P. Schmoldt, PhD - MTNet - DIAS

3. Mathematical description of electromagnetic relations
with ∇: Nabla operator, D: electric displacement field, Qe: electric charge density, E:
electric field, ∂t: partial derivative with respect to time, B: magnetic field, H: magnetising
field, and J f : electric current density of free charges. Maxwell found the four original
laws to be inconsistent and added the so-called Maxwellian term (∂t D) to Ampère’s Law
in order to provide a complete mathematical description of the physical relationship.
3.1.2. Ohm’s Law
Ohm’s Law describes the relationship between electric current J and electric field E for
the case of an ohmic conductor with conductivity σ, viz.
3.1.3. Vector calculus
J = σ E. (3.5)
Stokes theorem relates the rotation of a vector field F on a surface A to the flux of the
vector field through the boundary s of the volume
(∇ × F) · n dA = F · τ ds, (3.6)
A
with n and τ denoting the unit normal field and the unit tangential field of the surface,
respectively.
Gauss’s theorem (also referred to divergence theorem) relates the change of a vector
field F inside a volume V to the flux of the vector field through the surface A of the
volume
(∇ · F) dV = F · n dA, (3.7)
V
with n denoting the outward pointing unit normal field of the volume.
3.2. Deriving magnetotelluric parameters
The relation between electric response of a subsurface to an incident magnetic field is the
key element of MT, as electric conductivity (or its inverse the electric resistivity) can be
derived from the amplitude quotient and phase difference of the EM fields, using basic
equations (Sec. 3.1). Rewriting Equation 3.4 and using Ohm’s Law (Eq. 3.5) for the case
of a homogeneous, isotropic halfspace with use of the constitutive equations
32
A
s
H = B
µ , (3.8)
D = ε E, (3.9)