. This paper presents a detailed derivation of Jackson's main result, explains the significance of the missing term, and highlights the close connection between Jackson's findings and Dirac's original argument. (C) 2016 Sharif University of Technology. All rights reserved.

The issue of eddy currents induced by the rapid switching of magnetic field gradients is a long-standing problem in magnetic resonance imaging. A new method for dealing with this problem is presented whereby spatial harmonic components of the magnetic field are continuously sensed, through their temporal rates of change, and corrected. In this way, the effects of the eddy currents on multiple spatial harmonic components of the magnetic field can be detected and corrections applied during the rise time of the gradients. Sensing the temporal changes in each spatial harmonic is made possible with specially designed detection coils. However to make the design of these coils possible, general relationships between the spatial harmonics of the field, scalar potential, and vector potential are found within the quasi-static approximation. These relationships allow the vector potential to be found from the field – an inverse curl operation – and may be of use beyond the specific problem of detection coil design. Using the detection coils as sensors, methods are developed for designing a negative feedback system to control the eddy current effects and optimizing that system with respect to image noise and distortion. The design methods are successfully tested in a series of proof-of-principle experiments which lead to a discussion of how to incorporate similar designs into an operational MRI.

The issue of eddy currents induced by the rapid switching of magnetic field gradients is a long-standing problem in magnetic resonance imaging. A new method for dealing with this problem is presented whereby spatial harmonic components of the magnetic field are continuously sensed, through their temporal rates of change, and corrected. In this way, the effects of the eddy currents on multiple spatial harmonic components of the magnetic field can be detected and corrections applied during the rise time of the gradients. Sensing the temporal changes in each spatial harmonic is made possible with specially designed detection coils. However to make the design of these coils possible, general relationships between the spatial harmonics of the field, scalar potential, and vector potential are found within the quasi-static approximation. These relationships allow the vector potential to be found from the field – an inverse curl operation – and may be of use beyond the specific problem of detection coil design. Using the detection coils as sensors, methods are developed for designing a negative feedback system to control the eddy current effects and optimizing that system with respect to image noise and distortion. The design methods are successfully tested in a series of proof-of-principle experiments which lead to a discussion of how to incorporate similar designs into an operational MRI.

Atualmente o potencial vetor é geralmente tratado no eletromagnetismo clássico como um artifício para o cálculo dos campos elétricos e magnéticos sem um significado claro. No entanto, quando foi proposto na metade do século XIX, ele possuía um significado físico claro e desempenhava um papel central para Faraday, Maxwell e outros físicos britânicos. Um dos objetivos deste trabalho é entender como se deu esta mudança na interpretação do conceito de potencial vetor. Para isto foi realizado um estudo histórico analisando as diferentes interpretações para este conceito partindo dos trabalhos de Faraday sobre indução eletromagnética, onde propôs o conceito de estado eletrotônico. Analisamos as contribuições de William Thomson que fortemente inspiraram Maxwell a sugerir diferentes interpretações para o conceito em trabalhos publicados ao longo de cerca de duas décadas até a publicação do Treatise on Electricity and Magnestism em 1873. No final do século XIX a interpretação dada por Maxwell ao conceito de potencial vetor começou a ser questionada por vários físicos. Uma das questões envolvidas neste processo foi a realidade das grandezas físicas. Nomes como Heaviside, Hertz e outros defendiam que as grandezas dotadas de realidade física na teoria eletromagnética eram os campos elétrico e magnético e não o potencial vetor. Com essa nova visão desenvolveram uma nova teoria eletromagnética próxima da que conhecemos atualmente. No entanto, este processo não foi linear e aceito acriticamente. Ao longo do século XX foram publicados trabalhos propondo uma interpretação física para o potencial vetor, ainda no contexto clássico. O estudo histórico aqui desenvolvido priorizou a abordagem desenvolvida na Grã-Bretanha / Currently the vector potential generally is considered in the classical electromagnetic theory as an artifice for the calculation of the electric and magnetic fields and without a clear physical meaning. However, when it was proposed in the mid-nineteenth century, it used to have a clear physical meaning and played a central role for Faraday, Maxwell and other British physicists. One of the goals of this dissertation is to understand how the meanings attributed to the vector potential changed along years. In order to answer to this question, we developed a historical study analyzing the different interpretations for this concept starting with the works of Faraday on electromagnetic induction, where he introduced the concept of electrotonic state. We analyzed the contributions of William Thomson that inspired strongly Maxwell to suggest different interpretations for the concept in works published along the next two decades until the publication of the Treatise on Electricity and Magnestism in 1873. In the end of the nineteenth century Maxwells interpretations for the vector potential began to be questioned by several physicists. One of the issues involved in this questioning was the reality of the physical quantities. People as Heaviside, Hertz and others defended that electric and magnetic fields, not the vector potential, were quantities endowed with physical reality. With this new approach they developed a new electromagnetic theory closer to the currently accepted. Nevertheless, this process was not linear and uncritically accepted. Throughout the twentieth century papers and books were published defending a physical interpretation for the vector potential considering a classical context for the electromagnetic theory. The historical study developed here focused the developments in Great-Britain although some mentions to Continental physics are made

Lorentz TEM observations of magnetic nanoparticles contain information on the magnetic and electrostatic potentials of the sample. These potentials can be extracted from the electron wave phase shift by separating electrostatic and magnetic phase shifts, followed by 3D tomographic reconstructions. In past, Vector Field Electron Tomography (VFET) was utilized to perform the reconstruction. However, VFET is based on a conventional tomography method called filtered back-projection (FBP). Consequently, the VFET approach tends to produce inconsistencies that are prominent along the edges of the sample. We propose a model-based iterative reconstruction (MBIR) approach to improve the reconstruction of magnetic vector potential, A(r). In the case of scalar tomography, the MBIR method is known to yield better reconstructions than the conventional FBP approach, due to the fact that MBIR can incorporate prior knowledge about the system to be reconstructed. For the same reason, we seek to use the MBIR approach to optimize vector field tomographic reconstructions via incorporation of prior knowledge. We combine a forward model for image formation in TEM experiments with a prior model to formulate the tomographic problem as a maximum a posteriori probability estimation problem (MAP). The MAP cost function is minimized iteratively to deduce the vector potential. A detailed study of reconstructions from simulated as well as experimental data sets is provided to establish the superiority of the MBIR approach over the VFET approach.

Atualmente o potencial vetor é geralmente tratado no eletromagnetismo clássico como um artifício para o cálculo dos campos elétricos e magnéticos sem um significado claro. No entanto, quando foi proposto na metade do século XIX, ele possuía um significado físico claro e desempenhava um papel central para Faraday, Maxwell e outros físicos britânicos. Um dos objetivos deste trabalho é entender como se deu esta mudança na interpretação do conceito de potencial vetor. Para isto foi realizado um estudo histórico analisando as diferentes interpretações para este conceito partindo dos trabalhos de Faraday sobre indução eletromagnética, onde propôs o conceito de estado eletrotônico. Analisamos as contribuições de William Thomson que fortemente inspiraram Maxwell a sugerir diferentes interpretações para o conceito em trabalhos publicados ao longo de cerca de duas décadas até a publicação do Treatise on Electricity and Magnestism em 1873. No final do século XIX a interpretação dada por Maxwell ao conceito de potencial vetor começou a ser questionada por vários físicos. Uma das questões envolvidas neste processo foi a realidade das grandezas físicas. Nomes como Heaviside, Hertz e outros defendiam que as grandezas dotadas de realidade física na teoria eletromagnética eram os campos elétrico e magnético e não o potencial vetor. Com essa nova visão desenvolveram uma nova teoria eletromagnética próxima da que conhecemos atualmente. No entanto, este processo não foi linear e aceito acriticamente. Ao longo do século XX foram publicados trabalhos propondo uma interpretação física para o potencial vetor, ainda no contexto clássico. O estudo histórico aqui desenvolvido priorizou a abordagem desenvolvida na Grã-Bretanha / Currently the vector potential generally is considered in the classical electromagnetic theory as an artifice for the calculation of the electric and magnetic fields and without a clear physical meaning. However, when it was proposed in the mid-nineteenth century, it used to have a clear physical meaning and played a central role for Faraday, Maxwell and other British physicists. One of the goals of this dissertation is to understand how the meanings attributed to the vector potential changed along years. In order to answer to this question, we developed a historical study analyzing the different interpretations for this concept starting with the works of Faraday on electromagnetic induction, where he introduced the concept of electrotonic state. We analyzed the contributions of William Thomson that inspired strongly Maxwell to suggest different interpretations for the concept in works published along the next two decades until the publication of the Treatise on Electricity and Magnestism in 1873. In the end of the nineteenth century Maxwells interpretations for the vector potential began to be questioned by several physicists. One of the issues involved in this questioning was the reality of the physical quantities. People as Heaviside, Hertz and others defended that electric and magnetic fields, not the vector potential, were quantities endowed with physical reality. With this new approach they developed a new electromagnetic theory closer to the currently accepted. Nevertheless, this process was not linear and uncritically accepted. Throughout the twentieth century papers and books were published defending a physical interpretation for the vector potential considering a classical context for the electromagnetic theory. The historical study developed here focused the developments in Great-Britain although some mentions to Continental physics are made

The doctoral thesis is focused on research, design, implementation and testing of systems for the application of magnetic vector potential to biological materials. The main objective was to analyze and design systems which could generate magnetic vector potential without the presence of other unwanted fields or at least amplify its intensity. Moreover, the systems designed had to eliminate other foreign effects on the biological samples, especially the influence of waste heat from the coils. Toroidal coils were employed to generate the vector potential, because they confine the unwanted magnetic induction inside their core thanks to their shape. The thesis employed coils with two different outer diameters, specifically 102 and 600 mm. To excite the coils, four current pulse generators capable of delivering currents of up to 100 A were constructed. The systems’ generated fields were comprehensively analyzed with the help of finite-element simulations in ANSYS. To simplify the design phase, analytical equations for the calculation of vector potential intensity at an arbitrary point around the toroidal coils were also derived. A method employing electromagnetic shielding made of two different materials was developed to suppress the unwanted fields. To eliminate the influence of heat, the 102 mm system employed air cooling and the 600 mm system employed a closed water loop to equalize the temperatures of biological samples. The biological effects of both systems were tested on genetically modified bio-luminescent bacteria Escherichia coli K12 luxABCDEamp. The thesis was created in connection with the research project of The Ministry of Education, Youth and Sports of the Czech Republic named “Research into the effect of a combination of substances for targeted therapy and inhibitory action of the field pulse vector magnetic potential on oncogenous diseases”, No. 2B08063.

Les machines électriques sont modélisées pour prédire leurs performances et optimiser leur rendement. Cette modélisation peut être faite par des simulations avec la méthode des éléments finis. En particulier, les machines asynchrones nécessitent des simulations 3D pour prendre en compte les courants de Foucault et les têtes de bobines. Dans le logiciel Flux®, des formulations 3D basées sur le potentiel scalaire magnétique sont utilisées avec succès depuis de nombreuses années. Néanmoins, des coupures mathématiques artificielles sont nécessaires, lorsque le domaine n'est pas simplement connexe.Afin de se libérer de ces contraintes de connexité, des formulations en potentiel vecteur magnétique ont été étudiées et développées. En 3D, leur mise en œuvre nécessite l'utilisation d'éléments finis d'arêtes afin de respecter la nature des champs. Avec les éléments d'arêtes, les formulations sont généralement résolues avec une condition de jauge pour les solveurs directs comme pour les solveurs itératifs. De nouvelles formulations en potentiel vecteur magnétique auto-jaugées ont été développées permettant la prise en compte des bobines maillées et des bobines non maillées. La prise en compte du mouvement est relativement simple à mettre en œuvre pour les formulations en potentiel scalaire magnétique avec l'interpolation nodale.Avec les éléments d'arête, l'interpolation est plus délicate. C'est pourquoi la méthode des éléments avec joints a été développée pour prendre en compte le mouvement dans un cas général. / Electric machines are modeled in order to predict their performance and to optimize their output. This modeling can be done by simulation with the finite element method. In particular, induction machines require 3D simulation to take into account eddy currents and coils overhangs. In the Flux® software, 3D formulations based on magnetic scalar potential has been used with success for many years. Nevertheless, artificial mathematical cuts are necessary, when the domain is not simply connected.In order to avoid connection constraints, magnetic vector potential formulations have been studied and developed. In 3D, their implementation requires the use of edge elements to respect the nature of fields. With edge elements, formulations are generally solved with a gauge condition for direct solvers as well as for iterative solvers. New auto-gauged magnetic vector potential formulations have been developed to take into account meshed coils and non-meshed coils. Consideration of movement is relatively simple to implement for magnetic scalar potential formulations with nodal interpolation. With edge elements, the interpolation is more delicate. For this reason, the mortar method has been developed to take into account movement in a general case.

Submitted by Vasti Diniz (vastijpa@hotmail.com) on 2017-09-18T13:45:16Z No. of bitstreams: 1 arquivototal.pdf: 1340608 bytes, checksum: 8cceaa9c73ae05a5a903b4b78e3fa6ba (MD5) / Made available in DSpace on 2017-09-18T13:45:16Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1340608 bytes, checksum: 8cceaa9c73ae05a5a903b4b78e3fa6ba (MD5) Previous issue date: 2010-04-15 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / It is presented a review on three subjects: the origin and nature of topological defects, the non-relativistic and relativistic quantum mechanics, and the gravitational and eletrostatic selfforces that emerge from a conical spacetime surrounding a cosmic string. After setting up this theoretical framework, it is studied the behavior of a charged particle in the presence of a cosmic string, parallel to an uniform, constant magnetic field, which may be used as a model to a primordial large-scale magnetic field that permeates the universe. The geometry of a negative disclination is taken into account, this being a typically condensed matter physics topological defect equivalent to a cosmic string, where a wedge of material is inserted into the lattice. We computed exactly, the topological and electrostatic influences on the particles energy spectrum, and the phase shift for the charged scalar particle scattered states. Switching to a flat-spacetime context, spherically symmetric systems were studied, solving exactly, Klein-Gordon and Dirac equations which describe a scalar particle subject to a Coulomb vector potential and scalar central potentials. / Apresentamos uma revisão sobre a origem e a natureza dos defeitos topológicos, que surgem a partir de transições de fase que podem ter ocorrido no início do processo de formação do universo; sobre a mecânica quântica não-relativística e relativística; e sobre as auto-forças gravitacional e eletrostática que emergem da topologia cônica do espaço-tempo gerado pela corda cósmica. Utilizando estas ferramentas, estudamos o movimento de uma partícula carregada na presença de uma corda cósmica, paralela a um campo magnético uniforme, de magnitude constante, que poderia servir de modelo para um campo magnético primordial. Também consideramos a geometria anti-cônica de uma desclinação negativa, defeito topológico análogo à corda cósmica estudado em matéria condensada. Calculamos, exatamente, a influência da topologia e do campo magnético no espectro de energia da partícula e encontramos o ângulo de mudança de fase para seus estados espalhados. No espaço-tempo plano, estudamos sistemas com simetria esférica e investigamos a dinâmica de uma partícula escalar, resolvendo, exatamente, as equações de Klein-Gordon e Dirac, considerando potenciais centrais.

L'objectif principal de cette thèse concerne la mise en oeuvre d'une méthoded'éléments finis stabilisés pour la simulation des plasmas de fusion. Pour cela,nous avons d'abord dérivé les modèles magnétohydrodynamiques depuis lemodèle cinétique. Les modèles MHD sont généralement utilisés pour simuler lesinstabilités macroscopiques des plasmas. Nous nous sommes concentrés sur lemodèles de la MHD complète. Ensuite, l'approche numérique est décrite dans lecadre de la stabilisation Variationelle Multi-Échelles (VMS). Cette stabilisationvient ajouter un terme à la formulation faible pour mimer les effets des échellesnon-résolues sur celles résolues. Si les effets de ces sous-échelles ne sont paspris en compte lorsque l'on traite des écoulements dominés par convection,comme dans le cadre des plasmas de fusion, le schéma numérique conduit àdes résultats non-physiques. Une étude détaillée de l'instabilité de « Kinkinterne » a été faite ainsi qu'une étude préliminaire des plasmas avec point-Xayant pour but la validation du schéma numérique développé ici / The main objective of this thesis concerns the implementation of a robuststabilized finite element method for simulating fusion plasmas. For that, we firstderive the magnetohydrodynamic models from the kinetic model. MHD modelsare generally used for macroscopic simulations of plasma instabilities. Weconcentrate ou efforts on the full MHD model. Next, the numerical approach isdescribed in the context of the Variational Multi-Scale (VMS) stabilization. Thisstabilization comes to add a term to the weak formulation to mimics the effectsof the unresolved scales over the coarse scales. If the effects of these subscalesare not taken into account when dealing with fluxes dominated byconvection, as it is the cases for fusion plasmas, the numerical scheme canlead to unphysical results. A detailed study of the resistive internal kinkinstability has been done as well as an introductory study of the so called Xpointplasmas in order to validate the numerical scheme developed here