:PROPERTIES:
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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, which can be derived from [[id:e38d94f2-8332-4811-b7bd-060f80fcfa9b][special relativity]] and [[id:5388f4e8-7bb8-452e-b997-fe9892aefcf3][Coulomb's 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.
* 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.
** Relativistic Electrodynamics
The [[id:a871e62c-b4a0-4674-9dea-d377de2f780b][continuity equation]] has a Lorentz Invariant counterpart. We unify the charge density and the current density
under a four-vector quantity ~j~:
\begin{align*}
j = (\rho , J)
\end{align*}
Where ~J~ is the current density, and $$ \rho $$ is the charge density. The continuity equation can be reformulated
like so:
\begin{align*}
\partial_{\alpha} j^{\alpha} = 0
\end{align*}
Likewise, the electric scalar potential and the magnetic vector potential create an invariant four-vector:
\begin{align*}
a = (V, A)
\end{align*}
We notice that Gauss' Law:
\begin{align*}
\nabla^{2} V = \frac{\rho}{\epsilon_{0}}
\end{align*}
needs a more general formulation, as the potential and the charge density need to be replaced with Lorentz Invariant
quantities. We then use Ampere's law:
\begin{align}
\vec{\nabla} \times \vec{\nabla} \times \vec{A} = \mu_{0}\vec{J} + \mu_{0}\epsilon_{0}\frac{\partial(\nabla V)}{\partial t} \\
\vec{\nabla} \cdot \vec{\nabla} \times \vec{\nabla} \times \vec{A} = \nabla \cdot \mu_{0}\vec{J} + \mu_{0}\epsilon_{0}\frac{\partial(\nabla^{2} V)}{\partial t} \\
\vec{\nabla} \cdot \vec{\nabla} \times \vec{\nabla} \times \vec{A} = \mu_{0} \nabla\cdot\vec{J} + \mu_{0}\frac{\partial\rho}{\partial t} \\
\vec{\nabla} \cdot \vec{\nabla} \times \vec{\nabla} \times \vec{A} = \mu_{0} (\nabla\cdot\vec{J} + \frac{\partial\rho}{\partial t}) \\
\end{align}