3 - The constitutive relations

Summary

Introduction

Before (2.45) can be applied to the theory of wave propagation in a plasma, it is necessary to express the electric displacement D, and therefore the electric polarisation P, in terms of the electric intensity E. The resulting expressions are called the constitutive relations of the plasma and are derived in this chapter. The subject of wave propagation is resumed in ch. 4.

From now on, except in § 3.8, all fields are assumed to vary harmonically in time, and are designated by capital letters representing complex vectors, as explained in § 2.5. Time derivatives of some of the fields appear in the constitutive relations, and since ∂/∂t ≡ iω, it follows that the angular frequency ω appears in the expression for the electric permittivity. When this happens the medium is said to be time dispersive.

In this and the following chapter the plasma is assumed to be homogeneous. In later chapters the results are applied to an inhomogeneous plasma. This is justified provided that the plasma is sufficiently slowly varying. The meaning of ‘slowly varying’ is discussed in § 7.10. It is further assumed, in the present chapter, that the spatial variation of the fields can be ignored. This means that, over a distance large compared with N^−⅓, the fields can be treated as uniform. Thus spatial derivatives of the fields do not appear in the expression for the permittivity.