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/electric-field/index.pdc | 4 +- .../electromagnetic-wave-equation/index.pdc | 252 +++++++++++++++++++++ content/know/concept/heisenberg-picture/index.pdc | 4 +- content/know/concept/lorentz-force/index.pdc | 248 ++++++++++++++++++++ content/know/concept/magnetic-field/index.pdc | 4 +- content/know/concept/maxwells-equations/index.pdc | 211 +++++++++++++++++ 6 files changed, 717 insertions(+), 6 deletions(-) create mode 100644 content/know/concept/electromagnetic-wave-equation/index.pdc create mode 100644 content/know/concept/lorentz-force/index.pdc create mode 100644 content/know/concept/maxwells-equations/index.pdc (limited to 'content') diff --git a/content/know/concept/electric-field/index.pdc b/content/know/concept/electric-field/index.pdc index ce2c4fc..6162e0b 100644 --- a/content/know/concept/electric-field/index.pdc +++ b/content/know/concept/electric-field/index.pdc @@ -15,8 +15,8 @@ markup: pandoc The **electric field** $\vb{E}$ is a vector field that describes electric effects, -and is defined as the field that -correctly predicts the Lorentz force +and is defined as the field that correctly predicts +the [Lorentz force](/know/concept/lorentz-force/) on a particle with electric charge $q$: $$\begin{aligned} diff --git a/content/know/concept/electromagnetic-wave-equation/index.pdc b/content/know/concept/electromagnetic-wave-equation/index.pdc new file mode 100644 index 0000000..68fe062 --- /dev/null +++ b/content/know/concept/electromagnetic-wave-equation/index.pdc @@ -0,0 +1,252 @@ +--- +title: "Electromagnetic wave equation" +firstLetter: "E" +publishDate: 2021-09-09 +categories: +- Physics +- Electromagnetism +- Optics + +date: 2021-09-09T21:20:31+02:00 +draft: false +markup: pandoc +--- + +# Electromagnetic wave equation + +The electromagnetic wave equation describes +the propagation of light through various media. +Since an electromagnetic (light) wave consists of +an [electric field](/know/concept/electric-field/) +and a [magnetic field](/know/concept/magnetic-field/), +we need [Maxwell's equations](/know/concept/maxwells-equations/) +in order to derive the wave equation. + + +## Uniform medium + +We will use all of Maxwell's equations, +but we start with Ampère's circuital law for the "free" fields $\vb{H}$ and $\vb{D}$, +in the absence of a free current $\vb{J}_\mathrm{free} = 0$: + +$$\begin{aligned} + \nabla \cross \vb{H} + = \pdv{\vb{D}}{t} +\end{aligned}$$ + +We assume that the medium is isotropic, linear, +and uniform in all of space, such that: + +$$\begin{aligned} + \vb{D} = \varepsilon_0 \varepsilon_r \vb{E} + \qquad \quad + \vb{H} = \frac{1}{\mu_0 \mu_r} \vb{B} +\end{aligned}$$ + +Which, upon insertion into Ampère's law, +yields an equation relating $\vb{B}$ and $\vb{E}$. +This may seem to contradict Ampère's "total" law, +but keep in mind that $\vb{J}_\mathrm{bound} \neq 0$ here: + +$$\begin{aligned} + \nabla \cross \vb{B} + = \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdv{\vb{E}}{t} +\end{aligned}$$ + +Now we take the curl, rearrange, +and substitute $\nabla \cross \vb{E}$ according to Faraday's law: + +$$\begin{aligned} + \nabla \cross (\nabla \cross \vb{B}) + %= \nabla \cross \Big( \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdv{\vb{E}}{t} \Big) + = \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdv{t} (\nabla \cross \vb{E}) + %= \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdv{t} \Big( \!-\! \pdv{\vb{B}}{t} \Big) + = - \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdv[2]{\vb{B}}{t} +\end{aligned}$$ + +Using a vector identity, we rewrite the leftmost expression, +which can then be reduced thanks to Gauss' law for magnetism $\nabla \cdot \vb{B} = 0$: + +$$\begin{aligned} + - \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdv[2]{\vb{B}}{t} + &= \nabla (\nabla \cdot \vb{B}) - \nabla^2 \vb{B} + = - \nabla^2 \vb{B} +\end{aligned}$$ + +This describes $\vb{B}$. +Next, we repeat the process for $\vb{E}$: +taking the curl of Faraday's law yields: + +$$\begin{aligned} + \nabla \cross (\nabla \cross \vb{E}) + %= - \nabla \cross \pdv{\vb{B}}{t} + = - \pdv{t} (\nabla \cross \vb{B}) + %= - \pdv{t} \Big( \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdv{\vb{E}}{t} \Big) + = - \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdv[2]{\vb{E}}{t} +\end{aligned}$$ + +Which can be rewritten using same vector identity as before, +and then reduced by assuming that there is no net charge density $\rho = 0$ +in Gauss' law, such that $\nabla \cdot \vb{E} = 0$: + +$$\begin{aligned} + - \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdv[2]{\vb{E}}{t} + &= \nabla (\nabla \cdot \vb{E}) - \nabla^2 \vb{E} + = - \nabla^2 \vb{E} +\end{aligned}$$ + +We thus arrive at the following two (implicitly coupled) +wave equations for $\vb{E}$ and $\vb{B}$, +where we have defined the phase velocity $v \equiv 1 / \sqrt{\mu_0 \mu_r \varepsilon_0 \varepsilon_r}$: + +$$\begin{aligned} + \boxed{ + \pdv[2]{\vb{E}}{t} - \frac{1}{v^2} \nabla^2 \vb{E} + = 0 + } + \qquad \quad + \boxed{ + \pdv[2]{\vb{B}}{t} - \frac{1}{v^2} \nabla^2 \vb{B} + = 0 + } +\end{aligned}$$ + +Traditionally, it is said that the solutions are as follows, +where the wavenumber $|\vb{k}| = \omega / v$: + +$$\begin{aligned} + \vb{E}(\vb{r}, t) + &= \vb{E}_0 \exp\!(i \vb{k} \cdot \vb{r} - i \omega t) + \\ + \vb{H}(\vb{r}, t) + &= \vb{H}_0 \exp\!(i \vb{k} \cdot \vb{r} - i \omega t) +\end{aligned}$$ + +In fact, thanks to linearity, these solutions can be treated as +terms in a Fourier series, meaning that virtually +*any* function $f(\vb{k} \cdot \vb{r} - \omega t)$ is a valid solution. + + +## Non-uniform medium + +A useful generalization is to allow spatial change +in the relative permittivity $\varepsilon_r(\vb{r})$ +and the relative permeability $\mu_r(\vb{r})$. +We still assume that the medium is linear and isotropic, so: + +$$\begin{aligned} + \vb{D} + = \varepsilon_0 \varepsilon_r(\vb{r}) \vb{E} + \qquad \quad + \vb{B} + = \mu_0 \mu_r(\vb{r}) \vb{H} +\end{aligned}$$ + +Inserting these expressions into Faraday's and Ampère's laws +respectively yields: + +$$\begin{aligned} + \nabla \cross \vb{E} + = - \mu_0 \mu_r(\vb{r}) \pdv{\vb{H}}{t} + \qquad \quad + \nabla \cross \vb{H} + = \varepsilon_0 \varepsilon_r(\vb{r}) \pdv{\vb{E}}{t} +\end{aligned}$$ + +We then divide Ampère's law by $\varepsilon_r(\vb{r})$, +take the curl, and substitute Faraday's law, giving: + +$$\begin{aligned} + \nabla \cross \Big( \frac{1}{\varepsilon_r} \nabla \cross \vb{H} \Big) + = \varepsilon_0 \pdv{t} (\nabla \cross \vb{E}) + = - \mu_0 \mu_r \varepsilon_0 \pdv[2]{\vb{H}}{t} +\end{aligned}$$ + +Next, we exploit linearity by decomposing $\vb{H}$ and $\vb{E}$ +into Fourier series, with terms given by: + +$$\begin{aligned} + \vb{H}(\vb{r}, t) + = \vb{H}(\vb{r}) \exp\!(- i \omega t) + \qquad \quad + \vb{E}(\vb{r}, t) + = \vb{E}(\vb{r}) \exp\!(- i \omega t) +\end{aligned}$$ + +By inserting this ansatz into the equation, +we can remove the explicit time dependence: + +$$\begin{aligned} + \nabla \cross \Big( \frac{1}{\varepsilon_r} \nabla \cross \vb{H} \Big) \exp\!(- i \omega t) + = \mu_0 \varepsilon_0 \omega^2 \mu_r \vb{H} \exp\!(- i \omega t) +\end{aligned}$$ + +Dividing out $\exp\!(- i \omega t)$, +we arrive at an eigenvalue problem for $\omega^2$, +with $c = 1 / \sqrt{\mu_0 \varepsilon_0}$: + +$$\begin{aligned} + \boxed{ + \nabla \cross \Big( \frac{1}{\varepsilon_r(\vb{r})} \nabla \cross \vb{H}(\vb{r}) \Big) + = \Big( \frac{\omega}{c} \Big)^2 \mu_r(\vb{r}) \vb{H}(\vb{r}) + } +\end{aligned}$$ + +Compared to a uniform medium, $\omega$ is often not arbitrary here: +there are discrete eigenvalues $\omega$, +corresponding to discrete **modes** $\vb{H}(\vb{r})$. + +Next, we go through the same process to find an equation for $\vb{E}$. +Starting from Faraday's law, we divide by $\mu_r(\vb{r})$, +take the curl, and insert Ampère's law: + +$$\begin{aligned} + \nabla \cross \Big( \frac{1}{\mu_r} \nabla \cross \vb{E} \Big) + = - \mu_0 \pdv{t} (\nabla \cross \vb{H}) + = - \mu_0 \varepsilon_0 \varepsilon_r \pdv[2]{\vb{E}}{t} +\end{aligned}$$ + +Then, by replacing $\vb{E}(\vb{r}, t)$ with our plane-wave ansatz, +we remove the time dependence: + +$$\begin{aligned} + \nabla \cross \Big( \frac{1}{\mu_r} \nabla \cross \vb{E} \Big) \exp\!(- i \omega t) + = - \mu_0 \varepsilon_0 \omega^2 \varepsilon_r \vb{E} \exp\!(- i \omega t) +\end{aligned}$$ + +Which, after dividing out $\exp\!(- i \omega t)$, +yields an analogous eigenvalue problem with $\vb{E}(r)$: + +$$\begin{aligned} + \boxed{ + \nabla \cross \Big( \frac{1}{\mu_r(\vb{r})} \nabla \cross \vb{E}(\vb{r}) \Big) + = \Big( \frac{\omega}{c} \Big)^2 \varepsilon_r(\vb{r}) \vb{E}(\vb{r}) + } +\end{aligned}$$ + +Usually, it is a reasonable approximation +to say $\mu_r(\vb{r}) = 1$, +in which case the equation for $\vb{H}(\vb{r})$ +becomes a Hermitian eigenvalue problem, +and is thus easier to solve than for $\vb{E}(\vb{r})$. + +Keep in mind, however, that in any case, +the solutions $\vb{H}(\vb{r})$ and/or $\vb{E}(\vb{r})$ +must satisfy the two Maxwell's equations that were not explicitly used: + +$$\begin{aligned} + \nabla \cdot (\varepsilon_r \vb{E}) = 0 + \qquad \quad + \nabla \cdot (\mu_r \vb{H}) = 0 +\end{aligned}$$ + +This is equivalent to demanding that the resulting waves are *transverse*, +or in other words, +the wavevector $\vb{k}$ must be perpendicular to +the amplitudes $\vb{H}_0$ and $\vb{E}_0$. + + +## References +1. J.D. Joannopoulos, S.G. Johnson, J.N. Winn, R.D. Meade, + *Photonic crystals: molding the flow of light*, + 2nd edition, Princeton. diff --git a/content/know/concept/heisenberg-picture/index.pdc b/content/know/concept/heisenberg-picture/index.pdc index 2dd4118..9e4887d 100644 --- a/content/know/concept/heisenberg-picture/index.pdc +++ b/content/know/concept/heisenberg-picture/index.pdc @@ -37,9 +37,9 @@ converted to the Heisenberg picture by the following change of basis: $$\begin{aligned} \boxed{ - \ket{\psi_H} = \ket{\psi_S(0)} + \ket{\psi_H} \equiv \ket{\psi_S(0)} \qquad - \hat{L}_H(t) = \hat{U}^\dagger(t) \hat{L}_S(t) \hat{U}(t) + \hat{L}_H(t) \equiv \hat{U}^\dagger(t) \: \hat{L}_S(t) \: \hat{U}(t) } \end{aligned}$$ diff --git a/content/know/concept/lorentz-force/index.pdc b/content/know/concept/lorentz-force/index.pdc new file mode 100644 index 0000000..f0e9850 --- /dev/null +++ b/content/know/concept/lorentz-force/index.pdc @@ -0,0 +1,248 @@ +--- +title: "Lorentz force" +firstLetter: "L" +publishDate: 2021-09-08 +categories: +- Physics +- Electromagnetism + +date: 2021-09-08T17:00:32+02:00 +draft: false +markup: pandoc +--- + +# Lorentz force + +The **Lorentz force** is an empirical force used to define +the [electric field](/know/concept/electric-field/) $\vb{E}$ +and [magnetic field](/know/concept/magnetic-field/) $\vb{B}$. +For a particle with charge $q$ moving with velocity $\vb{u}$, +the Lorentz force $\vb{F}$ is given by: + +$$\begin{aligned} + \boxed{ + \vb{F} + = q (\vb{E} + \vb{u} \cross \vb{B}) + } +\end{aligned}$$ + + +## Uniform magnetic field + +Consider the simple case of a uniform magnetic field +$\vb{B} = (0, 0, B)$ in the $z$-direction, +without an electric field $\vb{E} = 0$. +If there are no other forces, +Newton's second law states: + +$$\begin{aligned} + \vb{F} + = m \dv{\vb{u}}{t} + = q \vb{u} \cross \vb{B} +\end{aligned}$$ + +Evaluating the cross product yields +three coupled equations for the components of $\vb{u}$: + +$$\begin{aligned} + \dv{u_x}{t} + = \frac{q B}{m} u_y + \qquad \quad + \dv{u_y}{t} + = - \frac{q B}{m} u_x + \qquad \quad + \dv{u_z}{t} + = 0 +\end{aligned}$$ + +Differentiating the first equation with respect to $t$, +and substituting $\dv*{u_y}{t}$ from the second, +we arrive at the following harmonic oscillator: + +$$\begin{aligned} + \dv[2]{u_x}{t} = - \omega_c^2 u_x +\end{aligned}$$ + +Where we have defined the **cyclotron frequency** $\omega_c$ as follows, +which is always positive: + +$$\begin{aligned} + \boxed{ + \omega_c + \equiv \frac{|q| B}{m} + } +\end{aligned}$$ + +Suppose we choose our initial conditions so that +the solution for $u_x(t)$ is given by: + +$$\begin{aligned} + u_x(t) + = - u_\perp \sin\!(\omega_c t) +\end{aligned}$$ + +Where $u_\perp \equiv \sqrt{u_x^2 + u_y^2}$ is the constant total transverse velocity. +Then $u_y(t)$ is found to be: + +$$\begin{aligned} + u_y(t) + = \frac{m}{q B} \dv{u_x}{t} + = - \frac{m \omega_c}{q B} u_\perp \cos\!(\omega_c t) + = - \mathrm{sgn}(q) \: u_\perp \cos\!(\omega_c t) +\end{aligned}$$ + +Where $\mathrm{sgn}$ is the signum function. +This tells us that the particle moves in a circular orbit, +and that the direction of rotation is determined by $q$. + +Integrating the velocity yields the position, +where we refer to the integration constants $x_{gc}$ and $y_{gc}$ +as the **guiding center**, around which the particle orbits or **gyrates**: + +$$\begin{aligned} + x(t) + = \frac{u_\perp}{\omega_c} \cos\!(\omega_c t) + x_{gc} + \qquad \quad + y(t) + = - \mathrm{sgn}(q) \: \frac{u_\perp}{\omega_c} \sin\!(\omega_c t) + y_{gc} +\end{aligned}$$ + +The radius of this orbit is known as the **Larmor radius** or **gyroradius** $r_L$, given by: + +$$\begin{aligned} + \boxed{ + r_L + \equiv \frac{u_\perp}{\omega_c} + = \frac{m u_\perp}{|q| B} + } +\end{aligned}$$ + +Finally, it is trivial to integrate the equation for the $z$-direction velocity $u_z$: + +$$\begin{aligned} + z(t) + = u_z t + z_{gc} +\end{aligned}$$ + +In conclusion, the particle's motion parallel to $\vb{B}$ +is not affected by the magnetic field, +while its motion perpendicular to $\vb{B}$ +is circular around an imaginary guiding center. +The end result is that particles follow a helical path +when moving through a uniform magnetic field. + + +## Uniform electric and magnetic field + +Let us now consider a more general case, +with constant uniform electric and magnetic fields $\vb{E}$ and $\vb{B}$, +which may or may not be perpendicular. +The equation of motion is then: + +$$\begin{aligned} + \vb{F} + = m \dv{\vb{u}}{t} + = q (\vb{E} + \vb{u} \cross \vb{B}) +\end{aligned}$$ + +If we take the dot product with the unit vector $\vu{B}$, +the cross product vanishes, leaving: + +$$\begin{aligned} + \dv{\vb{u}_\parallel}{t} + = \frac{q}{m} \vb{E}_\parallel +\end{aligned}$$ + +Where $\vb{u}_\parallel$ and $\vb{E}_\parallel$ are +the components of $\vb{u}$ and $\vb{E}$ +that are parallel to $\vb{B}$. +This equation is easy to integrate: +the guiding center accelerates according to $(q/m) \vb{E}_\parallel$. + +Next, let us define the perpendicular component $\vb{u}_\perp$ +such that $\vb{u} = \vb{u}_\parallel \vu{B} + \vb{u}_\perp$. +Its equation of motion is found by +subtracting $\vb{u}_\parallel$'s equation from the original: + +$$\begin{aligned} + m \dv{\vb{u}_\perp}{t} + = q (\vb{E} + \vb{u} \cross \vb{B}) - q \vb{E}_\parallel + = q (\vb{E}_\perp + \vb{u}_\perp \cross \vb{B}) +\end{aligned}$$ + +To solve this, we go to a moving coordinate system +by defining $\vb{u}_\perp = \vb{v}_\perp + \vb{w}_\perp$, +where $\vb{v}_\perp$ is a constant of our choice. +The equation is now as follows: + +$$\begin{aligned} + m \dv{t} (\vb{v}_\perp + \vb{w}_\perp) + = m \dv{\vb{w}_\perp}{t} + = q (\vb{E}_\perp + \vb{v}_\perp \cross \vb{B} + \vb{w}_\perp \cross \vb{B}) +\end{aligned}$$ + +We want to choose $\vb{v}_\perp$ such that the first two terms vanish, +or in other words: + +$$\begin{aligned} + 0 + = \vb{E}_\perp + \vb{v}_\perp \cross \vb{B} +\end{aligned}$$ + +To find $\vb{v}_\perp$, we take the cross product with $\vb{B}$, +and use the fact that $\vb{B} \cross \vb{E}_\perp = \vb{B} \cross \vb{E}$: + +$$\begin{aligned} + 0 + = \vb{B} \cross (\vb{E}_\perp + \vb{v}_\perp \cross \vb{B}) + = \vb{B} \cross \vb{E} + \vb{v}_\perp B^2 + \quad \implies \quad + \boxed{ + \vb{v}_\perp + = \frac{\vb{E} \cross \vb{B}}{B^2} + } +\end{aligned}$$ + +When $\vb{v}_\perp$ is chosen like this, +the perpendicular equation of motion is reduced to: + +$$\begin{aligned} + m \dv{\vb{w}_\perp}{t} + = q \vb{w}_\perp \cross \vb{B} +\end{aligned}$$ + +Which is simply the case we treated previously with $\vb{E} = 0$, +with a known solution +(assuming $\vb{B}$ still points in the positive $z$-direction): + +$$\begin{aligned} + w_x(t) + = - w_\perp \sin\!(\omega_c t) + \qquad + w_y(t) + = - \mathrm{sgn}(q) \: w_\perp \cos\!(\omega_c t) +\end{aligned}$$ + +However, this result is shifted by a constant $\vb{v}_\perp$, +often called the **drift velocity** $\vb{v}_d$, +at which the guiding center moves transversely. +Curiously, $\vb{v}_d$ is independent of $q$. + +Such a drift is not specific to an electric field. +In the equations above, $\vb{E}$ can be replaced +by a general force $\vb{F}/q$ (e.g. gravity) without issues. +In that case, $\vb{v}_d$ does depend on $q$. + +$$\begin{aligned} + \boxed{ + \vb{v}_d + = \frac{\vb{F} \cross \vb{B}}{q B^2} + } +\end{aligned}$$ + + + +## References +1. F.F. Chen, + *Introduction to plasma physics and controlled fusion*, + 3rd edition, Springer. diff --git a/content/know/concept/magnetic-field/index.pdc b/content/know/concept/magnetic-field/index.pdc index 2ad5fbf..3bfa90e 100644 --- a/content/know/concept/magnetic-field/index.pdc +++ b/content/know/concept/magnetic-field/index.pdc @@ -15,8 +15,8 @@ markup: pandoc The **magnetic field** $\vb{B}$ is a vector field that describes magnetic effects, -and is defined as the field -that correctly predicts the Lorentz force +and is defined as the field that correctly predicts +the [Lorentz force](/know/concept/lorentz-force/) on a particle with electric charge $q$: $$\begin{aligned} 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