diff options
Diffstat (limited to 'mindmap/Maxwell's Equations.org')
| -rw-r--r-- | mindmap/Maxwell's Equations.org | 78 |
1 files changed, 78 insertions, 0 deletions
diff --git a/mindmap/Maxwell's Equations.org b/mindmap/Maxwell's Equations.org new file mode 100644 index 0000000..187f7a5 --- /dev/null +++ b/mindmap/Maxwell's Equations.org @@ -0,0 +1,78 @@ +:PROPERTIES: +:ID: fde2f257-fa2e-469a-bc20-4d11714a515e +:END: +#+title: Maxwell's Equations +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* 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 symmetry. 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]]. |