From cd871dbaf485545a872f043ca4d76541be001f06 Mon Sep 17 00:00:00 2001 From: Lindsey Heagy Date: Sat, 5 Apr 2014 12:07:27 -0700 Subject: [PATCH] I think the in-text math should be straightened out now! --- docs/api_FDEM.rst | 34 +++++++++++++++++----------------- 1 file changed, 17 insertions(+), 17 deletions(-) diff --git a/docs/api_FDEM.rst b/docs/api_FDEM.rst index 43286e1d..2a8c2268 100644 --- a/docs/api_FDEM.rst +++ b/docs/api_FDEM.rst @@ -19,14 +19,14 @@ Electromagnetic phenomena are governed by Maxwell's equations. They describe the Fourier Transform Convention ---------------------------- -In order to examine Maxwell's equations in the frequency domain, we must first define our choice of harmonic time-dependence by choosing a Fourier transform convention. We use the \\(\\ e^{i\omega t} \\)\\ convention, so we define our Fourier Transform pair as +In order to examine Maxwell's equations in the frequency domain, we must first define our choice of harmonic time-dependence by choosing a Fourier transform convention. We use the \\(\\ e^{i \omega t} \\)\\ convention, so we define our Fourier Transform pair as .. math :: F(\omega) = \int_{-\infty}^{\infty} f(t) e^{- i \omega t} dt \\ - f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega) e^{i \omega t} d\omega + f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega) e^{i \omega t} d \omega -where \\ \omega\\ is angular frequency, \\t\\ is time, \\ F(\omega) \\ is the function defined in the frequency domain and \\ f(t) \\ is the function defined in the time domain. +where \\(\\omega\\) is angular frequency, \\(t\\) is time, \\(F(\omega)\\) is the function defined in the frequency domain and \\(f(t)\\) is the function defined in the time domain. Maxwell's Equations @@ -36,20 +36,20 @@ In the frequency domain, Maxwell's equations are given by .. math :: \curl \vec{E} = - i \omega \vec{B} \\ - \curl \vec{H} = \vec{J} + i \omega \vec{D} + \vec{J_s} \\ + \curl \vec{H} = \vec{J} + i \omega \vec{D} + \vec{J}_s \\ \div \vec{B} = 0 \\ \div \vec{D} = \rho_f where: -- \\(\\ \vec{E} \\)\\ : electric field (\\(\\V/m\\)\\) -- \\(\\ \vec{H} \\)\\ : magnetic field (\\(\\A/m\\)\\) -- \\(\\ \vec{B} \\)\\ : magnetic flux density (\\(\\Wb/m^2\\)\\) -- \\(\\ \vec{D} \\)\\ : electric displacement / electric flux density (\\(\\C/m^2\\)\\) -- \\(\\ \vec{J} \\)\\ : electric current density (\\(\\A/m^2\\)\\) -- \\(\\ \rho_f \\)\\ : free charge density -The source term is \\(\\ \vec{J_s} \\)\\ +- \\(\\vec{E}\\) : electric field (\\(V/m\\)) +- \\(\\vec{H}\\) : magnetic field (\\(A/m\\)) +- \\(\\vec{B}\\) : magnetic flux density (\\(Wb/m^2\\)) +- \\(\\vec{D}\\) : electric displacement / electric flux density (\\(C/m^2\\)) +- \\(\\vec{J}\\) : electric current density (\\(A/m^2\\)) +- \\(\\rho_f\\) : free charge density +The source term is \\(\\vec{J}_s\\) Constitutive Relations @@ -64,21 +64,21 @@ The fields and fluxes are related through the constitutive relations. At each fr \vec{D} = \varepsilon \vec{E} where -- \\(\\ \sigma \\)\\ : electrical conductivity (S/m) -- \\(\\ \mu \\)\\ : magnetic permeability (H/m) -- \\(\\ \varepsilon \\)\\ : dielectric permittivity (F/m) +- \\(\\sigma\\) : electrical conductivity \\(S/m\\) +- \\(\\mu\\) : magnetic permeability \\(H/m\\) +- \\(\\varepsilon\\) : dielectric permittivity \\(F/m\\) -\\(\\ \sigma \\)\\, \\(\\ \mu \\)\\, \\(\\ \varepsilon \\)\\ are physical properties which depend on the material. \\(\\ \sigma \\)\\ describes how easily electric current passes through a material, \\(\\ \mu \\)\\ describes how easily a material is magnetized, and \\(\\ \varepsilon \\)\\ describes how easily a material is electrically polarized. In most geophysical applications of EM, \\(\\ \sigma \\)\\ is the the primary physical property of interest, and \\(\\ \mu \\)\\, \\(\\ \varepsilon \\)\\ are assumed to have their free-space values \\(\\ \mu_0 = 4\pi \times 10^{-7} H/m \\)\\, \\(\\ \varepsilon_0 = 8.85 \times 10^{-12} F/m\\)\\ +\\(\\sigma\\), \\(\\mu\\), \\(\\varepsilon\\) are physical properties which depend on the material. \\(\\sigma\\) describes how easily electric current passes through a material, \\(\\mu\\) describes how easily a material is magnetized, and \\(\\varepsilon\\) describes how easily a material is electrically polarized. In most geophysical applications of EM, \\(\\sigma\\) is the the primary physical property of interest, and \\(\\mu\\), \\(\\varepsilon\\) are assumed to have their free-space values \\(\\mu_0 = 4\pi \times 10^{-7} H/m \\), \\(\\varepsilon_0 = 8.85 \times 10^{-12} F/m\\) For a more complete discussion of physical properties see `GPG `_ Quasi-static Approximation -------------------------- -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 +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 .. math :: \nabla \times \vec{E} = -i \omega \vec{B} \\ - \nabla \times \vec{H} = \vec{J} + \vec{J_s} + \nabla \times \vec{H} = \vec{J} + \vec{J}_s Fields from a Dipole