Basic electromagnetic laws and Maxwells equations

We will first consider the laws of Coulomb, Biot-Savart and Faraday, emphasizing their experimental origin and the areas in which they can be applied. The relationship between these laws and Maxwell s equations will then be described to further explore their physical meaning and especially the precise sources of electric and magnetic fields. 

Special attention will be paid to the set of equations which describes the quasistation-ary or quasistatic fields and provides an accurate model for induction logging, with the exception of dielectric logging where very high frequencies are used. 

We will finally consider the formulation of the Helmholtz equations for magnetic and electric vector potentials, which are useful in solving boundary value problems in a conducting medium. 

As a starting point, we will assume that the reader accepts the concept that an electric charge is the source of an electric field. As a consequence, the distribution of electric charges is the main factor in controlling the field. In describing electric fields, we will make use of such functional descriptions of charges as volume, surface and linear densities of charge. 

The volume density of charge is the most general way in which to describe a charge distribution, but in some particular cases, we might also wish to define such functions as a surface or a linear density of charge. Suppose that the volume density S is invariant in the direction perpendicular to the surface of the thin layer (see Fig. 1.1). The elementary volume charge can then be written as  

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