From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

---
 source/know/concept/dielectric-function/index.md | 46 ++++++++++++------------
 1 file changed, 23 insertions(+), 23 deletions(-)

(limited to 'source/know/concept/dielectric-function')

diff --git a/source/know/concept/dielectric-function/index.md b/source/know/concept/dielectric-function/index.md
index 30622f4..529ce2a 100644
--- a/source/know/concept/dielectric-function/index.md
+++ b/source/know/concept/dielectric-function/index.md
@@ -9,11 +9,11 @@ categories:
 layout: "concept"
 ---
 
-The **dielectric function** or **relative permittivity** $\varepsilon_r$
+The **dielectric function** or **relative permittivity** $$\varepsilon_r$$
 is a measure of how strongly a given medium counteracts
 [electric fields](/know/concept/electric-field/) compared to a vacuum.
-Let $\vb{D}$ be the applied external field,
-and $\vb{E}$ the effective field inside the material:
+Let $$\vb{D}$$ be the applied external field,
+and $$\vb{E}$$ the effective field inside the material:
 
 $$\begin{aligned}
     \boxed{
@@ -21,15 +21,15 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-If $\varepsilon_r$ is large, then $\vb{D}$ is strongly suppressed,
+If $$\varepsilon_r$$ is large, then $$\vb{D}$$ is strongly suppressed,
 because the material's electrons and nuclei move to create an opposing field.
-In order for $\varepsilon_r$ to be well defined, we only consider linear media,
-where the induced polarization $\vb{P}$ is proportional to $\vb{E}$.
+In order for $$\varepsilon_r$$ to be well defined, we only consider linear media,
+where the induced polarization $$\vb{P}$$ is proportional to $$\vb{E}$$.
 
-We would like to find an alternative definition of $\varepsilon_r$.
-Consider that the usual electric fields $\vb{E}$, $\vb{D}$, and $\vb{P}$
+We would like to find an alternative definition of $$\varepsilon_r$$.
+Consider that the usual electric fields $$\vb{E}$$, $$\vb{D}$$, and $$\vb{P}$$
 can each be written as the gradient of an electrostatic potential like so,
-where $\Phi_\mathrm{tot}$, $\Phi_\mathrm{ext}$ and $\Phi_\mathrm{ind}$
+where $$\Phi_\mathrm{tot}$$, $$\Phi_\mathrm{ext}$$ and $$\Phi_\mathrm{ind}$$
 are the total, external and induced potentials, respectively:
 
 $$\begin{aligned}
@@ -43,8 +43,8 @@ $$\begin{aligned}
     = \varepsilon_0 \nabla \Phi_\mathrm{ind}
 \end{aligned}$$
 
-Such that $\Phi_\mathrm{tot} = \Phi_\mathrm{ext} + \Phi_\mathrm{ind}$.
-Inserting this into $\vb{D} = \varepsilon_0 \varepsilon_r \vb{E}$
+Such that $$\Phi_\mathrm{tot} = \Phi_\mathrm{ext} + \Phi_\mathrm{ind}$$.
+Inserting this into $$\vb{D} = \varepsilon_0 \varepsilon_r \vb{E}$$
 then suggests defining:
 
 $$\begin{aligned}
@@ -57,10 +57,10 @@ $$\begin{aligned}
 
 ## From induced charge density
 
-A common way to calculate $\varepsilon_r$ is from
-the induced charge density $\rho_\mathrm{ind}$,
+A common way to calculate $$\varepsilon_r$$ is from
+the induced charge density $$\rho_\mathrm{ind}$$,
 i.e. the offset caused by the material's particles responding to the field.
-We start from [Gauss' law](/know/concept/maxwells-equations/) for $\vb{P}$:
+We start from [Gauss' law](/know/concept/maxwells-equations/) for $$\vb{P}$$:
 
 $$\begin{aligned}
     \nabla \cdot \vb{P}
@@ -77,8 +77,8 @@ $$\begin{aligned}
     = V(\vb{q}) \: \rho_\mathrm{ind}(\vb{q})
 \end{aligned}$$
 
-Where $V(\vb{q})$ represents Coulomb interactions,
-and $V(0) = 0$ to ensure overall neutrality:
+Where $$V(\vb{q})$$ represents Coulomb interactions,
+and $$V(0) = 0$$ to ensure overall neutrality:
 
 $$\begin{aligned}
     V(\vb{q})
@@ -89,7 +89,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 The [convolution theorem](/know/concept/convolution-theorem/)
-then gives us the solution $\Phi_\mathrm{ind}$ in the $\vb{r}$-domain:
+then gives us the solution $$\Phi_\mathrm{ind}$$ in the $$\vb{r}$$-domain:
 
 $$\begin{aligned}
     \Phi_\mathrm{ind}(\vb{r})
@@ -97,13 +97,13 @@ $$\begin{aligned}
     = \int_{-\infty}^\infty V(\vb{r} - \vb{r}') \: \rho_\mathrm{ind}(\vb{r}') \dd{\vb{r}'}
 \end{aligned}$$
 
-To proceed, we need to find an expression for $\rho_\mathrm{ind}$
-that is proportional to $\Phi_\mathrm{tot}$ or $\Phi_\mathrm{ext}$,
+To proceed, we need to find an expression for $$\rho_\mathrm{ind}$$
+that is proportional to $$\Phi_\mathrm{tot}$$ or $$\Phi_\mathrm{ext}$$,
 or some linear combination thereof.
 Such an expression must exist for a linear material.
 
-Suppose we can show that $\rho_\mathrm{ind} = C_\mathrm{ext} \Phi_\mathrm{ext}$,
-for some $C_\mathrm{ext}$, which may depend on $\vb{q}$. Then:
+Suppose we can show that $$\rho_\mathrm{ind} = C_\mathrm{ext} \Phi_\mathrm{ext}$$,
+for some $$C_\mathrm{ext}$$, which may depend on $$\vb{q}$$. Then:
 
 $$\begin{aligned}
     \Phi_\mathrm{tot}
@@ -115,8 +115,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Similarly, suppose we can show that $\rho_\mathrm{ind} = C_\mathrm{tot} \Phi_\mathrm{tot}$,
-for some quantity $C_\mathrm{tot}$, then:
+Similarly, suppose we can show that $$\rho_\mathrm{ind} = C_\mathrm{tot} \Phi_\mathrm{tot}$$,
+for some quantity $$C_\mathrm{tot}$$, then:
 
 $$\begin{aligned}
     \Phi_\mathrm{ext}
-- 
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