Basic Electromagnetics

Basic EM concepts Maxwell’s equations are a set of partial differential equations that, together with the Lorentz force law, Ohm’s law and the constitutive relationships form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern geo-electromagnetics. Electromagnetic phenomena are best and conventionally described by a suite of vector fields and their inter-relationships.

Vector fields of EM Theory The Electric field 𝑬 The electric field describes, in the 3-space, the equivalent of the electrical potential (voltage) in the one-space world of electric circuits. Generalized to the 3-space, the field has direction and magnitude. In SI units, the magnitude is measured in [V · m−1].

Vector fields of EM Theory The Magnetic field 𝑯 The magnetic field (in some descriptions called the “magnetizing field”) describes an analog of the current in the one-space world of electrical circuits. You might note that the magnetic field describes, essentially, a current per unit distance in the 3-space; in SI units, the magnitude of the magnetic field vector is measured in [A · m−1].

Vector fields of EM Theory The Magnetic induction 𝑩 The magnetic induction (sometimes called “magnetic intensity”) represents the effect of the magnetic field in a material or space having magnetic permeability. In SI physics, it is measured in units of [V·s·m-1] or [T], “tesla”. The magnetic permeability of “free space” (an idealized perfect vacuum) is an invariant in contemporary SI- physics, fixed to be μ0 = 4 × 10-7 V·s·A-1·m-1. One might regard this field as measuring the local “magnetic polarization”. In free space 𝑩 = µo 𝑯 .

Vector fields of EM Theory The Electric displacement 𝑫 The electric displacement (sometimes called dielectric displacement) is the electric field’s analog of 𝑩 . In SI physics, it is measured in units of [kg·s-2·V-1]. It measures, again, the local “electric polarization” according to the “dielectric permittivity” which is a derived or measured quantity in SI physics: 𝑫 = ɛo 𝑬 in free space. The best current value for ɛo = 8.854187817620... ×10-12 A2·s4·kg-1·m-3, determined as 𝒄 𝒐 𝟐 =𝟏/ ɛ 𝒐 𝟐 µ 𝒐 𝟐 ; 𝒄 𝒐 , the speed of light in free space.

Relationships between the vector fields Gauss’ law for electrics: Here, 𝝆 𝒇 represents the local density of “free charges” measured in units of [C·m-3] where [C] is the “coulomb”, the common SI-unit of electrical charge.

Relationships between the vector fields Gauss’ law for magnetics: You might note the similarity with Gauss’ law for electrics; electrical “charges” are separable in + and − charges while magnetic “poles” have never been found to be separable in nature, always found locally in balanced pairings of N and S poles of equal strength.

Faraday’s law of induction an electrical field can be generated through temporal variations in the magnetic induction.

Ampere’s law a magnetic field is generated by a current flow, 𝑱 𝒇 , [A·m-2] and temporal variation in the electrical displacement. In highly conductive materials like metals, extremely rapidly.

Constitutive relationships Ohm’s law where σ [S·m-2] is the measure of the electrical conductivity of the material volume in which the electrical field and derived current flux exists.

Constitutive relationships Material dielectrics and magnetics In analogy to Ohm’s law which relates the current density flux to the electric field in a conductor, we define 2 other relationships: where µ and ɛ are measures of the material’s magnetic permeability and dielectric permittivity.

Constitutive relationships The properties of geomaterials can be distinguished by their conductivity, dielectric permittivity and magnetic permeability. These properties may be complex. For example all of these properties may be best described as tensor, frequency dependent constants. Conductivity of rocks or formations often show a frequency-dependent tensor character:

Constitutive relationships Moreover, the above-described linear relationship between electric field and current flux density shown in terms of frequency, ω, often does not hold in the equivalent time domain. One of the major tools used in exploring for base metals, Induced Polarization, depends on a mineral deposit’s non-linear, time- dependent, tensor character of the equivalent ground conductivity.

Constitutive relationships Classically, in Physics courses on EM Theory, the equivalence of the time-domain representations of the theory and the frequency-domain descriptions is not fully explored. The detail of equivalence is extremely important in geophysical EM for real materials. In fact, one of the basic “laws” of physics, the law of causality, is commonly ignored. For example, one does not observe a distributed current flux in a material until a distribution of electric field is applied. This much complicates practical exploration geophysical electromagnetics. See Baranyi et al., 1987