Maxwell fundamental edits

content/maxwell1_fundamentals/appendix/appendix_index.rst

@@ -0,0 +1,16 @@
+.. _maxwell1_appendix_index:
+
+Appendix
+========
+
+Here, supplementary derivations and content relevant to *Maxwell 1: Fundaments* is provided.
+
+**Contents**
+
+.. toctree::
+    :maxdepth: 1
+
+    wave_eq_derivation
+    totalrefl_and_brewsterangl
+
+

content/maxwell1_fundamentals/reflection_and_refraction/totalrefl_and_brewsterangl.rst → content/maxwell1_fundamentals/appendix/totalrefl_and_brewsterangl.rst

@@ -1,16 +1,16 @@
 .. _totalrefl_and_brewsterangl:
 
-Total reflection and Brewster angle
-===================================
+Total Reflection and Brewster's Angle
+=====================================
 
 .. purpose::
 
     We first identify total reflection and brewster angle for a dielectric media, then relate them to conductive medium.
 
-Total reflection
+Total Reflection
 ----------------
 
-For a perfect dielectric, the conductivity is zero and the permeability is that of free space that is, :math:`\mu_1=\mu_2=\mu_0`. Snell's law of refraction shown in :ref:`snells_law` with the setup shown in :numref:`snellslaw_setup` then reduces to
+For a perfect dielectric, the conductivity is zero and the permeability is that of free space that is, :math:`\mu_1=\mu_2=\mu_0`. In this case, Snell's law reduces to:
 
 .. math::
     \frac{\text{sin} \theta_i}{\text{sin} \theta_t} = \frac{k_1}{k_2} = \Big(\frac{\epsilon_2}{\epsilon_1}\Big)^{1/2} = n_{12}
@@ -26,8 +26,8 @@ where :math:`n_{12}` is the relative index of refraction. If :math:`\epsilon_2 >
    Tranmission angle :math:`\theta_t` as a function of the incident angle :math:`\theta_i` when :math:`\sigma_1` = 1 S/m and :math:`\sigma_2` = 0.1 S/m. Magnetic permeability and dielectric permittivitivy assumed to be those of free-space (:math:`\epsilon = \epsilon_0` and :math:`\mu = \mu_0`)
 
 
-Brewster angle
---------------
+Brewster's Angle
+----------------
 
 From derived reflection coefficients for TE mode in :ref:`fresnel_equations`, the reflection coefficient for perfect dielectric can be written as
 

content/maxwell1_fundamentals/looking_for_more.rst → content/maxwell1_fundamentals/appendix/wave_eq_derivation.rst

@@ -1,36 +1,28 @@
-.. _maxwell1_looking_for_more:
+.. _maxwell1_appendix_wave_eq_derivation_time:
 
-Looking for more?
-=================
+Derivation of the Wave Equation in Time
+=======================================
 
-.. _time_domain_equations_details:
+Here, we derive the wave equations in time for the electric and magnetic fields.To accomplish this, we begin with :ref:`Faraday's Law <faraday>` and :ref:`Ampere-Maxwell's Law <ampere_maxwell>`:
 
-Maxwell's equations in time
----------------------------
+.. include:: ../../equation_bank/faraday_time.rst
 
-To derive the wave equations in the time domain, we begin with :ref:`Faraday's
-Law <faraday>` and :ref:`Ampere-Maxwell's Law <ampere_maxwell>`:
+.. include:: ../../equation_bank/ampere_maxwell_time.rst
 
-.. include:: ../equation_bank/faraday_time.rst
+as well as the three constitutive relations:
 
-.. include:: ../equation_bank/ampere_maxwell_time.rst
-
-and the three constitutive relations:
-
-.. include:: ../equation_bank/ohms_law_time.rst
+.. include:: ../../equation_bank/ohms_law_time.rst
 
 .. math:: \mathbf{d} = \epsilon \mathbf{e}
         :name: depse
 
 .. math:: \mathbf{b} = \mu \mathbf{h}
         :name: bmuh
 
-The derivations will be done for  first :math:`\mathbf{e}` and then for :math:`\mathbf{h}`.
+Derivation for the Electric Field
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
 
-Derivation for electric field
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-To derive the wave equation which uses only :math:`\mathbf{e}`, we first take
+To derive the wave equation for :math:`\mathbf{e}`, we first take
 the curl of Faraday's Law, shown in equation :eq:`faraday_time`:
 
 .. math:: \boldsymbol{\nabla} \times (\boldsymbol{\nabla} \times \mathbf{e}) = - \boldsymbol{\nabla} \times \frac{\partial \mathbf{b}}{\partial t}
@@ -43,7 +35,7 @@ The appropriate constitutive relations can be substituted into Equation
 .. math:: \boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \mathbf{e} = - \boldsymbol{\nabla} \times \left (  \frac{\partial}{\partial t} (\mu \mathbf{h}) \right )
         :name: hme2
 
-Because we assume a homogenous space, the physical properties :math:`\mu`,
+Assuming the physical properties are homogeneous throughout the domain, :math:`\mu`,
 :math:`\epsilon`, and :math:`\sigma` can be moved out front of the derivative
 terms. This simplifies the above expressions:
 
@@ -86,11 +78,11 @@ into :eq:`hme5`, we get the following expression:
 
 This is the wave equation for the electric field in the time domain.
 
-Derivation for magnetic field
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+Derivation for the Magnetic Field
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
 
 To derive the wave equation for :math:`\mathbf{h}`, we repeat the above
-derivation but now start with taking the curl of Ampere's Law, shown in
+derivation but start by taking the curl of Ampere's Law, shown in
 equation :eq:`ampere_maxwell_time`:
 
 .. math:: \boldsymbol{\nabla} \times (\boldsymbol{\nabla} \times \mathbf{h}) = \boldsymbol{\nabla} \times \mathbf{j} + \boldsymbol{\nabla} \times \frac{\partial \mathbf{d}}{\partial t}
@@ -139,74 +131,6 @@ into the wave equation. The following shows these derivations.
 .. math:: \boldsymbol{\nabla}^2 \mathbf{h} - \epsilon \mu \frac{\partial^2 \mathbf{h}}{\partial t^2} - \sigma \mu \frac{\partial \mathbf{h}}{\partial t} = 0
         :name: hmh6
 
-Equation :eq:`hmh6` is then the wave equation for the magnetic field in the tim domain.
-
-Summary
-^^^^^^^
-
-We now have two wave equations or second-order differential equations; one for
-the electric field and one for the magnetic field, summarized in Equations
-:eq:`hme7` and :eq:`hmh7`.
-
-.. math::  \boldsymbol{\nabla}^2 \mathbf{e} - \mu \sigma \frac{\partial \mathbf{e}}{\partial t} - \mu \epsilon \frac{\partial^2 \mathbf{e}}{\partial t^2}  = 0
-        :name: hme7
-
-.. math:: \boldsymbol{\nabla}^2 \mathbf{h} - \mu \sigma \frac{\partial \mathbf{h}}{\partial t} - \mu \epsilon \frac{\partial^2 \mathbf{h}}{\partial t^2}  = 0
-        :name: hmh7
-
-Solving the wave equation when :math:`\sigma=0`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-In the case where :math:`\sigma` is zero, Equations :eq:`hme7` and :eq:`hmh7` become:
-
-.. math::  \boldsymbol{\nabla}^2 \mathbf{e} - \mu \epsilon \frac{\partial^2 \mathbf{e}}{\partial t^2}  = 0
-        :name: hme13
-
-.. math:: \boldsymbol{\nabla}^2 \mathbf{h} - \mu \epsilon \frac{\partial^2 \mathbf{h}}{\partial t^2}  = 0
-        :name: hmh14
-
-The following derivation will be the same for both the magnetic and electric
-fields. Let's use the magnetic field for this analysis. In 1D, Equation
-:eq:`hmh14` is now written as:
-
-.. math:: \frac{\partial^2 \mathbf{h}}{\partial z^2} = \mu \epsilon \frac{\partial^2 \mathbf{h}}{\partial t^2},
-        :name: hmh15
-
-for which there is a solution of the form:
-
-.. math:: \mathbf{h} = \mathbf{h}_0 \cos \left ( 2\pi \frac{z-vt}{\lambda} \right ),
-        :name: slwave
-
-where :math:`v` is the speed of the sinosoidal wave and :math:`\lambda` is its
-wavelength. We can check this solution by taking the derivatives with respect
-to :math:`z` and :math:`t` to see if we get back to Equation :eq:`hmh15`.
-
-.. math:: \frac{\partial^2 \mathbf{h}}{\partial z^2} = \frac{\partial}{\partial z} \left [ - \mathbf{h}_0 \sin \left(  2\pi \frac{z-vt}{\lambda} \right) \left( \frac{2\pi}{\lambda}\right) \right ] = - \mathbf{h}_0 \cos \left(  2\pi \frac{z-vt}{\lambda} \right) \left( \frac{2\pi}{\lambda}\right)^2
-
-.. math:: \frac{\partial^2 \mathbf{h}}{\partial t^2} =\frac{\partial}{\partial t} \left [ - \mathbf{h}_0 \sin \left ( 2\pi \frac{z-vt}{\lambda} \right ) \left ( \frac{-2\pi v}{\lambda} \right) \right ] = - \mathbf{h}_0 \cos \left ( 2\pi \frac{z-vt}{\lambda} \right ) \left ( \frac{-2\pi v}{\lambda} \right)^2
-
-We now substitude these solutions into Equation :eq:`hmh15` and simplify:
-
-.. math:: \left (\frac{2\pi}{\lambda} \right)^2 = \epsilon \mu \left (\frac{-2\pi v}{\lambda} \right)^2
-
-.. math:: 1 = \epsilon \mu v^2
-
-.. math:: v = \sqrt{\frac{1}{\epsilon \mu}}
-        :name: v
-
-This shows that the solution to Equation :eq:`hmh15` is a wave of the form
-given in Equation :eq:`slwave` if Equation :eq:`v` holds true. In free space,
-we can quickly evaluate Equation :eq:`v`, knowing that :math:`\mu = 4\pi
-\times 10^{-7} \frac{T \cdot m}{A}` and :math:`\epsilon = 8.85 \times 10^{-12}
-\frac{F}{m}`:
-
-.. math:: v = \sqrt{\frac{1}{( 4\pi \times 10^{-7})(8.85 \times 10^{-12})}} = 3 \times 10^8
-
-The units of :math:`v` work out as following:
-
-.. math:: \sqrt{ \frac{1}{\left[\frac{T \cdot m}{A} \right] \left [ \frac{F}{m} \right ]} } = \left ( \frac{A}{T \cdot F} \right) ^ {1/2} = \left ( \frac{\frac{C}{s}}{\frac{V\cdot s}{m^2} \cdot F} \right) ^ {1/2} =
+Equation :eq:`hmh6` is then the wave equation for the magnetic field in the time domain.
 
-.. math:: \left ( \frac{\frac{F \cdot V}{s}}{\frac{V\cdot s}{m^2} \cdot F} \right) ^ {1/2} = \left ( \frac{F\cdot V}{s} \frac{m^2}{V \cdot F \cdot s} \right ) ^{1/2} = \frac{m}{s}
 
-Thus, the velocity of the wave is :math:`3 \times 10^8` m/s, which is the
-speed of light! The same derivation can be done using the electric field.

content/maxwell1_fundamentals/dipole_sources_in_homogeneous_media/electric_dipole_frequency/fields.rst

@@ -10,48 +10,7 @@ Visualization of the Electromagnetic Fields
     By completing this exercise, you will become comfortable with the numerical modeling tools provided and gain a fundamental understanding of the fields which are caused by a harmonic electrical current dipole.
 
 
-
-Introduction
-------------
-
-Here, we will show you how to use the widget and walk you through some research questions.
-
-
-
-**Getting Started**
-
-Within the jupyter notebook, there are 3 tools:
-
-
-.. figure:: images/E_source_widget_geometry.png
-		:align: right
-		:figwidth: 50%
-		:name: widget_geometry
-
-		Visualization of problem geometry in 3D.
-
-
-**Step 1: Choosing Geometric Parameters**
-
-These toggles allow the user to adjust the problem geometry and visualize it in 3D.
-Relative to the source dipole, the user may specify observation locations on a profile line or comprising a plane.
-An example of the problem geometry is shown on the right.
-
-
-**Step 2: Visualizing the Fields**
-
-These toggles allow the user to change physical parameters relevant to the problem and visualize different components of the electric field, magnetic field and current density.
-As any of the physical parameters or plotting parameters are changed, the fields are automatically plotted on the profile line and plane defined in Step 1.
-An example of the fields on a profile line and on a plane is shown below.
-
-
-.. figure:: images/E_source_plane_example.png
-		:align: center
-		:figwidth: 100%
-		:name: field_example
-
-		Visualization of the fields. (Left panel) Vector plot of :math:`E_x` at locations on a plane. (Right panel) In-phase component of :math:`E_x` along a profile line.
-
+**Link to the modeling app**
 
 Research Questions
 ------------------

content/maxwell1_fundamentals/dipole_sources_in_homogeneous_media/index.rst

@@ -13,6 +13,8 @@ Dipole Sources in Homogeneous Media
     	- Asymptotic expressions are provided for several cases.
     	- Numerical modeling tools are made available for investigating the dependency of the electric and magnetic fields on various parameters.
 
+**Introduction**
+
 Dipole sources are fundamental electromagnetic sources which exist at a single point in space.
 Although true dipole sources do not exist in nature, they do very well at approximating the electromagnetic sources used for many geophysical applications.
 In geophysics, there are two types of dipole sources: electrical current dipole sources and magnetic dipole sources.

content/maxwell1_fundamentals/dipole_sources_in_homogeneous_media/magnetic_dipole_frequency/fields.rst

@@ -10,31 +10,7 @@ Visualization of the Electromagnetic Fields
     By completing this exercise, you will become comfortable with the numerical modeling tools provided and gain a fundamental understanding of the fields which are caused by a harmonic magnetic dipole.
 
 
-
-Introduction
-------------
-
-Here, we will show you how to use the widget and walk you through some research questions.
-
-
-
-**Getting Started**
-
-Within the jupyter notebook, there are 3 tools:
-
-**Step 1: Choosing Geometric Parameters**
-
-These toggles allow the user to adjust the problem geometry and visualize it in 3D.
-Relative to the source dipole, the user may specify observation locations on a profile line or comprising a plane.
-An example of the problem geometry is shown on the right.
-
-
-**Step 2: Visualizing the Fields**
-
-These toggles allow the user to change physical parameters relevant to the problem and visualize different components of the electric field, magnetic field and current density.
-As any of the physical parameters or plotting parameters are changed, the fields are automatically plotted on the profile line and plane defined in Step 1.
-An example of the fields on a profile line and on a plane is shown below.
-
+**Link to the app**
 
 Research Questions
 ------------------
@@ -68,24 +44,20 @@ Now slowly increase the frequency by factors of 10. When you reach 1000 Hz, noti
 
 **The Inductive Response:**
 
-According to `Faraday's law<faraday>`, the effects of EM induction increase as frequency increases. Set the conductivity to 0.1 S/m and choose a point (x,y,z) = (40m, 0m, 0m). Examine the x,y and z components of the electric and magnetic fields.
+According to :ref:`Faraday's law<faraday>`, the effects of EM induction increase as frequency increases. Set the conductivity to 0.1 S/m and choose a point (x,y,z) = (40m, 0m, 0m). Examine the x,y and z components of the electric and magnetic fields.
 
 	- At what frequency do the effects of EM induction become significant?
 	- Now increase the background conductivity to 1 S/m and examine the same location. At what frequency do the effects of EM induction become significant?
 	- Now choose a location closer to the dipole source (x,y,z) = (10m, 0m, 0m). At what frequency do the effects of EM induction become significant compared to the primary field?
 
 **Magnetic Permeability and Dielectric Permittivity**
 
-Set the log-conductivity to .
-
+Set the log-conductivity to 0.01 S/m.
 
 	- Try increasing the relative permeability (:math:`\mu_r`). Do you notice any significant changes in the shape and amplitude of the electric and magnetic fields?
 	- Now try increasing the relative permittivity (:math:`\varepsilon_r`). When you do this at low frequencies, do you notice any significant changes in the shape and amplitude of the electric and magnetic fields? How about when you do this at high frequencies?
 
 
-.. **Hypothetical Scenario 1:**
-
-.. *I put this here in case we wanted to make a hypthetical scenario where these equations could be used to solve a practical problem.*
 
 
 

content/maxwell1_fundamentals/formative_laws/details.rst → content/maxwell1_fundamentals/formative_laws/coulomb.rst

@@ -1,7 +1,7 @@
-.. _fundamental_laws_details:
+.. _fundamental_laws_coulomb:
 
-Details
-=======
+Gauss's Law from Coulomb's Law
+==============================
 
 
 .. _gauss_electric_equivalence_to_coulombs_law:

content/maxwell1_fundamentals/formative_laws/gauss_electric.rst

@@ -78,22 +78,58 @@ integrands, giving the differential form of Gauss's law:
 It can be shown that Gauss' law for electric fields is equivalent to Coulomb's
 law (see :ref:`gauss_electric_equivalence_to_coulombs_law`)
 
+Gauss's Law in Matter
+---------------------
+
+Gauss's law for electric fields is most easily understood by neglecting :ref:`electric displacement<dielectric_permittivity_index>` (:math:`\mathbf{d}`). In matter, the :ref:`dielectric permittivity<dielectric_permittivity_index>` may not be equal to the permittivity of free-space (i.e. :math:`\varepsilon \neq \varepsilon_0`). In matter, the density of electric charges can be separated into a "free" charge density (:math:`\rho_f`) and a "bounded" charge density (:math:`\rho_b`), such that:
+
+.. math::
+	\rho = \rho_f + \rho_b
+	:label: gauss_law_charge_decomp
+
+The free-charge density refers to charges which flow freely under the application of an electric field; i.e. they produce a current which is divergence-free. The bounded-charge density refers to electrical charges attributed to electrical polarization (:math:`\mathbf{p}`). By combining Eqs. :eq:`Gauss_e_diff` and :eq:`gauss_law_charge_decomp` with our definition for :ref:`electrical polarization<dielectric_permittivity_index>`, we find that:
+
+.. math::
+	\nabla \cdot \mathbf{d} - \nabla \cdot \mathbf{p} = \rho_f + \rho_b
+	:label:
+
+By using the constitutive relationship :math:`\mathbf{d} = \varepsilon \mathbf{e}` and separating the previous equation into bounded and free contributions, we find that:
+
+.. math::
+	-\nabla \cdot \mathbf{p} = \rho_b
+	:label:
+
+and
+
+.. math::
+	\nabla \cdot \mathbf{d} = \rho_f
+	:label:
+
+The above equation is the **differential form of Gauss's equation in matter**. Meanwhile, the **integral form of Gauss's equations in matter** is given by:
+
+.. math::
+	\int_V \nabla \cdot \mathbf{d} \; dV = \oint_S \mathbf{d} \cdot \mathbf{\hat n} \; da = Q_f
+
+where :math:`Q_f` is the total enclosed free charge.
+
 Units
 -----
 
-+-----------------------+---------------------+------------------------------------+---------------------------------------+
-|     Surface area      |  :math:`\text{S}`   | :math:`\text{m}^{2}`               |      Square meter                     |
-+-----------------------+---------------------+------------------------------------+---------------------------------------+
-|     Volume            |  :math:`V`          | :math:`\text{m}^{3}`               |                  Cubic meter          |
-+-----------------------+---------------------+------------------------------------+---------------------------------------+
-|     Electric charge   | :math:`\text{q, Q}` | :math:`\text{C}`                   |            Coulomb                    |
-+-----------------------+---------------------+------------------------------------+---------------------------------------+
-|Electric charge density| :math:`\rho`        |:math:`\frac{\text{C}}{\text{m}^3}` |  Coulomb per cubic meter              |
-+-----------------------+---------------------+------------------------------------+---------------------------------------+
-|     Electric field    | :math:`\mathbf{e}`  |:math:`\frac{\text{V}}{\text{m}}`   | Volt per meter                        |
-+-----------------------+---------------------+------------------------------------+---------------------------------------+
-|Electrical permittivity|:math:`\varepsilon_0`| :math:`\frac{\text{F}}{\text{m}}`  | Farad per meter                       |
-+-----------------------+---------------------+------------------------------------+---------------------------------------+
++-----------------------+-----------------------------+---------------------+-------------------------+
+|     Surface area      |  :math:`\text{S}`           | m :math:`\! ^{2}`   | Square meter            |
++-----------------------+-----------------------------+---------------------+-------------------------+
+|     Volume            |  :math:`V`                  | m :math:`\! ^{3}`   |  Cubic meter            |
++-----------------------+-----------------------------+---------------------+-------------------------+
+|     Electric charge   | :math:`q, Q, Q_f`           | C                   |  Coulomb                |
++-----------------------+-----------------------------+---------------------+-------------------------+
+|Electric charge density| :math:`\rho, \rho_f, \rho_b`| C/m :math:`\! ^{3}` | Coulomb per cubic meter |
++-----------------------+-----------------------------+---------------------+-------------------------+
+|     Electric field    | :math:`\mathbf{e}`          | V/m                 | Volt per meter          |
++-----------------------+-----------------------------+---------------------+-------------------------+
+|Electric displacement  | :math:`\mathbf{d}`          | A/m :math:`\! ^{2}` | Volt per meter          |
++-----------------------+-----------------------------+---------------------+-------------------------+
+|Dielectric permittivity|:math:`\varepsilon`          | F/m                 | Farad per meter         |
++-----------------------+-----------------------------+---------------------+-------------------------+
 
 **Conversions**