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Lecture 1
1. Definition of a plasma, as a state of the matter - mixture of charge (and neutral particles)
satisfying two conditions:
a) quasinetrality, ne ≈ ni , where nα is the density of a charged particle;
b) collective behavior - motion of a charged particle is "feeling" by all other
particles via Coulomb interactions;
2. We have considered and developed the concept of the temperature of a plasma (or plasma
components - charged and neutral particles) as the averaged kinetic energy of plasma
particles. In this treatment we have assumed the equilibrium state of a plasma and thus used
the Maxwellian distribution of particles. As one of the results, the temperature can be
measured in the energy units, in particular, eV. 1eV  11600 K  104 K .
3. The more detailed investigation of the quasineutrality condition. The estimation is based
on the assumption that the electrostatic potential energy of a test charged particle is balanced
by its thermal energy. This results in some characteristic spatial scale - the smallest spatial
scale of a plasma, the Debye length, λD (or rD ). The corresponding temporal scale is so-called
plasma frequency. These quantities are usually related to the electron component of a plasma.
Thus, for a linear scales L >> λD , the condition of quasineutrality holds, in the opposite limit,
one has to use the single particle model.
4. We now can estimate the number of charged particles N D in so called Debye sphere with a
linear scale given by λD . To meet the condition of the collective behavior, we need N D >> 1 .
5. The motion of a charged particle in a given (prescribed) electric and magnetic fields single particle approximation ( no self-consistent description of fields and particle motion,
since the electric and magnetic fields of a particle are not taken into account).
6. We have considered the motion of a charged particle in a given magnetic field. The
parameters of a problem: the mass of a particle, the particle charge, and the magnetic field.
Very important: these parameters (usually in some combination) must be present in the
solution of the problem.
Result: the motion of a charged particle along the magnetic field and perpendicular to it is
different. In the plane perpendicular to the magnetic field, the charged particle is gyrating
around the magnetic field line with a centre (so called guiding centre) placed on the line,
whereas along the magnetic field the particle is moving with the constant velocity. The
frequency of this gyration is the so-called cyclotron (or Larmor) frequency, ωc , the radius of
a gyration is known as the Larmor radius, rL . The total trajectory of a motion of a charged
particle in the magnetic field has the form of a spiral.
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Conclusion. The main effect of the magnetic field on the random motion of a charged
particles is the restriction of the motion in the plane perpendicular to the field.
7. The combination of the magnetic and the electric field (with a component perpendicular to
the magnetic field), results in the new motion of a guiding centre, so called drift motion
which is superimposed on the cyclotron gyration and is due to the electric field component.
This drift motion is known as the electric drift, or E × B drift. The main characteristic of this
drift is that the electric drift velocity does not depend on the charge and mass of the particle.
Therefore, electrons and ions are drifting together. Special attention should be given to the
direction of the electric drift velocity, namely, the particles are drifting in the direction which
is neither the direction of the magnetic field, nor the electric field.
We thus can decompose the motion of a charged particle into a "fast" motion, i.e. cyclotron
gyration, performed with the cyclotron frequency and "slow" motion performed with the drift
velocity. In order to consider the drift motion itself, we must averaged the corresponding
equations with respect to the "fast" motion.
8. In general, if we have the external force with the component perpendicular to the magnetic
field, this force results in the drift motion of a guiding centre of a charged particle.
Questions.
1. Formulate the conditions for the state of a matter to be treated as a plasma.
2. Equilibrium state of a plasma ( or its components) - how you understand it? Give some
examples.
3. Define the distribution function of a plasma particles, general form and the physical sense
of it, what is the Maxwellian distribution.
4. How you can determine the temperature of a plasma? Units of the plasma temperature.
5. What assumption you are using to estimate the Debye length? Why it is often treated as the
smallest length scale of a plasma? Give the definition of the corresponding temporal scale.
6. On what spatial scales the condition of quasineutrality is held?
7. How you understand the condition N D >> 1 ?
8. Single particle approximation. What sense of this approximation?
9. How you are coming to the conclusion that the motion of a charged particle is different
along and perpendicular to the magnetic field?
10. Define the cyclotron frequency, the Larmor radius and the general form of the trajectory
of a motion of charged particle in a given constant magnetic field.
11. What is main effect of the magnetic field on the random motion of charged particles?
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12. Consider the motion of a charged particle in a combination of an electric and magnetic
fields. What is the effect of electric field component parallel to the magnetic field on this
motion?
13. Explain why the drift motion is appearing and the direction of this motion. Physical
picture and the roll of the electric field component perpendicular to the magnetic field.
14. Explain the concept of the "fast" and "slow" motion separation for the particle moving in
the combination of electric and magnetic fields.
15. What mathematical tool one has to use in order to "extract" the drift motion? Explain
result.
16. What is the general condition for the drift motion of guiding center to be excited?
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