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} -- cgit v1.2.3