# 5.1: Introduction

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In Chapter 4, the subject is electromechanical kinematics. Field sources are physically constrained to have predetermined spatial distributions and the relative motion is prescribed. As a result, in atypical example, the electromechanical dynamics can be in capsulated in a lumped-parameter model. In this and the next chapter, the mechanics remain kinematic, in that the material deformations are again prescribed. However, now material may be suffering relative deformations, represented by a given velocity field \(\overrightarrow{v}(\overrightarrow{r},t)\). More important, in this and the next chapter, electrodes and wires are no longer used to constrain the "free" field sources. Rather, the distribution of free charge and current is now determined by.the field laws themselves, augmented by conservation laws and constitutive relations.

The physical situations now considered are electroquasistatic and the sources are therefore charge densities. In Chapter 6, magnetoquasistatic systems are of interest, the relevant sources are the free current density and magnetization density, and the subject is magnetic diffusion in the face of material convection.

In the next section, equations are deduced that represent the fate of each species of charge.Throughout this chapter, the charge carriers are dominated in their motions by collisions with neutral particles and with each other. On the average, collisions are so frequent that the inertia of each carrier can be ignored. Such collision-dominated carrier motions are introduced in Sees. 3.2 and 3.3,where the observation is made that it is only if the particle inertia is ignorable that the electrical force on the carrier can be taken as instantaneously transmitted to the media through which it moves.If the carrier inertia is important, the carrier densities constitute mechanical continua in their own right. Such examples are the electron beam in vacuum and the ions and electrons that constitute a "cold"plasma. These models are therefore appropriately included in Chaps. 7 and 8, where fluids and fluid-like continua are studied.

The conservation of charge equations, together with the electroquasistatic field laws and the specified material deformation, constitute a description of the way in which the fields and their sources self-consistently evolve. Whether to gain insights concerning the implications of these equations, or to solve these equations in a specific situation, characteristic coordinates are valuable.Thus, the characteristic approach to partial differential equations is introduced in the context of charge-charrier migration, relaxation and convection. The method of characteristics will be used extensively to describe other phenomena involving propagation in later sections and chapters.

Examples treated in Secs. 5.4 and 5.5, which illustrate "imposed field and flow" dynamics of systems of carriers, involve a space charge due to the charge carriers that is ignorable in its contribution to the field. The impact charging of macroscopic particles treated in Sec. 5.5 results in a model widely used in atmospheric sciences, macroscopic particle physics and air-pollution control.

When space-charge effects are significant, it is necessary to be more specialized in the treatment. In Sec. 5.6 only one species of charge carrier is presumed to be significant. The unipolar carriers might be ions injected by a corona discharge into a neutral gas or into a highly insulating liquid. They might also be charged macroscopic particles carrying a constant charge per particle and migrating through a gas or liquid. Section 5.7 considers steady-flow one-dimensional unipolar conduction and its relation to the d-c family of energy converters.

Bipolar conduction, discussed in Secs. 5.8 and 5.9, has as a limiting model ohmic conduction;These sectiona have two major objectives, to illustrate charge migration and convection phenomena with more than one species of carrier, and to put the ohmic conduction model in perspective. In Sec. 5.10, charge relaxation is described in general terms by again resorting to the method of characteristics. The remaining sections are based on the ohmic conduction model.

The transfer relations for regions of uniform conductivity are discussed in Sec. 5.12 and applied to important illustrative physical situations in Secs. 5.13 and 5.14. These case studies are profit-ably contrasted with their magnetic counterparts developed in Secs. 6.4 and 6.5.

Temporal transients, initiated from spatially periodic initial conditions, are considered in Sec. 5.15. Just as the natural modes are closely related to the driven response of lumped-parameter linear systems, the natural modes of the continuum systems discussed in terms of their responses to spatially periodic drives in Secs. 5.13 and 5.14 are found to be closely related to the natural modes for distributed systems. This section, which is the first to illustrate the third category of response for linear systems that are uniform in at least one direction, as presaged in Sec. 1.2, also illustrates how heterogeneous systems of uniform ohmic conductors (which support a charge relaxation processing each bulk region) can display charge diffusion in the system taken as a whole. This type of diffusion should be discriminated from diffusion at the carrier (microscopic) level. Diffusion in the latter sense is included in Sec. 5.2 so that the domain of validity of migration and convection processes in which diffusion is neglected can be appreciated. Molecular diffusion and its effect on charge evolution, introduced in Sec. 5.2, is largely delayed until Chapter 10.

Finally, in Sec. 5.16, the response of an Ohmic moving sheet is used to introduce the fourth type of continuum linear response eluded to in Sec. 1.2, a spatially transient response to a drive that is temporarily in the sinusoidal steady state.