From aecb9413e3e881d59b12dfa6debcced74c31337c Mon Sep 17 00:00:00 2001 From: Sidonie Bouthors Date: Tue, 14 Jan 2025 20:38:10 +0100 Subject: [PATCH] feat: publish maxwell equations --- content/share/maxwell-equations.mdx | 238 ++++++++++++++++++++++++++++ 1 file changed, 238 insertions(+) create mode 100644 content/share/maxwell-equations.mdx diff --git a/content/share/maxwell-equations.mdx b/content/share/maxwell-equations.mdx new file mode 100644 index 0000000..ce4a4b3 --- /dev/null +++ b/content/share/maxwell-equations.mdx @@ -0,0 +1,238 @@ +--- +title: Maxwell's Equations +description: A quick cheatsheet on Maxwell's Equations. Written as part of my notes for the PHYS-114 Course @ EPFL +date: 2023-01-05 +tags: [epfl, electromagnetism] +published: true +--- + +# Maxwell's Equations + + + This document is a quick cheatsheet on Maxwell's Equations. + It is not meant to be a comprehensive guide, but rather a quick reference, based on my notes from the EPFL's [PHYS-114 Course](https://edu.epfl.ch/coursebook/fr/physique-generale-electromagnetisme-PHYS-114) on Electromagnetism. + + +## Deriving the Equations + +### First Equation + +Gauss's Law (Electric flux through a closed surface) + +$$ +\Phi_E = \frac{Q}{\varepsilon_0} +$$ + +$$ +\iint_{\partial \Omega} \mathbf E \cdot d \mathbf s = \frac{Q}{\varepsilon_0} = 4 \pi k_e Q +$$ + +$$ +\iint_{\partial \Omega} \mathbf E \cdot d \mathbf s = 4\pi k_e \iiint_{\Omega} \rho \; dV +$$ + +Using the Divergence Theorem : + +$$ +\iiint_{\Omega} \nabla \cdot \mathbf E \; dV = 4\pi k_e \iiint_{\Omega} \rho \; dV +$$ +$$ +\nabla \cdot \mathbf E = 4\pi k_e \rho +$$ +$$ +\nabla \cdot \mathbf E = \frac{\rho}{\varepsilon_0} +$$ + +### Second Equation + +Gauss's Law for Magnetism (Magnetic flux through a closed surface) +$$ +\Phi_B = \iint_{\partial \Omega} \mathbf B \cdot d \mathbf s +$$ +$$ +\iint_{\partial \Omega} \mathbf B \cdot d \mathbf s = 0 +$$ +Using the Divergence Theorem : +$$ +\iiint_{\Omega} \nabla \cdot \mathbf B \; dV = 0 +$$ +$$ +\nabla \cdot \mathbf B = 0 +$$ + +This is equivalent to saying : +- Magnetic monopoles / charges do not exist (base entity is the dipole) +- Magnetic field lines have neither a beginning nor an end + +### Third Equation + +Faraday's Law (electromotive force, emf) +$$ +\mathcal{E} = - \partial_t \Phi_B +$$ +$$ +\mathcal{E} = \int_{\partial\Sigma} \mathbf E \cdot d \mathbf l = - \partial_t \Phi_B = - \partial_t \iint_{\Sigma} \mathbf B \cdot d \mathbf s +$$ +$$ +\int_{\partial\Sigma} \mathbf E \cdot d \mathbf l = - \partial_t \iint_{\Sigma} \mathbf B \cdot d \mathbf s +$$ + +Using Stokes Theorem : +$$ +\iint_{\Sigma} \nabla \times \mathbf E \; d \mathbf s = - \partial_t \iint_{\Sigma} \mathbf B \cdot d \mathbf s +$$ +$$ +\nabla \times \mathbf E = - \partial_t \mathbf B +$$ + +Faraday's Law : + +The electromotive force around a closed path is equal to the negative of the time rate of change of the magnetic flux enclosed by the path. + +### Fourth Equation + +Ampere's Law +$$ +\int_{\partial \Sigma} \mathbf B \cdot d \mathbf l = \mu_0 I_{\text{encl}} = \mu_0 \iint_{\Sigma} \mathbf j \cdot d \mathbf s +$$ +Maxwell's equation has the following component added to it : +$$ +\mu_0 \varepsilon_0 \;\partial_t \iint_{\Sigma} \mathbf E \cdot d \mathbf s$$ +$$\int_{\partial \Sigma} \mathbf B \cdot d \mathbf l = \mu_0 \left( \iint_{\Sigma} \mathbf j \cdot d \mathbf s + \varepsilon_0 \; \partial_t \iint_{\Sigma} \mathbf E \cdot d \mathbf s \right) +$$ + +Using Stoke's Theorem : +$$ +\iint_{\Sigma} \nabla \times \mathbf B \; d \mathbf s = \mu_0 \left( \iint_{\Sigma} \mathbf j \cdot d \mathbf s + \varepsilon_0 \;\partial_t \iint_{\Sigma} \mathbf E \cdot d \mathbf s \right) +$$ +$$ +\nabla \times \mathbf B = \mu_0 \left( \mathbf j + \varepsilon_0 \; \partial_t \mathbf E\right) +$$ + +Ampère's Law : + +The original law states that magnetic fields relate to electric current, Maxwell's addition states that they also relate to changing electric fields + +Note : +$\mu_0 = 4 \pi k_M$ + +## Differential Forms + +$\nabla \cdot \mathbf E = \frac{\rho}{\varepsilon_0}$ + +$\nabla \cdot \mathbf B = 0$ + +$\nabla \times \mathbf E = - \partial_t \mathbf B$ + +$\nabla \times \mathbf B = \mu_0 \left( \mathbf j + \varepsilon_0 \; \partial_t \mathbf E\right)$ + +## Integral Forms + +**First Equation** +$$ +\iint_{\partial \Omega} \mathbf E \cdot d \mathbf s = \frac{1}{\varepsilon_0} \iiint_{\Omega} \rho \; dV = \frac{Q}{\varepsilon_0} +$$ +**Second Equation** +$$ +\iint_{\partial \Omega} \mathbf B \cdot d \mathbf s = 0 +$$ +**Third Equation** +$$ +\int_{\partial\Sigma} \mathbf E \cdot d \mathbf l = - \partial_t \iint_{\Sigma} \mathbf B \cdot d \mathbf s +$$ +**Fourth Equation** +$$ +\int_{\partial \Sigma} \mathbf B \cdot d \mathbf l = \mu_0 \left( \iint_{\Sigma} \mathbf j \cdot d \mathbf s + \varepsilon_0 \; \partial_t \iint_{\Sigma} \mathbf E \cdot d \mathbf s \right) +$$ + +## In Empty Space + +$\nabla \cdot \mathbf E = 0$ + +$\nabla \cdot \mathbf B = 0$ + +$\nabla \times \mathbf E = - \partial_t \mathbf B$ + +$\nabla \times \mathbf B = \mu_0 \varepsilon_0 \; \partial_t \mathbf E$ + + +## Notation + +Here are some remarks on the notation used that may be useful : + +$\partial_t \mathbf F$ is $\frac{d\mathbf F}{dt}$ + +$\nabla \cdot F$ is the divergence of $F$ + +$\nabla \times F$ is the curl (rotationel) of $F$ + +### Divergence Theorem +In 2 dimensions (useless here) +$$ +\iint_{\Sigma} \nabla \cdot F \; ds = \int_{\partial \Sigma} \langle F, \nu \rangle \; dl +$$ +In 3 dimensions +$$ +\iiint_{\Omega} \nabla \cdot F \; dV = \iint_{\partial \Omega} \langle F, \nu \rangle \; ds +$$ +$\nu$ is the outwards pointing unit normal at each point on the boundary + +$\langle F, \nu \rangle \; ds = F \cdot (\nu \; ds) = F \cdot d\mathbf s$ + +If we use a surface $S$ such that the normal to the surface is either perpendicular or parallel to $F$ : +- The perpendicular parts have zero flux through the surface +- The parallel parts have a flux through the surface simply equal to their value ($\mathbf F \cdot d\mathbf s$ becomes $F \; ds$) + +### Stoke's Theorem +$$ +\iint_{\Sigma} \nabla \times F \; ds = \int_{\partial \Sigma} F \cdot dl +$$ + +### Constants and Variables + +$\mathbf E$ : electric field + +$\mathbf B$ : magnetic field + +$\rho$ : electric charge density (total charge per unit volume) + +$\mathbf j$ : current density (total current per unit area) + + +$Q$ : total electric charge + +$$ +Q = \iiint_{\Omega} \rho \; dV +$$ + +$I$ : net electric current + +$\mathcal{E}$ : emf (electromotive force) + +$I_{\text{encl}}$ : total current through the loop + +$\varepsilon_0$ : permittivity of free space + +$\mu_0$ : permeability of free space + +$k_e$ : Coulomb constant + +$$ +k_e = \frac{1}{4 \pi \varepsilon_0} +$$ +$k_M$ : Magic constant +$$ +k_M = \frac{\mu_0}{4 \pi} +$$ +$$ +\frac{k_M}{k_e} = \frac{1}{c^2} = \mu_0 \varepsilon_0 +$$ + +$\Omega$ : any volume with closed boundary surface $\partial \Omega$ + +$\Sigma$ : any surface with closed boundary curve $\partial \Sigma$ + + + All integrals $\int_{\partial \Sigma}$ and $\iint_{\partial \Omega}$ could have been written using the loop notation $\oint$, which indicates a **closed** boundary (curve or surface). + Indeed all boundaries of $\Omega$ and $\Sigma$ in this document are **closed** boundaries + \ No newline at end of file