DISTRIBUTION OF CHARGED PARTICLES TRAPPED IN A VARYING STRONG MAGNETIC FIELD (ONE-DIMENSIONAL CASE) WITH APPLICATIONS TO TRAPPED RADIATION

1964 ◽  
Vol 42 (6) ◽  
pp. 1185-1194 ◽  
Author(s):  
Tomiya Watanabe
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Distribution Function ◽  
Strong Magnetic Field ◽  
Three Dimensional ◽  
Charged Particles ◽  
Dimensional Case ◽  
One Dimensional ◽  
Canonical Variables ◽  
Trapped Radiation

The differential equation for the distribution function of charged particles trapped in a strong magnetic field is discussed for the case when the charged particles are constrained in a tube of magnetic lines of force with a small normal cross section (the one-dimensional case). The distribution function is not defined by the six canonical variables, the variables in geometrical space and velocity space, as is done usually when referring to the Boltzmann equation. Instead, it is defined by the space coordinates of the guiding center of a representative particle together with its speed and pitch angle (i.e. five variables, in the three-dimensional case). In some problems, this type of description makes the correspondence between the physical picture and its mathematical description much easier. Several problems relating to trapped radiation are discussed using the differential equation.

Landau–Kelly representation of statistical thermodynamics of a quantum plasma and electron emission from metals

2020 ◽  
Vol 86 (3) ◽  
Author(s):  
L. N. Tsintsadze ◽  
G. M. Peradze ◽  
N. L. Tsintsadze
Keyword(s):  
Magnetic Field ◽  
Distribution Function ◽  
Strong Magnetic Field ◽  
Electron Emission ◽  
Three Dimensional ◽  
Thermionic Emission ◽  
Quantum Plasma ◽  
Dimensional System ◽  
Perpendicular Component ◽  
The Magnetic Field

We have investigated the influence of a strong magnetic field on various aspects of a quantum Fermi plasma. Due to the strong magnetic field, the distribution function becomes anisotropic. First, we consider non-degenerate quantum, Landau and Kelly distribution function. It was found that the adiabatic equation is similar to the adiabatic equation for a Maxwell distribution function, when we include the magnetic field in the energy expression. Using the Kelly distribution for a degenerate, quantum Fermi gas, parallel and perpendicular components of the pressure were derived. It was found that perpendicular component of pressure never becomes zero and three-dimensional system always stay three-dimensional. Lastly, we investigated electron emission from metals and have shown the influence of the magnetic field. We calculated thermionic emission, the so-called Richardson effect. In addition, we investigate the influence of external electromagnetic radiation on the electron current density (Hallwachs effect) from metals.

scholarly journals The on-axis magnetic well and Mercier's criterion for arbitrary stellarator geometries

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
P. Kim ◽  
R. Jorge ◽  
W. Dorland
Keyword(s):  
Magnetic Field ◽  
Original Work ◽  
Three Dimensional ◽  
Radial Distance ◽  
Analytical Form ◽  
One Dimensional ◽  
Modern Notation ◽  
Magnetic Well ◽  
First Time ◽  
Dimensional Integral

A simplified analytical form of the on-axis magnetic well and Mercier's criterion for interchange instabilities for arbitrary three-dimensional magnetic field geometries is derived. For this purpose, a near-axis expansion based on a direct coordinate approach is used by expressing the toroidal magnetic flux in terms of powers of the radial distance to the magnetic axis. For the first time, the magnetic well and Mercier's criterion are then written as a one-dimensional integral with respect to the axis arclength. When compared with the original work of Mercier, the derivation here is presented using modern notation and in a more streamlined manner that highlights essential steps. Finally, these expressions are verified numerically using several quasisymmetric and non-quasisymmetric stellarator configurations including Wendelstein 7-X.

On approach to determine the internal potential and gravitational energy of ellipsoid

2021 ◽  
Vol 8 (3) ◽  
pp. 359-367
Author(s):  
М. M. Fys ◽  
◽  
А. M. Brydun ◽  
М. I. Yurkiv ◽  
◽  
...  
Keyword(s):  
Distribution Function ◽  
Mass Distribution ◽  
Special Form ◽  
Iterative Process ◽  
Three Dimensional ◽  
Gravitational Energy ◽  
One Dimensional ◽  
Piecewise Continuous ◽  
The Masses ◽  
Internal Potential

Formulas are derived for the calculation of the potential of bodies, which surface is a sphere or an ellipsoid, and the distribution function has a special form: a piecewise continuous one-dimensional function and a three-dimensional mass distribution. For each of these cases, formulas to calculate both external and internal potentials are derived. With their help, further the expressions are given for calculation of the potential (gravitational) energy of the masses of such bodies and their corresponding distributions. For spherical bodies, the exact and approximate relations for determining the energy are provided, which makes it possible to compare the iterative process and the possibility of its application to an ellipsoid. The described technique has been tested by a specific numerical example.

On the minimizers of the Ginzburg–Landau energy for high kappa: the one-dimensional case

1997 ◽  
Vol 8 (4) ◽  
pp. 331-345 ◽  
Author(s):  
AMANDINE AFTALION
Keyword(s):  
Magnetic Field ◽  
Asymptotic Behaviour ◽  
Dimensional Case ◽  
Numerical Computations ◽  
Ginzburg Landau ◽  
Landau Model ◽  
One Dimensional ◽  
Landau Parameter ◽  
Convergence Results ◽  
The One

The Ginzburg–Landau model for superconductivity is examined in the one-dimensional case. First, putting the Ginzburg–Landau parameter κ formally equal to infinity, the existence of a minimizer of this reduced Ginzburg–Landau energy is proved. Then asymptotic behaviour for large κ of minimizers of the full Ginzburg–Landau energy is analysed and different convergence results are obtained, according to the exterior magnetic field. Numerical computations illustrate the various behaviours.

Three-Dimensional Numerical Analysis for Convection of Air in a Bore Space of a Superconducting Magnet

2003 ◽  
Author(s):  
G. Tomita ◽  
M. Kaneda ◽  
T. Tagawa ◽  
H. Ozoe
Keyword(s):  
Magnetic Field ◽  
Heat Flux ◽  
Natural Convection ◽  
Numerical Analysis ◽  
Strong Magnetic Field ◽  
Three Dimensional ◽  
Superconducting Magnet ◽  
Constant Heat Flux ◽  
Constant Heat ◽  
Magnetizing Force

Three-dimensional numerical computations were carried out for the natural convection of air in a horizontal cylindrical enclosure in a magnetic field, which is modeled for a bore space of a horizontal superconducting magnet. The enclosure was cooled from the circumferential sidewall at the constant heat flux and vertical end walls were thermally insulated. A strong magnetic field was considered by a one-turn electric coil with the concentric and twice diameter of the cylinder. Without a magnetic field, natural convection occurs along the circumferential sidewall. When a magnetic field was applied, magnetizing force induced the additional convection, that is, the cooled air at the circumferential wall was attracted to the location of a coil. Consequently, the temperature around the coil decreased extensively.

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