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authorPrefetch2022-10-20 18:25:31 +0200
committerPrefetch2022-10-20 18:25:31 +0200
commit16555851b6514a736c5c9d8e73de7da7fc9b6288 (patch)
tree76b8bfd30f8941d0d85365990bcdbc5d0643cabc /source/know/concept/dielectric-function
parente5b9bce79b68a68ddd2e51daa16d2fea73b84fdb (diff)
Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'
Diffstat (limited to 'source/know/concept/dielectric-function')
-rw-r--r--source/know/concept/dielectric-function/index.md46
1 files changed, 23 insertions, 23 deletions
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}