diff options
author | Prefetch | 2021-10-20 11:50:20 +0200 |
---|---|---|
committer | Prefetch | 2021-10-20 11:50:20 +0200 |
commit | 3170fc4b5c915669cf209a521e551115a9bd0809 (patch) | |
tree | 4cb70f6715be60b383fd2a23662d9795f56fef62 | |
parent | 2a5024543ed99f569fbade8744e6c8001f2edb02 (diff) |
Expand knowledge base
-rw-r--r-- | content/know/concept/archimedes-principle/index.pdc | 6 | ||||
-rw-r--r-- | content/know/concept/boltzmann-relation/index.pdc | 96 | ||||
-rw-r--r-- | content/know/concept/coulomb-logarithm/index.pdc | 9 | ||||
-rw-r--r-- | content/know/concept/debye-length/index.pdc | 156 | ||||
-rw-r--r-- | content/know/concept/euler-equations/index.pdc | 6 | ||||
-rw-r--r-- | content/know/concept/hydrostatic-pressure/index.pdc | 2 | ||||
-rw-r--r-- | content/know/concept/lindhard-function/index.pdc | 7 | ||||
-rw-r--r-- | content/know/concept/maxwell-bloch-equations/index.pdc | 15 | ||||
-rw-r--r-- | content/know/concept/maxwells-equations/index.pdc | 54 | ||||
-rw-r--r-- | content/know/concept/rutherford-scattering/index.pdc | 3 | ||||
-rw-r--r-- | content/know/concept/two-fluid-equations/index.pdc | 178 |
11 files changed, 517 insertions, 15 deletions
diff --git a/content/know/concept/archimedes-principle/index.pdc b/content/know/concept/archimedes-principle/index.pdc index 0837cc9..3a063ec 100644 --- a/content/know/concept/archimedes-principle/index.pdc +++ b/content/know/concept/archimedes-principle/index.pdc @@ -47,9 +47,9 @@ $$\begin{aligned} = - \int_V \nabla p \dd{V} \end{aligned}$$ -The last step follows from Gauss' theorem. -We replace $\nabla p$ by assuming -[hydrostatic equilibrium](/know/concept/hydrostatic-pressure/), +Where we have used the divergence theorem. +Assuming [hydrostatic equilibrium](/know/concept/hydrostatic-pressure/), +we replace $\nabla p$, leading to the definition of the **buoyant force**: $$\begin{aligned} diff --git a/content/know/concept/boltzmann-relation/index.pdc b/content/know/concept/boltzmann-relation/index.pdc new file mode 100644 index 0000000..ddaa22f --- /dev/null +++ b/content/know/concept/boltzmann-relation/index.pdc @@ -0,0 +1,96 @@ +--- +title: "Boltzmann relation" +firstLetter: "B" +publishDate: 2021-10-18 +categories: +- Physics +- Plasma physics + +date: 2021-10-18T15:25:39+02:00 +draft: false +markup: pandoc +--- + +# Boltzmann relation + +In a plasma where the ions and electrons are both in thermal equilibrium, +and in the absence of short-lived induced electromagnetic fields, +their densities $n_i$ and $n_e$ can be predicted. + +By definition, a particle in an [electric field](/know/concept/electric-field/) $\vb{E}$ +experiences a [Lorentz force](/know/concept/lorentz-force/) $\vb{F}_e$. +This corresponds to a force density $\vb{f}_e$, +such that $\vb{F}_e = \vb{f}_e \dd{V}$. +For the electrons, we thus have: + +$$\begin{aligned} + \vb{f}_e + = q_e n_e \vb{E} + = - q_e n_e \nabla \phi +\end{aligned}$$ + +Meanwhile, if we treat the electrons as a gas +obeying the ideal gas law $p_e = k_B T_e n_e$, +then the pressure $p_e$ leads to another force density $\vb{f}_p$: + +$$\begin{aligned} + \vb{f}_p + = - \nabla p_e + = - k_B T_e \nabla n_e +\end{aligned}$$ + +At equilibrium, we demand that $\vb{f}_e = \vb{f}_p$, +and isolate this equation for $\nabla n_e$, yielding: + +$$\begin{aligned} + k_B T_e \nabla n_e + = q_e n_e \nabla \phi + \quad \implies \quad + \nabla n_e + = \frac{q_e \nabla \phi}{k_B T_e} n_e + = \nabla \bigg( \frac{q_e \phi}{k_B T_e} \bigg) n_e +\end{aligned}$$ + +This equation is straightforward to integrate, +leading to the following expression for $n_e$, +known as the **Boltzmann relation**, +due to its resemblance to the statistical Boltzmann distribution +(see [canonical ensemble](/know/concept/canonical-ensemble/)): + +$$\begin{aligned} + \boxed{ + n_e(\vb{r}) + = n_{e0} \exp\!\bigg( \!-\! \frac{q_e \phi(\vb{r})}{k_B T_e} \bigg) + } +\end{aligned}$$ + +Where the linearity factor $n_{e0}$ represents +the electron density for $\phi = 0$. +We can do the same for ions instead of electrons, +leading to the following ion density $n_i$: + +$$\begin{aligned} + \boxed{ + n_i(\vb{r}) + = n_{i0} \exp\!\bigg( \!-\! \frac{q_i \phi(\vb{r})}{k_B T_i} \bigg) + } +\end{aligned}$$ + +However, due to their larger mass, +ions are much slower to respond to fluctuations in the above equilibrium. +Consequently, after a perturbation, +the ions spend much more time in a transient non-equilibrium state +than the electrons, so this formula for $n_i$ is only valid +if the perturbation is sufficiently slow, +allowing the ions to keep up. +Usually, electrons do not suffer the same issue, +thanks to their small mass and fast response. + + +## References +1. P.M. Bellan, + *Fundamentals of plasma physics*, + 1st edition, Cambridge. +2. M. Salewski, A.H. Nielsen, + *Plasma physics: lecture notes*, + 2021, unpublished. diff --git a/content/know/concept/coulomb-logarithm/index.pdc b/content/know/concept/coulomb-logarithm/index.pdc index 649806b..71b13a8 100644 --- a/content/know/concept/coulomb-logarithm/index.pdc +++ b/content/know/concept/coulomb-logarithm/index.pdc @@ -143,8 +143,8 @@ We know that the deflection grows for smaller $b$, so it would be reasonable to choose $b_\mathrm{large}$ as the lower limit. For very large $b$, the plasma shields the particles from each other, thereby nullifying the deflection, -so as upper limit -we choose the Debye length $\lambda_D$, +so as upper limit we choose +the [Debye length](/know/concept/debye-length/) $\lambda_D$, i.e. the plasma's self-shielding length. We thus find: @@ -157,12 +157,15 @@ $$\begin{aligned} \end{aligned}$$ Here, $\ln\!(\Lambda)$ is known as the **Coulomb logarithm**, -with $\Lambda$ defined as follows: +with the **plasma parameter** $\Lambda$ defined below, +equal to $9/2$ times the number of particles +in a sphere with radius $\lambda_D$: $$\begin{aligned} \boxed{ \Lambda \equiv \frac{\lambda_D}{b_\mathrm{large}} + = 6 \pi n \lambda_D^3 } \end{aligned}$$ diff --git a/content/know/concept/debye-length/index.pdc b/content/know/concept/debye-length/index.pdc new file mode 100644 index 0000000..19d7188 --- /dev/null +++ b/content/know/concept/debye-length/index.pdc @@ -0,0 +1,156 @@ +--- +title: "Debye length" +firstLetter: "D" +publishDate: 2021-10-18 +categories: +- Physics +- Plasma physics + +date: 2021-10-15T20:28:31+02:00 +draft: false +markup: pandoc +--- + +# Debye length + +If a charged object is put in a plasma, +it repels like charges and attracts opposite charges, +leading to a **Debye sheath** around the object's surface +with a net opposite charge. +This has the effect of **shielding** the object's presence +from the rest of the plasma. + +We start from [Gauss' law](/know/concept/maxwells-equations/) +for the [electric field](/know/concept/electric-field/) $\vb{E}$, +expressing $\vb{E}$ as the gradient of a potential $\phi$, +i.e. $\vb{E} = -\nabla \phi$, +and splitting the charge density into ions $n_i$ and electrons $n_e$: + +$$\begin{aligned} + \nabla^2 \phi(\vb{r}) + = - \frac{1}{\varepsilon_0} \Big( q_i n_i(\vb{r}) + q_e n_e(\vb{r}) + q_t \delta(\vb{r}) \Big) +\end{aligned}$$ + +The last term represents a *test particle*, +which will be shielded. +This particle is a point charge $q_t$, +whose density is simply a [Dirac delta function](/know/concept/dirac-delta-function/) $\delta(\vb{r})$, +and is not included in $n_i$ or $n_e$. + +For a plasma in thermal equilibrium, +we have the [Boltzmann relations](/know/concept/boltzmann-relation/) +for the densities: + +$$\begin{aligned} + n_i(\vb{r}) + = n_{i0} \exp\!\bigg( \!-\! \frac{q_i \phi(\vb{r})}{k_B T_i} \bigg) + \qquad \quad + n_e(\vb{r}) + = n_{e0} \exp\!\bigg( \!-\! \frac{q_e \phi(\vb{r})}{k_B T_e} \bigg) +\end{aligned}$$ + +We assume that electrical interactions are weak compared to thermal effects, +i.e. $k_B T \gg q \phi$ in both cases. +Then we Taylor-expand the Boltzmann relations to first order: + +$$\begin{aligned} + n_i(\vb{r}) + \approx n_{i0} \bigg( 1 - \frac{q_i \phi(\vb{r})}{k_B T_i} \bigg) + \qquad \quad + n_e(\vb{r}) + \approx n_{e0} \bigg( 1 - \frac{q_e \phi(\vb{r})}{k_B T_e} \bigg) +\end{aligned}$$ + +Inserting this back into Gauss' law, +we arrive at the following equation for $\phi(\vb{r})$, +where we have assumed quasi-neutrality such that $q_i n_{i0} = q_e n_{e0}$: + +$$\begin{aligned} + \nabla^2 \phi + &= - \frac{1}{\varepsilon_0} + \bigg( q_i n_{i0} - n_{i0} \frac{q_i^2 \phi}{k_B T_i} + q_e n_{e0} - n_{e0} \frac{q_e^2 \phi}{k_B T_e} + q_t \delta(\vb{r}) \bigg) + \\ + &= \bigg( \frac{n_{i0} q_i^2}{\varepsilon_0 k_B T_i} + \frac{n_{e0} q_e^2}{\varepsilon_0 k_B T_e} \bigg) \phi + - \frac{q_t}{\varepsilon_0} \delta(\vb{r}) +\end{aligned}$$ + +We now define the **ion** and **electron Debye lengths** +$\lambda_{Di}$ and $\lambda_{De}$ as follows: + +$$\begin{aligned} + \boxed{ + \frac{1}{\lambda_{Di}^2} + \equiv \frac{n_{i0} q_i^2}{\varepsilon_0 k_B T_i} + } + \qquad \quad + \boxed{ + \frac{1}{\lambda_{De}^2} + \equiv \frac{n_{e0} q_e^2}{\varepsilon_0 k_B T_e} + } +\end{aligned}$$ + +And then the **total Debye length** $\lambda_D$ is defined as the sum of their inverses, +and gives the rough thickness of the Debye sheath: + +$$\begin{aligned} + \boxed{ + \frac{1}{\lambda_D^2} + \equiv \frac{1}{\lambda_{Di}^2} + \frac{1}{\lambda_{De}^2} + = \frac{n_{i0} q_i^2 T_e + n_{e0} q_e^2 T_i}{\varepsilon_0 k_B T_i T_e} + } +\end{aligned}$$ + +With this, the equation can be put in the form below, +suggesting exponential decay: + +$$\begin{aligned} + \nabla^2 \phi(\vb{r}) + &= \frac{1}{\lambda_D^2} \phi(\vb{r}) + - \frac{q_t}{\varepsilon_0} \delta(\vb{r}) +\end{aligned}$$ + +This has the following solution, +known as the **Yukawa potential**, +which decays exponentially, +representing the plasma's **self-shielding** +over a characteristic distance $\lambda_D$: + +$$\begin{aligned} + \boxed{ + \phi(r) + = \frac{q_t}{4 \pi \varepsilon_0 r} \exp\!\Big( \!-\!\frac{r}{\lambda_D} \Big) + } +\end{aligned}$$ + +Note that $r$ is a scalar, +i.e. the potential depends only on the radial distance to $q_t$. +This treatment only makes sense +if the plasma is sufficiently dense, +such that there is a large number of particles +in a sphere with radius $\lambda_D$. +This corresponds to a large [Coulomb logarithm](/know/concept/coulomb-logarithm/) $\ln\!(\Lambda)$: + +$$\begin{aligned} + 1 \ll \frac{4 \pi}{3} n_0 \lambda_D^3 = \frac{2}{9} \Lambda +\end{aligned}$$ + +The name *Yukawa potential* originates from particle physics, +but can in general be used to refer to any potential (electric or energetic) +of the following form: + +$$\begin{aligned} + V(r) + = \frac{A}{r} \exp\!(-B r) +\end{aligned}$$ + +Where $A$ and $B$ are scaling constants that depend on the problem at hand. + + + +## References +1. P.M. Bellan, + *Fundamentals of plasma physics*, + 1st edition, Cambridge. +2. M. Salewski, A.H. Nielsen, + *Plasma physics: lecture notes*, + 2021, unpublished. diff --git a/content/know/concept/euler-equations/index.pdc b/content/know/concept/euler-equations/index.pdc index 0088d4f..b531260 100644 --- a/content/know/concept/euler-equations/index.pdc +++ b/content/know/concept/euler-equations/index.pdc @@ -57,7 +57,7 @@ Next, we want to find another expression for $\va{f^*}$. We know that the overall force $\va{F}$ on an arbitrary volume $V$ of the fluid is the sum of the gravity body force $\va{F}_g$, and the pressure contact force $\va{F}_p$ on the enclosing surface $S$. -Using Gauss' theorem, we then find: +Using the divergence theorem, we then find: $$\begin{aligned} \va{F} @@ -91,7 +91,7 @@ $$\begin{aligned} The last ingredient is **incompressibility**: the same volume must simultaneously be flowing in and out of an arbitrary enclosure $S$. -Then, by Gauss' theorem: +Then, by the divergence theorem: $$\begin{aligned} 0 @@ -131,7 +131,7 @@ but the size of their lumps does not change (incompressibility). To update the equations, we demand conservation of mass: the mass evolution of a volume $V$ is equal to the mass flow through its boundary $S$. -Applying Gauss' theorem again: +Applying the divergence theorem again: $$\begin{aligned} 0 diff --git a/content/know/concept/hydrostatic-pressure/index.pdc b/content/know/concept/hydrostatic-pressure/index.pdc index 001a198..d47d77f 100644 --- a/content/know/concept/hydrostatic-pressure/index.pdc +++ b/content/know/concept/hydrostatic-pressure/index.pdc @@ -54,7 +54,7 @@ $$\begin{aligned} If we now consider a *closed* surface, which encloses a "blob" of the fluid, -then we can use Gauss' theorem to get a volume integral: +then we can use the divergence theorem to get a volume integral: $$\begin{aligned} \va{F} diff --git a/content/know/concept/lindhard-function/index.pdc b/content/know/concept/lindhard-function/index.pdc index d38dc2e..96244c9 100644 --- a/content/know/concept/lindhard-function/index.pdc +++ b/content/know/concept/lindhard-function/index.pdc @@ -13,6 +13,11 @@ markup: pandoc # Lindhard function +The **Lindhard function** describes the response of an electron gas +to an external perturbation, +and can be regarded as a quantum-mechanical +alternative to the [Drude model](/know/concept/drude-model/). + We start from the [Kubo formula](/know/concept/kubo-formula/) for the electron density operator $\hat{n}$, which describes the change in $\expval{\hat{n}}$ @@ -396,7 +401,7 @@ $$\begin{aligned} \end{aligned}$$ Therefore, by inserting all the above expressions, -we arrive at the following dielectric function $\varepsilon_r$ +we arrive at the Lindhard dielectric function $\varepsilon_r$ for a non-interacting electron gas in a uniform potential: $$\begin{aligned} diff --git a/content/know/concept/maxwell-bloch-equations/index.pdc b/content/know/concept/maxwell-bloch-equations/index.pdc index ae7d119..020a120 100644 --- a/content/know/concept/maxwell-bloch-equations/index.pdc +++ b/content/know/concept/maxwell-bloch-equations/index.pdc @@ -393,9 +393,18 @@ It is trivial to show that $\vb{E}$ and $\vb{P}$ can be replaced by $\vb{E}^{+}$ and $\vb{P}^{+}$. It is also simple to convert the dipole $\vb{p}^{+}$ and inversion $d$ -into their macroscopic versions $\vb{P}^{+}$ and $D$, -simply by averaging over the atoms per unit of volume. -We thus arrive at the **Maxwell-Bloch equations**: +into their macroscopic versions $\vb{P}^{+}$ and $D$: + +$$\begin{aligned} + \vb{P}^{+}(\vb{r}, t) + = \sum_{n} \vb{p}^{+}_n \: \delta(\vb{r} \!-\! \vb{r}_n) + \qquad \quad + D(\vb{r}, t) + = \sum_{n} d_n \: \delta(\vb{r} \!-\! \vb{r}_n) +\end{aligned}$$ + +We thus arrive at the **Maxwell-Bloch equations**, +which are relevant for laser theory: $$\begin{aligned} \boxed{ diff --git a/content/know/concept/maxwells-equations/index.pdc b/content/know/concept/maxwells-equations/index.pdc index 1551311..967372d 100644 --- a/content/know/concept/maxwells-equations/index.pdc +++ b/content/know/concept/maxwells-equations/index.pdc @@ -209,3 +209,57 @@ $$\begin{aligned} = \vb{J}_M + \vb{J}_P = \nabla \cross \vb{M} + \pdv{\vb{P}}{t} \end{aligned}$$ + + +## Redundancy of Gauss' laws + +In fact, both of Gauss' laws are redundant, +because they are already implied by Faraday's and Ampère's laws. +Suppose we take the divergence of Faraday's law: + +$$\begin{aligned} + 0 + = \nabla \cdot \nabla \cross \vb{E} + = - \nabla \cdot \pdv{\vb{B}}{t} + = - \pdv{t} (\nabla \cdot \vb{B}) +\end{aligned}$$ + +Since the divergence of a curl is always zero, +the right-hand side must vanish. +We know that $\vb{B}$ can vary in time, +so our only option to satisfy this is to demand that $\nabla \cdot \vb{B} = 0$. +We thus arrive arrive at Gauss' law for magnetism from Faraday's law. + +The same technique works for Ampère's law. +Taking its divergence gives us: + +$$\begin{aligned} + 0 + = \frac{1}{\mu_0} \nabla \cdot \nabla \cross \vb{B} + = \nabla \cdot \vb{J} + \varepsilon_0 \pdv{t} (\nabla \cdot \vb{E}) +\end{aligned}$$ + +We integrate this over an arbitrary volume $V$, +and apply the divergence theorem: + +$$\begin{aligned} + 0 + &= \int_V \nabla \cdot \vb{J} \dd{V} + \pdv{t} \int_V \varepsilon_0 \nabla \cdot \vb{E} \dd{V} + \\ + &= \oint_S \vb{J} \cdot \dd{S} + \pdv{t} \int_V \varepsilon_0 \nabla \cdot \vb{E} \dd{V} +\end{aligned}$$ + +The first integral represents the current (charge flux) +through the surface of $V$. +Electric charge is not created or destroyed, +so the second integral *must* be the total charge in $V$: + +$$\begin{aligned} + Q + = \int_V \varepsilon_0 \nabla \cdot \vb{E} \dd{V} + \quad \implies \quad + \nabla \cdot \vb{E} + = \frac{\rho}{\varepsilon_0} +\end{aligned}$$ + +And we thus arrive at Gauss' law from Ampère's law and charge conservation. diff --git a/content/know/concept/rutherford-scattering/index.pdc b/content/know/concept/rutherford-scattering/index.pdc index 81bb133..c89b477 100644 --- a/content/know/concept/rutherford-scattering/index.pdc +++ b/content/know/concept/rutherford-scattering/index.pdc @@ -14,7 +14,8 @@ markup: pandoc # Rutherford scattering **Rutherford scattering** or **Coulomb scattering** -is an elastic pseudo-collision of two electrically charged particles. +is an [elastic pseudo-collision](/know/concept/elastic-collision/) +of two electrically charged particles. It is not a true collision, and is caused by Coulomb repulsion. The general idea is illustrated below. diff --git a/content/know/concept/two-fluid-equations/index.pdc b/content/know/concept/two-fluid-equations/index.pdc new file mode 100644 index 0000000..cd77f5e --- /dev/null +++ b/content/know/concept/two-fluid-equations/index.pdc @@ -0,0 +1,178 @@ +--- +title: "Two-fluid equations" +firstLetter: "T" +publishDate: 2021-10-19 +categories: +- Physics +- Plasma physics + +date: 2021-10-18T10:12:20+02:00 +draft: false +markup: pandoc +--- + +# Two-fluid equations + +The **two-fluid model** describes a plasma as two separate but overlapping fluids, +one for ions and one for electrons. +Instead of tracking individual particles, +it gives the dynamics of fluid elements $\dd{V}$ (i.e. small "blobs"). +These blobs are assumed to be much larger than +the [Debye length](/know/concept/debye-length/), +such that electromagnetic interactions between nearby blobs can be ignored. + +From Newton's second law, we know that the velocity $\vb{v}$ +of a particle with mass $m$ and charge $q$ is as follows, +when subjected only to the [Lorentz force](/know/concept/lorentz-force/): + +$$\begin{aligned} + m \dv{\vb{v}}{t} + = q (\vb{E} + \vb{v} \cross \vb{B}) +\end{aligned}$$ + +From here, the derivation is similar to that of the +[Navier-Stokes equations](/know/concept/navier-stokes-equations/). +We replace the time derivative with a +[material derivative](/know/concept/material-derivative/) $\mathrm{D}/\mathrm{D}t$, +and define a blob's velocity $\vb{u}$ +as the average velocity of the particles inside it, leading to: + +$$\begin{aligned} + m n \frac{\mathrm{D} \vb{u}}{\mathrm{D} t} + = q n (\vb{E} + \vb{u} \cross \vb{B}) +\end{aligned}$$ + +Where we have multiplied by the number density $n$ of the particles. +Due to particle collisions in the fluid, +stresses become important. Therefore, we include +the [Cauchy stress tensor](/know/concept/cauchy-stress-tensor/) $\hat{P}$, +leading to the following two equations: + +$$\begin{aligned} + m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t} + &= q_i n_i (\vb{E} + \vb{u}_i \cross \vb{B}) + \nabla \cdot \hat{P}_i{}^\top + \\ + m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t} + &= q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) + \nabla \cdot \hat{P}_e{}^\top +\end{aligned}$$ + +Where the subscripts $i$ and $e$ refer to ions and electrons, respectively. +Finally, we also account for momentum transfer between ions and electrons +due to [Rutherford scattering](/know/concept/rutherford-scattering/), +leading to these **two-fluid momentum equations**: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t} + &= q_i n_i (\vb{E} + \vb{u}_i \cross \vb{B}) + \nabla \cdot \hat{P}_i{}^\top - f_{ie} m_i n_i (\vb{u}_i - \vb{u}_e) + \\ + m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t} + &= q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) + \nabla \cdot \hat{P}_e{}^\top - f_{ei} m_e n_e (\vb{u}_e - \vb{u}_i) + \end{aligned} + } +\end{aligned}$$ + +Where $f_{ie}$ is the mean frequency at which an ion collides with electrons, +and vice versa for $f_{ei}$. +For simplicity, we assume that the plasma is isotropic +and that shear stresses are negligible, +in which case the stress term can be replaced +by the gradient $- \nabla p$ of a scalar pressure $p$: + +$$\begin{aligned} + m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t} + &= q_i n_i (\vb{E} + \vb{u}_i \cross \vb{B}) - \nabla p_i - f_{ie} m_i n_i (\vb{u}_i - \vb{u}_e) + \\ + m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t} + &= q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) - \nabla p_e - f_{ei} m_e n_e (\vb{u}_e - \vb{u}_i) +\end{aligned}$$ + +Next, we demand that matter is conserved. +In other words, the rate at which particles enter/leave a volume $V$ +must be equal to the flux through the enclosing surface $S$: + +$$\begin{aligned} + 0 + &= \pdv{t} \int_V n \dd{V} + \oint_S n \vb{u} \cdot \dd{\vb{S}} + = \int_V \Big( \pdv{n}{t} + \nabla \cdot (n \vb{u}) \Big) \dd{V} +\end{aligned}$$ + +Where we have used the divergence theorem. +Since $V$ is arbitrary, we can remove the integrals, +leading to the following **continuity equations**: + +$$\begin{aligned} + \boxed{ + \pdv{n_i}{t} + \nabla \cdot (n_i \vb{u}_i) + = 0 + \qquad \quad + \pdv{n_e}{t} + \nabla \cdot (n_e \vb{u}_e) + = 0 + } +\end{aligned}$$ + +Currently, we have 8 equations (2 scalar continuity, 2 vector momentum), +but 16 unknowns $\vb{u}_i$, $\vb{u}_e$, $\vb{E}$, $\vb{B}$, $n_i$, $n_e$, $p_i$ and $p_e$. +We would like to close this system, so we need 8 more. +An obvious choice is [Maxwell's equations](/know/concept/maxwells-equations/), +in particular Faraday's and Ampère's law +(since Gauss' laws are in fact redundant; see the article on Maxwell's equations): + +$$\begin{aligned} + \boxed{ + \nabla \cross \vb{E} = - \pdv{\vb{B}}{t} + \qquad \quad + \nabla \cross \vb{B} = \mu_0 \Big( n_i q_i \vb{u}_i + n_e q_e \vb{u}_e + \varepsilon_0 \pdv{\vb{E}}{t} \Big) + } +\end{aligned}$$ + +Now we have 14 equations, so we need 2 more, for the pressures $p_i$ and $p_e$. +This turns out to be the thermodynamic **equation of state**: +for quasistatic, reversible, adiabatic compression +of a gas with constant heat capacities (i.e. a *calorically perfect* gas), +it turns out that: + +$$\begin{aligned} + \dv{t} \big( p V^\gamma \big) = 0 + \qquad \quad + \gamma + \equiv \frac{C_P}{C_V} + = \frac{N + 2}{N} +\end{aligned}$$ + +Where $\gamma$ is the *heat capacity ratio*, +and can be calculated from the number of degrees of freedom $N$ +of each particle in the gas. +In a fully ionized plasma, $N = 3$. + +The density $n \propto 1/V$, +so since $p V^\gamma$ is constant in time, +for some (known) constant $C$: + +$$\begin{aligned} + \dv{t} \Big( \frac{p}{n^\gamma} \Big) = 0 + \quad \implies \quad + p = C n^\gamma +\end{aligned}$$ + +In the two-fluid model, we thus have the following two equations of state, +giving us a set of 16 equations for 16 unknowns: + +$$\begin{aligned} + \boxed{ + p_i = C_i n_i^\gamma + \qquad \quad + p_e = C_e n_e^\gamma + } +\end{aligned}$$ + + + +## References +1. F.F. Chen, + *Introduction to plasma physics and controlled fusion*, + 3rd edition, Springer. +2. M. Salewski, A.H. Nielsen, + *Plasma physics: lecture notes*, + 2021, unpublished. |