2  Electromagnetic fields

In electromagnetics we deal with the following fields:

All considered fields are functions of space \(\mathbf r\) and time \(t\), i.e.,

\[ \mathbf e(\mathbf r, t), \mathbf h(\mathbf r, t), \mathbf d(\mathbf r, t), \mathbf b(\mathbf r, t), \mathbf j(\mathbf r, t) \tag{2.1}\] or a function of the angular frequency \(\omega = 2 \pi f\), such that \[ \mathbf E(\mathbf r, \omega), \mathbf H(\mathbf r, \omega), \mathbf D(\mathbf r, \omega), \mathbf B(\mathbf r, \omega), \mathbf J(\mathbf r, \omega). \tag{2.2}\]

In the latter case, the time dependency of any field \(\mathbf F\) is always defined as \[ \mathbf F(\mathbf r, \omega) = \mathbf F_0(\mathbf r) e^{i \omega t}, \] and the quantity of interest is \(\mathbf F_0\).

Convention: Upper case letters: Frequency domain, lower case letters: Time domain.

See 3.2 for a definition of the Fourier transform.

2.1 Material properties

In electromagnetics we have the following material properties:

  • electrical conductivity \(\sigma\)
  • dielectrical permittivity \(\varepsilon\)
  • magnetic permeability \(\mu\).

In the context of geo-electromagnetics, these parameters are associated with particular rock properties which are studied in petrophysics.

2.2 Simplifications

As we know from theroretical physics, the relations between the fields and the associated parameters are very general and allow, e.g., strong frequency-dependency or non-linearities.

In geo-electromagnetics, however, we can allow for a few simplifications.

All rock parameters are supposed to be

  • linear with respect to the fields
  • stationary, and
  • isotropic.

We will see later that anisotropy is a quite general rock property which needs to be considered in the interpretation of geo-electromagnetic field data.

Moreover, we will first study the general properties of the EM induction in a uniform full-space by neglecting any spatial inhomogeneities of the parameters.