14.1.Theoretical Approaches to Plasma Dynamics
The general area of study embracing the interaction of ionized gases with time-dependent electromagnetic fields is called plasma dynamics. For many of the problems in this area, and these are the more important and interesting ones, it is impossible to treat a plasma adequately in terms of a known conventionally as kinetic theory. The motions of individual ions and electrons must be studied; their collisions with other particels must be taken into account throught solution of the Boltzmann transport equation. Thus a rigorous formulation for plasma problems exists, but their solution is neglect some of the terms in the Boltzmann equation. There are, however three approximate formulations that provide considerable insight into what is happening inside the plasma.
The first of these methods is equilibrium theory which rests on the premise that collisions between charged particles are sufficient to maintain the well-known Maxwell-Bolzmann velocity distribution for particles in the body of the plasma:
Nj(v)dvx dvy dvz = N0j dvx dvy dvz
Where N0j is the number of particles of type j per unit volume in the plasma, vx, vy, and vz are the components of velocity, mp is the mass of type j particles, and T is the absolute temperature. Kinetic and transport properties may then be calculated in terms of this velocity distribution.
14.2.Electrical Neutrality in A Plasma
One of the most important properties of a plasma is its tendency to remain electrically neutral-that is, its tendency to balance positive and negative space charge in each macroscopic volume element. A slight imbalance in the space-charge densities gives rise to strong electrostatic forces that act, wherver possible, in the direction of restoring neutrality. On the other hand, if a plasma is deliberately subjected to an external electric field, the space-charge densities will adjust themselves so that the major part of the plasma is shielded from the field.
Let us consider a rather simple example. Suppose a point charge +Q is introduced into a plasma, thereby subjecting the plasma to an electric field. Electrons find it energetically favorable to move close to the charge, whereas positive ions tend to move away.
Ne = No exp
Where is the local potential, is the reference potential (plasma potential), T is the absolute temperature of the plasma, and k is Boltzmann’s constant. N0 is the electronic density in regions where .
14.3.Particle Orbits and Drift Motion in A Plasma
The orbit of a charged particle q moving in a prescribed electric and magnetic field may be calculated directly from the force equation:
F = q(E+v x B)
We shall find it convenient to start with relatively simple field configurations, and then to generalize to fields which are slowly vaarying in space.
A constant electric field applied to a plasma is not particulary interesting because the plasma adjust itself by developing a thin sheath of other hand, a constant magnetic field causes the particles to gyrate about the field lines without altering the space-charge distribution.
14.4.Magnetic Mirrors
The results of the preceding section show that a slowly converging magnetic field can, in principle, confine a plasma. At right angels to the principal field direction the particles are bent into circular orbits; along the principal direction of the field the particles are slowed down and finally reflected by the converging field lines. Such a field configuration is called a magnetic mirror.
14.5.The Hydromagnetic Equations
Collective motions of the particles in a plasma, such as the “pinch effect” and plasma oscillations, are handled best in the hydromagnetic formulation. According to this description, the plasma is regarded as a classical fluid that obeys the convetional equations of hydrodynamics. The fuild, however, is an electrical conductor, and thus electromagnetic forces must be taken into account explicitly.
The force on a unit volume of the plasma may be written as
Fv = J x B
Where J is the current density and p is the fluid pressure. Other forces, such as gravitational and viscous forces, may also be included, but are neglected here in the interest of simplicity.
14.6.The Pinch Effect
The tendency of a high-current discharge through a plasma to constrict itself laterally is known as the “pinch effect”. The basic mechanism causing the pinch is the interaction of a current with its own magnetic field or, equivalently, the attraction between parallel current filaments. The pinch effect was first predicated by Bennett, and later independently by Tonks.
14.7.Magnetic Confinement Systems for Controlled Thermonuclear Fusion
An important quantity for the design of fusion reactor systems is the ratio of kinetic pressure (kinetic plus magnetic pm). This ratio is given the symbol :
where N is the sum of the ion and electron densities in the plasma. Fusion reactors are generally characterized by their -value. Low refers to valuesless than 0.01 and high values lie between 0.1 and 1.0. the plasma is either deuterium or a deuterium-tritium plasma with temperature in excess of 108K; its density is in the range 1019m-3 to 1022m-3. Confinement does not have to be absolute, but must be for a long enough period so that more energy is produced in the thermonuclear reaction than is consumed in establishing the plasma conditions. confinement is believed to be adequate when the Lawson condition is met:
Ni
Where Ni is the ion density in the plasma.
14.8.Plasma Oscillations and Wave Motion
One of the interesting properties of a plasma is its ability to sustain oscillations and propagate waves. Various types of oscillatory behavior are possible, and because of the nonlinear character of the hydrodynamic equations, these oscillations can be quite complex. We find it expedient to been observed in controlled experiments.
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