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slight improvement on intro / background and second attempt at fixing the math
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@@ -9,11 +9,13 @@
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Frequency Domain Electromagnetics
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*********************************
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Intro Here
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Electromagnetic (EM) geophysical methods are used in a variety of applications from resource exploration, including for hydrocarbons and minerals, to environmental applications, such as groundwater monitoring.
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Background
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==========
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Electromagnetic (EM) geophysical methods are used in a variety of applications from resource exploration, including for hydrocarbons and minerals, to environmental applications, such as groundwater monitoring.
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Electromagnetic phenomena are governed by Maxwell's equations. They describe the behavior of EM fields and fluxes. Electromagnetic theory for geophysical applications by Ward and Hohmann (1988) is a highly recommended resource on this topic.
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Fourier Transform Convention
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----------------------------
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@@ -31,7 +33,7 @@ Maxwell's Equations
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===================
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In the frequency domain, Maxwell's equations are given by
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.. math::
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.. math ::
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\curl \vec{E} = - i \omega \vec{B} \\
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\curl \vec{H} = \vec{J} + i \omega \vec{D} + \vec{J_s} \\
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@@ -54,7 +56,7 @@ Constitutive Relations
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----------------------
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The fields and fluxes are related through the constitutive relations. At each frequency, they are given by
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.. math::
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.. math ::
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\vec{J} = \sigma \vec{E} \\
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\vec{B} = \mu \vec{H} \\
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@@ -74,7 +76,7 @@ Quasi-static Approximation
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--------------------------
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For the frequency range typical of most geophysical surveys, the contribution of the electric displacement is negligible compared to the electric current density. In this case, we use the \emph{Quasi-static approximation} and assume that this term can be neglected, giving
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.. math::
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.. math ::
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\nabla \times \vec{E} = -i \omega \vec{B} \\
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\nabla \times \vec{H} = \vec{J} + \vec{J_s}
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