From 942035bfe0c19be78efe1452d88b85490f035aab Mon Sep 17 00:00:00 2001 From: Prefetch Date: Sun, 12 Sep 2021 19:22:22 +0200 Subject: Expand knowledge base --- content/know/concept/maxwells-equations/index.pdc | 211 ++++++++++++++++++++++ 1 file changed, 211 insertions(+) create mode 100644 content/know/concept/maxwells-equations/index.pdc (limited to 'content/know/concept/maxwells-equations') diff --git a/content/know/concept/maxwells-equations/index.pdc b/content/know/concept/maxwells-equations/index.pdc new file mode 100644 index 0000000..1551311 --- /dev/null +++ b/content/know/concept/maxwells-equations/index.pdc @@ -0,0 +1,211 @@ +--- +title: "Maxwell's equations" +firstLetter: "M" +publishDate: 2021-09-09 +categories: +- Physics +- Electromagnetism + +date: 2021-09-09T21:20:18+02:00 +draft: false +markup: pandoc +--- + +# Maxwell's equations + +In physics, **Maxwell's equations** govern +all macroscopic electromagnetism, +and notably lead to the +[electromagnetic wave equation](/know/concept/electromagnetic-wave-equation/), +which describes the existence of light. + + +## Gauss' law + +**Gauss' law** states that the electric flux $\Phi_E$ through +a closed surface $S(V)$ is equal to the total charge $Q$ +contained in the enclosed volume $V$, +divided by the vacuum permittivity $\varepsilon_0$: + +$$\begin{aligned} + \Phi_E + = \oint_{S(V)} \vb{E} \cdot \dd{\vb{A}} + = \frac{1}{\varepsilon_0} \int_{V} \rho \dd{V} + = \frac{Q}{\varepsilon_0} +\end{aligned}$$ + +Where $\vb{E}$ is the [electric field](/know/concept/electric-field/), +and $\rho$ is the charge density in $V$. +Gauss' law is usually more useful when written in its vector form, +which can be found by applying the divergence theorem +to the surface integral above. +It states that the divergence of $\vb{E}$ is proportional to $\rho$: + +$$\begin{aligned} + \boxed{ + \nabla \cdot \vb{E} = \frac{\rho}{\varepsilon_0} + } +\end{aligned}$$ + +This law can just as well be expressed for +the displacement field $\vb{D}$ +and polarization density $\vb{P}$. +We insert $\vb{E} = (\vb{D} - \vb{P}) / \varepsilon_0$ +into Gauss' law for $\vb{E}$, multiplied by $\varepsilon_0$: + +$$\begin{aligned} + \rho + = \nabla \cdot \big( \vb{D} - \vb{P} \big) + = \nabla \cdot \vb{D} - \nabla \cdot \vb{P} +\end{aligned}$$ + +To proceed, we split the net charge density $\rho$ +into a "free" part $\rho_\mathrm{free}$ +and a "bound" part $\rho_\mathrm{bound}$, +respectively corresponding to $\vb{D}$ and $\vb{P}$, +such that $\rho = \rho_\mathrm{free} + \rho_\mathrm{bound}$. +This yields: + +$$\begin{aligned} + \boxed{ + \nabla \cdot \vb{D} = \rho_{\mathrm{free}} + } + \qquad \quad + \boxed{ + \nabla \cdot \vb{P} = - \rho_{\mathrm{bound}} + } +\end{aligned}$$ + +By integrating over an arbitrary volume $V$ +we can get integral forms of these equations: + +$$\begin{aligned} + \Phi_D + &= \oint_{S(V)} \vb{D} \cdot \dd{\vb{A}} + = \int_{V} \rho_{\mathrm{free}} \dd{V} + = Q_{\mathrm{free}} + \\ + \Phi_P + &= \oint_{S(V)} \vb{P} \cdot \dd{\vb{A}} + = - \int_{V} \rho_{\mathrm{bound}} \dd{V} + = - Q_{\mathrm{bound}} +\end{aligned}$$ + + +## Gauss' law for magnetism + +**Gauss' law for magnetism** states that magnetic flux $\Phi_B$ +through a closed surface $S(V)$ is zero. +In other words, all magnetic field lines entering +the volume $V$ must leave it too: + +$$\begin{aligned} + \Phi_B + = \oint_{S(V)} \vb{B} \cdot \dd{\vb{A}} + = 0 +\end{aligned}$$ + +Where $\vb{B}$ is the [magnetic field](/know/concept/magnetic-field/). +Thanks to the divergence theorem, +this can equivalently be stated in vector form as follows: + +$$\begin{aligned} + \boxed{ + \nabla \cdot \vb{B} = 0 + } +\end{aligned}$$ + +A consequence of this law is the fact that magnetic monopoles cannot exist, +i.e. there is no such thing as "magnetic charge", +in contrast to electric charge. + + +## Faraday's law of induction + +**Faraday's law of induction** states that a magnetic field $\vb{B}$ +that changes with time will induce an electric field $E$. +Specifically, the change in magnetic flux through a non-closed surface $S$ +creates an electromotive force around the contour $C(S)$. +This is written as: + +$$\begin{aligned} + \oint_{C(S)} \vb{E} \cdot \dd{\vb{l}} + = - \dv{t} \int_{S} \vb{B} \cdot \dd{\vb{A}} +\end{aligned}$$ + +By using Stokes' theorem on the contour integral, +the vector form of this law is found to be: + +$$\begin{aligned} + \boxed{ + \nabla \times \vb{E} = - \pdv{\vb{B}}{t} + } +\end{aligned}$$ + + +## Ampère's circuital law + +**Ampère's circuital law**, with Maxwell's correction, +states that a magnetic field $\vb{B}$ +can be induced along a contour $C(S)$ by two things: +a current density $\vb{J}$ through the enclosed surface $S$, +and a change of the electric field flux $\Phi_E$ through $S$: + +$$\begin{aligned} + \oint_{C(S)} \vb{B} \cdot d\vb{l} + = \mu_0 \Big( \int_S \vb{J} \cdot d\vb{A} + \varepsilon_0 \dv{t} \int_S \vb{E} \cdot d\vb{A} \Big) +\end{aligned}$$ +$$\begin{aligned} + \boxed{ + \nabla \times \vb{B} = \mu_0 \Big( \vb{J} + \varepsilon_0 \pdv{\vb{E}}{t} \Big) + } +\end{aligned}$$ + +Where $\mu_0$ is the vacuum permeability. +This relation also exists for the "bound" fields $\vb{H}$ and $\vb{D}$, +and for $\vb{M}$ and $\vb{P}$. +We insert $\vb{B} = \mu_0 (\vb{H} + \vb{M})$ +and $\vb{E} = (\vb{D} - \vb{P})/\varepsilon_0$ +into Ampère's law, after dividing it by $\mu_0$ for simplicity: + +$$\begin{aligned} + \nabla \cross \big( \vb{H} + \vb{M} \big) + &= \vb{J} + \pdv{t} \big( \vb{D} - \vb{P} \big) +\end{aligned}$$ + +To proceed, we split the net current density $\vb{J}$ +into a "free" part $\vb{J}_\mathrm{free}$ +and a "bound" part $\vb{J}_\mathrm{bound}$, +such that $\vb{J} = \vb{J}_\mathrm{free} + \vb{J}_\mathrm{bound}$. +This leads us to: + +$$\begin{aligned} + \boxed{ + \nabla \times \vb{H} = \vb{J}_{\mathrm{free}} + \pdv{\vb{D}}{t} + } + \qquad \quad + \boxed{ + \nabla \times \vb{M} = \vb{J}_{\mathrm{bound}} - \pdv{\vb{P}}{t} + } +\end{aligned}$$ + +By integrating over an arbitrary surface $S$ +we can get integral forms of these equations: + +$$\begin{aligned} + \oint_{C(S)} \vb{H} \cdot d\vb{l} + &= \int_S \vb{J}_{\mathrm{free}} \cdot \dd{\vb{A}} + \dv{t} \int_S \vb{D} \cdot \dd{\vb{A}} + \\ + \oint_{C(S)} \vb{M} \cdot d\vb{l} + &= \int_S \vb{J}_{\mathrm{bound}} \cdot \dd{\vb{A}} - \dv{t} \int_S \vb{P} \cdot \dd{\vb{A}} +\end{aligned}$$ + +Note that $\vb{J}_\mathrm{bound}$ can be split into +the **magnetization current density** $\vb{J}_M = \nabla \cross \vb{M}$ +and the **polarization current density** $\vb{J}_P = \pdv*{\vb{P}}{t}$: + +$$\begin{aligned} + \vb{J}_\mathrm{bound} + = \vb{J}_M + \vb{J}_P + = \nabla \cross \vb{M} + \pdv{\vb{P}}{t} +\end{aligned}$$ -- cgit v1.2.3