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Diffstat (limited to 'source/know/concept/maxwells-equations/index.md')
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diff --git a/source/know/concept/maxwells-equations/index.md b/source/know/concept/maxwells-equations/index.md index ea052ce..36eb7b2 100644 --- a/source/know/concept/maxwells-equations/index.md +++ b/source/know/concept/maxwells-equations/index.md @@ -17,10 +17,10 @@ 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 $\varepsilon_0$: +**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 $$\varepsilon_0$$: $$\begin{aligned} \Phi_E @@ -29,12 +29,12 @@ $$\begin{aligned} = \frac{Q}{\varepsilon_0} \end{aligned}$$ -Where $\vb{E}$ is the [electric field](/know/concept/electric-field/), -and $\rho$ is the charge density in $V$. +Where $$\vb{E}$$ is the [electric field](/know/concept/electric-field/), +and $$\rho$$ is the charge density in $$V$$. Gauss' law is usually more useful when written in its vector form, which can be found by applying the divergence theorem to the surface integral above. -It states that the divergence of $\vb{E}$ is proportional to $\rho$: +It states that the divergence of $$\vb{E}$$ is proportional to $$\rho$$: $$\begin{aligned} \boxed{ @@ -43,10 +43,10 @@ $$\begin{aligned} \end{aligned}$$ 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 $\varepsilon_0$: +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 $$\varepsilon_0$$: $$\begin{aligned} \rho @@ -54,11 +54,11 @@ $$\begin{aligned} = \nabla \cdot \vb{D} - \nabla \cdot \vb{P} \end{aligned}$$ -To proceed, we split the net charge density $\rho$ -into a "free" part $\rho_\mathrm{free}$ -and a "bound" part $\rho_\mathrm{bound}$, -respectively corresponding to $\vb{D}$ and $\vb{P}$, -such that $\rho = \rho_\mathrm{free} + \rho_\mathrm{bound}$. +To proceed, we split the net charge density $$\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} @@ -71,7 +71,7 @@ $$\begin{aligned} } \end{aligned}$$ -By integrating over an arbitrary volume $V$ +By integrating over an arbitrary volume $$V$$ we can get integral forms of these equations: $$\begin{aligned} @@ -89,10 +89,10 @@ $$\begin{aligned} ## Gauss' law for magnetism -**Gauss' law for magnetism** states that magnetic flux $\Phi_B$ -through a closed surface $S(V)$ is zero. +**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: +the volume $$V$$ must leave it too: $$\begin{aligned} \Phi_B @@ -100,7 +100,7 @@ $$\begin{aligned} = 0 \end{aligned}$$ -Where $\vb{B}$ is the [magnetic field](/know/concept/magnetic-field/). +Where $$\vb{B}$$ is the [magnetic field](/know/concept/magnetic-field/). Thanks to the divergence theorem, this can equivalently be stated in vector form as follows: @@ -117,10 +117,10 @@ in contrast to electric charge. ## Faraday's law of induction -**Faraday's law of induction** states that a magnetic field $\vb{B}$ -that changes with time will induce an electric field $E$. -Specifically, the change in magnetic flux through a non-closed surface $S$ -creates an electromotive force around the contour $C(S)$. +**Faraday's law of induction** states that a magnetic field $$\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} @@ -141,37 +141,38 @@ $$\begin{aligned} ## Ampère's circuital law **Ampère's circuital law**, with Maxwell's correction, -states that a magnetic field $\vb{B}$ -can be induced along a contour $C(S)$ by two things: -a current density $\vb{J}$ through the enclosed surface $S$, -and a change of the electric field flux $\Phi_E$ through $S$: +states that a magnetic field $$\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}$$ -Where $\mu_0$ is the vacuum permeability. -This relation also exists for the "bound" fields $\vb{H}$ and $\vb{D}$, -and for $\vb{M}$ and $\vb{P}$. -We insert $\vb{B} = \mu_0 (\vb{H} + \vb{M})$ -and $\vb{E} = (\vb{D} - \vb{P})/\varepsilon_0$ -into Ampère's law, after dividing it by $\mu_0$ for simplicity: +Where $$\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}$$ -To proceed, we split the net current density $\vb{J}$ -into a "free" part $\vb{J}_\mathrm{free}$ -and a "bound" part $\vb{J}_\mathrm{bound}$, -such that $\vb{J} = \vb{J}_\mathrm{free} + \vb{J}_\mathrm{bound}$. +To proceed, we split the net current density $$\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} @@ -184,7 +185,7 @@ $$\begin{aligned} } \end{aligned}$$ -By integrating over an arbitrary surface $S$ +By integrating over an arbitrary surface $$S$$ we can get integral forms of these equations: $$\begin{aligned} @@ -195,9 +196,9 @@ $$\begin{aligned} &= \int_S \vb{J}_{\mathrm{bound}} \cdot \dd{\vb{A}} - \dv{}{t}\int_S \vb{P} \cdot \dd{\vb{A}} \end{aligned}$$ -Note that $\vb{J}_\mathrm{bound}$ can be split into -the **magnetization current density** $\vb{J}_M = \nabla \cross \vb{M}$ -and the **polarization current density** $\vb{J}_P = \ipdv{\vb{P}}{t}$: +Note that $$\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 = \ipdv{\vb{P}}{t}$$: $$\begin{aligned} \vb{J}_\mathrm{bound} @@ -221,8 +222,8 @@ $$\begin{aligned} Since the divergence of a curl is always zero, the right-hand side must vanish. -We know that $\vb{B}$ can vary in time, -so our only option to satisfy this is to demand that $\nabla \cdot \vb{B} = 0$. +We know that $$\vb{B}$$ can vary in time, +so our only option to satisfy this is to demand that $$\nabla \cdot \vb{B} = 0$$. We thus arrive arrive at Gauss' law for magnetism from Faraday's law. The same technique works for Ampère's law. @@ -234,7 +235,7 @@ $$\begin{aligned} = \nabla \cdot \vb{J} + \varepsilon_0 \pdv{}{t}(\nabla \cdot \vb{E}) \end{aligned}$$ -We integrate this over an arbitrary volume $V$, +We integrate this over an arbitrary volume $$V$$, and apply the divergence theorem: $$\begin{aligned} @@ -245,9 +246,9 @@ $$\begin{aligned} \end{aligned}$$ The first integral represents the current (charge flux) -through the surface of $V$. +through the surface of $$V$$. Electric charge is not created or destroyed, -so the second integral *must* be the total charge in $V$: +so the second integral *must* be the total charge in $$V$$: $$\begin{aligned} Q |