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Diffstat (limited to 'source/know/concept/debye-length')
| -rw-r--r-- | source/know/concept/debye-length/index.md | 36 |
1 files changed, 18 insertions, 18 deletions
diff --git a/source/know/concept/debye-length/index.md b/source/know/concept/debye-length/index.md index a42d137..e226ad9 100644 --- a/source/know/concept/debye-length/index.md +++ b/source/know/concept/debye-length/index.md @@ -16,10 +16,10 @@ 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 $n_e$: +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 $$n_e$$: $$\begin{aligned} \nabla^2 \phi(\vb{r}) @@ -28,9 +28,9 @@ $$\begin{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 $n_e$. +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/) @@ -45,7 +45,7 @@ $$\begin{aligned} \end{aligned}$$ We assume that electrical interactions are weak compared to thermal effects, -i.e. $k_B T \gg q \phi$ in both cases. +i.e. $$k_B T \gg q \phi$$ in both cases. Then we Taylor-expand the Boltzmann relations to first order: $$\begin{aligned} @@ -57,8 +57,8 @@ $$\begin{aligned} \end{aligned}$$ Inserting this back into Gauss' law, -we arrive at the following equation for $\phi(\vb{r})$, -where we have assumed quasi-neutrality such that $q_i n_{i0} = q_e n_{e0}$: +we arrive at the following equation for $$\phi(\vb{r})$$, +where we have assumed quasi-neutrality such that $$q_i n_{i0} = q_e n_{e0}$$: $$\begin{aligned} \nabla^2 \phi @@ -70,7 +70,7 @@ $$\begin{aligned} \end{aligned}$$ We now define the **ion** and **electron Debye lengths** -$\lambda_{Di}$ and $\lambda_{De}$ as follows: +$$\lambda_{Di}$$ and $$\lambda_{De}$$ as follows: $$\begin{aligned} \boxed{ @@ -84,7 +84,7 @@ $$\begin{aligned} } \end{aligned}$$ -And then the **total Debye length** $\lambda_D$ is defined as the sum of their inverses, +And then the **total Debye length** $$\lambda_D$$ is defined as the sum of their inverses, and gives the rough thickness of the Debye sheath: $$\begin{aligned} @@ -108,7 +108,7 @@ This has the following solution, known as the **Yukawa potential**, which decays exponentially, representing the plasma's **self-shielding** -over a characteristic distance $\lambda_D$: +over a characteristic distance $$\lambda_D$$: $$\begin{aligned} \boxed{ @@ -117,13 +117,13 @@ $$\begin{aligned} } \end{aligned}$$ -Note that $r$ is a scalar, -i.e. the potential depends only on the radial distance to $q_t$. +Note that $$r$$ is a scalar, +i.e. the potential depends only on the radial distance to $$q_t$$. This treatment only makes sense if the plasma is sufficiently dense, such that there is a large number of particles -in a sphere with radius $\lambda_D$. -This corresponds to a large [Coulomb logarithm](/know/concept/coulomb-logarithm/) $\ln\!(\Lambda)$: +in a sphere with radius $$\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 @@ -138,7 +138,7 @@ $$\begin{aligned} = \frac{A}{r} \exp(-B r) \end{aligned}$$ -Where $A$ and $B$ are scaling constants that depend on the problem at hand. +Where $$A$$ and $$B$$ are scaling constants that depend on the problem at hand. |