Uma espira condutora movendo-se com velocidade constante através de uma região com um campo magnético constitui-se em uma das situações mais utilizadas em livros didáticos de física em conexão com a lei de Faraday Lenz, servindo para evidenciar a consistência do eletromagnetismo com o princípio da conservação da energia. Da análise dessa situação é frequentemente extraída a conclusão de que correntes não são induzidas em espiras perfeitamente condutoras porque nestas uma corrente induzida teria de ser infinita e portanto violaria o princípio de conservação de energia. No presente artigo apresentamos uma discussão sobre este problema e mostramos que um modelo para a espira que inclua sua autoindutância não apresenta divergência da corrente e leva a resultados compatíveis com a conservação da energia.

A conductive loop of wire moving with constant velo city in a magnetic field is one of the most used examples in physics textbooks in connection to the Faraday-Lenz law, highlighting the consistency of electromagnetism with the principIe of conservation of energy. It is frequently concluded that currents are not induced in a perfectly conducting loop because an induced current would have to be infinite and therefore would violate the principIe of energy conservation. ln this paper we discuss this problem and show that a model that includes the loop self-inductance prevents the divergence of the current and leads to results compatible with the conservation of energy.

We utilize a kinetic approach to the problem of wave propagation in dusty plasmas, taking into account the variation of the charge of the dust particles due to inelastic collisions with electrons and ions. The components of the dielectric tensor are written in terms of a finite and an infinite series, containing all effects of harmonics and Larmor radius. The formulation is quite general and valid for the whole range of frequencies above the plasma frequency of the dust particles, which are assumed motionless. The formulation is employed to the study of electrostatic waves propagating along the direction of the ambient magnetic field, in the case for which ions and electrons are described by bi-Maxwellian distributions. The results obtained in a numerical analysis corroborate previous analysis, about the important role played by the dust charge variation, particularly on the imaginary part of the dispersion relation, and about the very minor role played in the case of electrostatic waves by some additional terms appearing in the components of the dielectric tensor, which are entirely due to the occurrence of the dust charge variation.

A kinetic approach to the problem of wave propagation in dusty plasmas, which takes into account the variation of the charge of the dust particles due to inelastic collisions with electrons and ions, is utilized as a starting point for the development of a new formulation, which writes the components of the dielectric tensor in terms of a finite and an infinite series, containing all effects of harmonics and Larmor radius. The formulation is quite general and valid for the whole range of frequencies above the plasma frequency of the dust particles, which were assumed motionless. The formulation is employed to the study of electrostatic waves propagating along the direction of the ambient magnetic field, in the case for which ions and electrons are described by Maxwellian distributions. The results obtained in a numerical analysis corroborate previous analysis, about the important role played by the inelastic collisions between electrons and ions and the dust particles, particularly on the imaginary part of the dispersion relation. The numerical analysis also show that additional terms in the components of the dielectric tensor, which are entirely due these inelastic collisions, play a very minor role in the case of electrostatic waves, under the conditions considered in the present analysis.

A kinetic formulation developed to analyze wave propagation in dusty plasmas, which takes into account the variation of the charge of the dust particles due to inelastic collisions with electrons and ions, is utilized to study the propagation and damping of electrostatic waves with wave number exactly parallel to the external magnetic field and Maxwellian distributions for the electrons and ions in the equilibrium. It is shown that, due to the presence of the dust, the damping of Langmuir waves in the region of large wavelengths is increased as compared to conventional Landau damping. Langmuir waves in the occurrence of collisional charging of dust particles also feature weak damping for small wavelengths, which vanishes if the effect of collisional charging of the dust particles is neglected in the dispersion relation. It is also shown that the damping of ion-acoustic waves is modified by the presence of the dust, and that some damping effect due to the dust particles remains even if the effect of collisional charging of dust particles is neglected in the dispersion relation.

In the present paper we study the effects due to the occurrence of radial transport of particles in a tokamak on the efficiency of current drive due to combined action of lower hybrid waves and electron cyclotron waves, in the presence of an internal transport barrier. The results are obtained by numerical solution of the Fokker-Planck equation which rules the evolution of the electron distribution function. We assume that the radial transport of particles can be due to magnetic or to electrostatic fluctuations, and compare the two situations. In both cases the efficiency of current drive is shown to increase with the increase of the fluctuations which originate the transport. The current drive efficiency is shown to depend weakly on the radial position of the barrier, with a slightly more pronounced dependence in the case of magnetic fluctuations.

The preservation of Onsager symmetry for the effective dielectric tensor is discussed for a homogeneous plasma immersed in a inhomogeneous magnetic field, using the unperturbed orbits correct up to order kB, which is the scalelength of the field inhomogeneity. General features of the calculation of the components of the tensor are discussed and detailed calculations are developed for the zz component, which is shown to satisfy the conditions for Onsager symmetry, in agreement with previous results obtained using less precise expressions for the unperturbed orbits.

In the present paper we study the effects of occurrence of radial transport of particles in a tokamak, and the effects of the presence of an Internal Transport Barrier (ITB), on the current drive efficiency and power deposition profiles in the case of lower hybrid waves generating an extended tail in the electron distribution function. The results are obtained by numerical solution of the Fokker-Planck equation which rules the evolution of the electron distribution function. We assume that the radial transport of particles is due to magnetic or electrostatic fluctuations, and introduce a model to describe the ITB, with adjustable parameters. The presence of an Edge Transport Barrier (ETB) is simulated by Neumann boundary conditions at the plasma edge. The results obtained show very different behavior for current drive whether we have electrostatic or magnetic transport origin. The change in the plasma current due to magnetic transport has been observed to be more significant than the change due to electrostatic transport, basically because the magnetic transport is more effective in diffusing high-velocity particles of the electron tail.

We compare and discuss several approximations to the dispersion relation for electrostatic waves in inhomogeneous plasmas, either obtained directly from Poisson's equation, or from the dielectric constant obtained using a dielectric tensor derived using the plane wave approximation, or from the dielectric constant derived using the effective dielectric tensor.

We present a detailed derivation of the effective dielectric constant to be used in the dispersion relation for electrostatic waves in the case of a plasma immersed in a inhomogeneous magnetic field, with inhomogeneity perpendicular to the direction of the magnetic field.

The procedures used to obtain general expressions for the components of the effective dielectric tensor for electromagnetic waves in inhomogeneous magnetized plasmas are briefly reviewed, and the relationship between these expressions and their counterparts which can be obtained assuming electrostatic fluctuations is discussed. It is argued that a general formulation formerly available in the literature, which do not satisfy Onsager symmetry in the case of electrostatic fluctuations, is not the suitable form for description of dielectric properties in the electrostatic case, which require a dielectric constant. A general expression for an effective dielectric constant is therefore provided, obtained from the effective dielectric tensor, which satisfy Onsager symmetry.

This paper discusses the interaction between energetic ions and lower hybrid waves in tokamak plasmas, presenting a numerical analysis which employs a quasilinear formalism appropriated for the situation in which ion stochastic diffusion occurs. The results show that significant wave-particle interaction may take place when a population of energetic ions in present in the plasma, in partial agreement with evidence from experimental results available in the literature.

In the present paper we consider the interaction between energetic ions and lower hybrid waves in tokamak plasmas and study the parametric dependence of the threshold condition for stochastic ion diffusion. It is shown that the threshold condition as obtained in the work by Karney [C. F. F. Karney, Phys. Fluids 22, 2188 (1979)] may be not easily satisfied in present day large tokamaks, but can attained in small tokamaks with relatively modest levels of wave power.

We use quasilinear theory to investigate the influence of losses of energetic particles on the behaviour of electron cyclotron absorption during lower hybrid plus electron cyclotron current drive. Our analysis takes into account the quasilinear effects of the interaction of electron cyclotron radiation with a lower hybrid produced tail in the presence of collisions and a loss term for energetic particles. We study the behaviour of different quantities, as the parallel electron distribution function and the perpendicular temperature, for several values of the electron confinement time, an important parameter of the model loss term. The absorption of extraordinary mode electron cyclotron waves by tail electrons results to be substantially reduced when the loss term is taken into account in the quasilinear diffusion equation, as compared to the case where these losses are neglected.