Concept# Lorentz force

Summary

In physics (specifically in electromagnetism), the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity v in an electric field E and a magnetic field B experiences a force (in SI units) of
\mathbf{F} = q\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right).
It says that the electromagnetic force on a charge q is a combination of a force in the direction of the electric field E proportional to the magnitude of the field and the quantity of charge, and a force at right angles to the magnetic field B and the velocity v of the charge, proportional to the magnitude of the field, the charge, and the velocity. Variations on this basic formula describe the magnetic force on a current-carrying wire (sometimes called Laplace force), the electromotive force in a wire loop moving through a magnetic field (an as

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In one of its acceptation, the word quench is synonym of destruction. And this is even more consistent with reality in the case of the Large Hadron Collider dipole magnets, whose magnetic field and stored energy are unprecedented: the uncontrolled transition from the superconducting to the resistive state can be the origin of dramatic events. This is why the protection of magnets is so important, and why so many studies and investigations have been carried out on quench origin. The production, cold testing and installation of the 1232 arc dipole magnets is completed. They have fulfilled all the requirements and the operation reliability of these magnets has already been partially confirmed. From an academic standpoint, nevertheless, the anomalous mechanical behaviour, which was sometimes observed during power tests, has not yet been given a clear explanation. The work presented in this thesis aims at providing an instrument to better understand the reasons for such anomalies, by means of finite element modelling of the cross-section of the dipole cold mass. During the investigation on quench phenomenology and its characterization, a distinction can be done between the two main quench origins during cold test without beam: the local degradation of the conductor and the frictional heating resulting from mechanical disturbances (such as conductor motion under the effects of the Lorentz forces). Concerning the second type, it is illustrated how important a good positioning of the cables is in a magnet cross-section and which is the fundamental role of azimuthal pre-stress. There are numerous studies of the consequences of conductor motion under the effect of electro-magnetic forces and of the loss of pre-stress during energization. However, no model has ever been able to reproduce in detail and predict such phenomena. The present model, developed in ANSYS® environment, was initiated with the idea of representing the real behaviour of an LHC-type dipole coil, by taking into account each turn individually, reproducing the non-linear and hysteretic mechanical behaviour observed on a stack of insulated cables and inserting friction between mating surfaces. The representation of the mechanical complexity of the composite material is certainly one of the originalities of this study. To validate the model, a comparison with elastic modulus measurements, systematically performed in industry, was carried out, both for single layers and for assembled poles. The agreement is certainly worth the effort lavished and justifies the following steps in simulation, which are the modelling of the collaring mechanism and the cool-down process. These are other important and original elements. The last one, in particular, requires progressively changing the mechanical properties of the superconductor, following the temperature profile. This implied some simplifications to comply with the enhanced convergence difficulties, but does not invalidate the goodness of the description and the results obtained. This is a faithful reproduction of a magnet life-cycle, uncommon in this kind of studies.

We provide a mathematical analysis and a numerical framework for magnetoacoustic tomography with magnetic induction. The imaging problem is to reconstruct the conductivity distribution of biological tissue from measurements of the Lorentz force induced tissue vibration. We begin with reconstructing from the acoustic measurements the divergence of the Lorentz force, which is acting as the source term in the acoustic wave equation. Then we recover the electric current density from the divergence of the Lorentz force. To solve the nonlinear inverse conductivity problem, we introduce an optimal control method for reconstructing the conductivity from the electric current density. We prove its convergence and stability. We also present a point fixed approach and prove its convergence to the true solution. A new direct reconstruction scheme involving a partial differential equation is then proposed based on viscosity-type regularization to a transport equation satisfied by the electric current density field. We prove that solving such an equation yields the true conductivity distribution as the regularization parameter approaches zero. Finally, we test the three schemes numerically in the presence of measurement noise, quantify their stability and resolution, and compare their performance. © 2015 Elsevier Inc.

We provide a mathematical analysis and a numerical framework for Lorentz force electrical conductivity imaging. Ultrasonic vibration of a tissue in the presence of a static magnetic field induces an electrical current by the Lorentz force. This current can be detected by electrodes placed around the tissue; it is proportional to the velocity of the ultrasonic pulse, but depends nonlinearly on the conductivity distribution. The imaging problem is to reconstruct the conductivity distribution from measurements of the induced current. To solve this nonlinear inverse problem, we first make use of a virtual potential to relate explicitly the current measurements to the conductivity distribution and the velocity of the ultrasonic pulse. Then, by applying a Wiener filter to the measured data, we reduce the problem to imaging the conductivity from an internal electric current density. We first introduce an optimal control method for solving such a problem. A new direct reconstruction scheme involving a partial differential equation is then proposed based on viscosity-type regularization to a transport equation satisfied by the current density field. We prove that solving such an equation yields the true conductivity distribution as the regularization parameter approaches zero. We also test both schemes numerically in the presence of measurement noise, quantify their stability and resolution, and compare their performance. © 2014 Elsevier Masson SAS.

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