1.1 Units and fundamental equations

The subject treated in this dissertation lends itself to analysis largely by classical electrodynamics. As a result, Maxwell's equations are fundamental to much of the theoretical discussion. The Système Internationale or ``rationalized MKS'' units are used throughout. In differential form, Maxwell's noble laws for electromagnetic fields are:

$$\nabla \cdot \ensuremath{{\bf B}} =$$ 0 (1.1)

$$\nabla \cdot \ensuremath{{\bf E}} = \frac{\rho}{\ensuremath{\epsilon_{}}}$$ (1.2)

$$\nabla\times \ensuremath{{\bf B}} = \ensuremath{\mu_{}}{\bf J} + \ensuremath{\mu_{}}\ensuremath{\epsilon_{}}\ensuremath{\frac{\partial \ensuremath{{\bf E}}}{\partial t}}$$ (1.3)

$$\nabla\times \ensuremath{{\bf E}} = -\ensuremath{\frac{\partial \ensuremath{{\bf B}}}{\partial t}}$$ (1.4)

Here the permittivity of free space $\ensuremath{\epsilon_{}}=8.854\times10^{-12}$ Fm$^{-1}$ and the permeability of free space $\ensuremath{\mu_{}}=4\pi\times10^{-7}$ Hm$^{-1}$ satisfy $c^{-2}=\ensuremath{\epsilon_{}}\ensuremath{\mu_{}}$, where $c$ is the speed of light in vacuum.

The force exerted on a non-relativistic charged particle of mass $m$ and charge $q$ due to these fields is given by the Lorentz force equation,

$$m \ensuremath{\frac{\partial \bf v}{\partial t}}= q({\bf E} + {\bf v}\times{\bf B}).$$ (1.5)

Under conditions of high charged particle density and number, a group of charged particles takes on the properties of a plasma and may be described by its collective behavior. By defining a distribution function $f({\bf r},{\bf v},t)$ describing the time-dependent occupation of phase space for each kind of charged particle, the Boltzmann equation

$$\ensuremath{\frac{\partial f}{\partial t}}+ [{\bf v}\cdot\nabla_{... ..._{\bf v}]f = \left(\ensuremath{\frac{\partial f}{\partial t}}\right)_{\rm coll}$$ (1.6)

follows from the Liouville theorem and is a fundamental point of departure for the study of plasma heating. In (1.6), $\nabla_{\bf r}$ and $\nabla_{\bf v}$ are gradient operators which act on the position and velocity spaces, respectively, of $f({\bf r},{\bf v},t)$, and $\left(\ensuremath{\frac{\partial f}{\partial t}}\right)_{\rm coll}$ is a term which accounts for all collisional changes to the distribution function.