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#+title: Maxwell's Equations
#+author: Preston Pan
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* Introduction
Maxwell's Equations are a set of four differential equations in multiple dimensions that produce a complete classical
theory of electromagnetic phenomena.

* Derivations
These are the derivations of all four laws in their differential forms based on the [[id:2a543b79-33a0-4bc8-bd1c-e4d693666aba][inverse square]] law for [[id:32f0b8b1-17bc-4c91-a824-2f2a3bbbdbd1][electrostatics]] ([[id:5388f4e8-7bb8-452e-b997-fe9892aefcf3][Coulomb's Law]]), the [[id:658f3916-6b7f-4897-85c6-9acc82b13214][Lorentz Force]],
as well as the [[id:a871e62c-b4a0-4674-9dea-d377de2f780b][continuity equation]] and electromagnetic induction (which are just special cases of [[id:e38d94f2-8332-4811-b7bd-060f80fcfa9b][special relativity]]) as initial assumptions:
** Gauss' Law
This is given by the divergence of an [[id:2a543b79-33a0-4bc8-bd1c-e4d693666aba][inverse square]] field, specifically for an electric field which is the same as in [[id:32f0b8b1-17bc-4c91-a824-2f2a3bbbdbd1][electrostatics]]:
\begin{align*}
\vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon_{0}}
\end{align*}
** Divergence of Magnetic Field
The divergence of the magnetic field is the same as in [[id:5c36d0f1-06ad-436a-a56f-5ecc198b9b3e][magnetostatics]]:
\begin{align*}
\vec{\nabla} \cdot \vec{B} = 0
\end{align*}
** Ampere's Law with Modifications
The [[id:5c36d0f1-06ad-436a-a56f-5ecc198b9b3e][magnetostatic]] magnetic field is given by the Bio-Savart Law:
\begin{align*}
\vec{B} = \frac{\mu_{0}}{4\pi}\int_{V}\frac{\vec{J} \times \hat{r}}{r^{2}}d\tau
\end{align*}
Now the curl of this field is given by [[id:5c36d0f1-06ad-436a-a56f-5ecc198b9b3e][magnetostatics]]:
\begin{align*}
\vec{\nabla} \times \vec{B} = \mu_{0}\vec{J}
\end{align*}
However, if you take the divergence of this equation, the left hand side reduces to zero by the definition of the [[id:4bfd6585-1305-4cf2-afc0-c0ba7de71896][del operator]], but the
right hand side does not always:
\begin{align*}
\vec{\nabla} \cdot \mu_{0}\vec{J} = \mu_{0} (\vec{\nabla} \cdot \vec{J}) \neq 0
\end{align*}

Given this problem, a correction is given via the [[id:a871e62c-b4a0-4674-9dea-d377de2f780b][continuity equation]]:
\begin{align*}
\vec{\nabla} \cdot \vec{J} = -\frac{\partial \rho}{\partial t} \\
\epsilon_{0}(\vec{\nabla} \cdot \vec{E}) = \rho \\
\vec{\nabla} \cdot \vec{J} = -\epsilon_{0}\vec{\nabla} \cdot \frac{\partial\vec{E}}{\partial t}
\end{align*}
So therefore when we account for the fact that $\vec{\nabla} \cdot \vec{\nabla} \times \vec{B} = 0$
\begin{align*}
\vec{\nabla} \times \vec{B} = \mu_{0}\vec{J} + \mu_{0}\epsilon_{0}\frac{\partial\vec{E}}{\partial t}
\end{align*}
** Faraday's Law of Induction
By definition of electromagnetic induction (and to make Ampere's law consistent with relativity):
\begin{align*}
\vec{\nabla} \times \vec{E} = - \frac{\partial\vec{B}}{\partial t}
\end{align*}
Instead of assuming induction as an axiom, it is possible to fix Ampere's equation with the [[id:a871e62c-b4a0-4674-9dea-d377de2f780b][continuity equation]] first, and then
assume Lorentz Covariance. This explanation is a work in progress. Though, a possible explanation is through the fact
that this is the simplest way for Maxwell's equation to describe waves in a vacuum.
* Implications
Maxwell's Equations can be used to calculate all electromagnetic phenomena on the macro scale all the way down to the atom.
In practice, solving Maxwell's Equations can be analytically impossible, so several simplifying assumptions are often made.
To recap, these are the four equations:
\begin{align}
\vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon_{0}} \\
\vec{\nabla} \times \vec{E} = -\frac{\partial\vec{B}}{\partial t} \\
\vec{\nabla} \cdot \vec{B} = 0 \\
\vec{\nabla} \times \vec{B} = \mu_{0}\vec{J} + \mu_{0}\epsilon_{0}\frac{\partial\vec{E}}{\partial t}
\end{align}
* Speed of Light
Maxwell's Equations can be shown to reproduce the speed of light in a vacuum, where:
\begin{align*}
\mu_{0}\epsilon_{0} = \frac{1}{c^{2}}
\end{align*}
* Relativity
It is known that Maxwell's Equations are consistent with [[id:e38d94f2-8332-4811-b7bd-060f80fcfa9b][special relativity]] and can be expressed
in terms of curved spacetime. In fact, if relativity is taken as an axiom, it can be proven that the electric
and magnetic fields are descriptions of the same phenomena; this can be taken as a specific example of a [[id:1b1a8cff-1d20-4689-8466-ea88411007d7][duality]].
The result is a Lorentz Invariant theory of Electromagnetism.
** Gauss-Ampere Equation
The [[id:a871e62c-b4a0-4674-9dea-d377de2f780b][continuity equation]] has a Lorentz Invariant counterpart, as predicted by [[id:e38d94f2-8332-4811-b7bd-060f80fcfa9b][special relativity]]. We take a look at
Maxwell's equations (specifically Gauss' law and Ampere's law) in the uncondensed form:
\begin{align}
\partial_{x}E_{x} + \partial_{y}E_{y} + \partial_{z}E_{z} = \frac{\rho}{\epsilon_{0}} \\
\partial_{x}E_{x} + \partial_{y}E_{y} + \partial_{z}E_{z} = \mu_{0}c^{2}\rho \\
\frac{1}{c}(\partial_{x}E_{x} + \partial_{y}E_{y} + \partial_{z}E_{z}) = \mu_{0}j_{t}
\end{align}
Where $j_{t} = c\rho$. Once we formulate a covariant form of the continuity equation, this will become more clear.
And now the unexpanded version of Ampere's law:
\begin{align}
\partial_{y}E_{z} - \partial_{z}E_{y} = \mu_{0}j_{x} + \frac{1}{c}\partial_{t}E_{x} \\
\partial_{z}E_{x} - \partial_{x}E_{z}  = \mu_{0}j_{y} + \frac{1}{c}\partial_{t}E_{y} \\
\partial_{x}E_{y} - \partial_{y}E_{x}  = \mu_{0}j_{z} + \frac{1}{c}\partial_{t}E_{z}
\end{align}
(where $\partial_{t} = \frac{1}{c}\frac{\partial}{\partial t}$). We can now construct a 4-dimensional rank-2 tensor equation for these equations:
\begin{align}
D \cdot
\begin{pmatrix}
0 && \frac{1}{c}E_{x} && \frac{1}{c}E_{y} && \frac{1}{c}E_{z} \\
-\frac{1}{c}E_{x} && 0 && E_{z} && -E_{y} \\
-\frac{1}{c}E_{y} && -E_{z} && 0 && E_{x} \\
-\frac{1}{c}E_{z} && E_{y} && -E_{x} && 0 \\
\end{pmatrix}
= \mu_{0}
\begin{pmatrix}
j_{t} \\
j_{x} \\
j_{y} \\
j_{z}
\end{pmatrix}
\end{align}
where the right hand side is an emergent four-vector $(c\rho, j)$, and $D$ is an operator that takes the t, x, y,
and z derivative of each respective column and then sums the rows to make a vector. We can reformulate
the [[id:a871e62c-b4a0-4674-9dea-d377de2f780b][continuity equation]] in terms of this four-vector:
\begin{align*}
\vec{\nabla} \cdot \vec{j} = 0
\end{align*}
Where $\vec{\nabla}$ in this case represents the del operator but with a forth time dimension.
If we condense the notation, we can write:
\begin{align*}
D \cdot M = \vec{j}
\end{align*}
** Gauss-Faraday Equation
By the same logic, we can use the remaining two equations (the divergence and curl of $B$) to construct this tensor:
\begin{align}
D \cdot
\begin{pmatrix}
0 && -B_{x} && -B_{y} && -B_{z} \\
B_{x} && 0 && -B_{z} && B_{y} \\
B_{y} && B_{z} && 0 && -B_{x} \\
B_{z} && -B_{y} && B_{x} && 0 \\
\end{pmatrix}
= \vec{0}
\end{align}
we can write this in compact form:
\begin{align*}
D \cdot M' = \vec{0}
\end{align*}
$M'$ is the dual tensor of $M$, where all the $E_{n}$ are swapped with $B_{n}$, and it is negative. Note that
all these tensors are antisymmetric; they are symmetric under reflection across their diagonals and by flipping the signs.
** Covariant Form
The covariant form can be reached by gauge-fixing. Gauge fixing is a process wherein we utilize gauge invariance
(invariance to a scalar addition to a potential) in order to fix a single scalar and remove a redundancy. Remember that
in [[id:63713308-0ff7-433f-8103-8b64ba9bdbe1][electrostatics]]:
\begin{align}
\vec{E} = -\vec{\nabla}V
\end{align}
and in [[id:5c36d0f1-06ad-436a-a56f-5ecc198b9b3e][magnetostatics]]:
\begin{align}
\vec{B} = \vec{\nabla} \times \vec{A}
\end{align}
but in electrodynamics:
\begin{align}
\vec{E} = -\vec{\nabla}V - \frac{\partial \vec{A}}{\partial t} \\
\vec{B} = \vec{\nabla} \times \vec{A}
\end{align}
because any [[id:6f2aba40-5c9f-406b-a1fa-13018de55648][conservative force]] field added to $\vec{A}$ preserves the properties of $\vec{B}$ when you take the [[id:b25e0e44-c764-4f0a-a5ad-7f9d79c7660d][curl]].
You can verify that if we take it for granted that $(\frac{V}{c}, A)$ is covariant, we can formulate the above two tensor
equations in terms of a single equation:
\begin{align}
D \cdot (M + M') = \mu_{0}\vec{j} \\
\Box A = \mu_{0}\vec{j}
\end{align}
where $\Box = \partial^{2}_{t} - \vec{\nabla}$. This single equation is the covariant Maxwell's equation. Truly a beautiful sight.