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-rw-r--r--source/know/concept/electric-field/index.md76
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diff --git a/source/know/concept/electric-field/index.md b/source/know/concept/electric-field/index.md
index 38c1ff6..2433edf 100644
--- a/source/know/concept/electric-field/index.md
+++ b/source/know/concept/electric-field/index.md
@@ -8,43 +8,43 @@ categories:
layout: "concept"
---
-The **electric field** $\vb{E}$ is a vector field
+The **electric field** $$\vb{E}$$ is a vector field
that describes electric effects,
and is defined as the field that correctly predicts
the [Lorentz force](/know/concept/lorentz-force/)
-on a particle with electric charge $q$:
+on a particle with electric charge $$q$$:
$$\begin{aligned}
\vb{F}
= q \vb{E}
\end{aligned}$$
-This definition implies that the direction of $\vb{E}$
+This definition implies that the direction of $$\vb{E}$$
is from positive to negative charges,
since opposite charges attracts and like charges repel.
-If two opposite point charges with magnitude $q$
+If two opposite point charges with magnitude $$q$$
are observed from far away,
they can be treated as a single object called a **dipole**,
-which has an **electric dipole moment** $\vb{p}$ defined like so,
-where $\vb{d}$ is the vector going from
-the negative to the positive charge (opposite direction of $\vb{E}$):
+which has an **electric dipole moment** $$\vb{p}$$ defined like so,
+where $$\vb{d}$$ is the vector going from
+the negative to the positive charge (opposite direction of $$\vb{E}$$):
$$\begin{aligned}
\vb{p} = q \vb{d}
\end{aligned}$$
Alternatively, for consistency with [magnetic fields](/know/concept/magnetic-field/),
-$\vb{p}$ can be defined from the aligning torque $\vb{\tau}$
-experienced by the dipole when placed in an $\vb{E}$-field.
-In other words, $\vb{p}$ satisfies:
+$$\vb{p}$$ can be defined from the aligning torque $$\vb{\tau}$$
+experienced by the dipole when placed in an $$\vb{E}$$-field.
+In other words, $$\vb{p}$$ satisfies:
$$\begin{aligned}
\vb{\tau} = \vb{p} \times \vb{E}
\end{aligned}$$
-Where $\vb{p}$ has units of $\mathrm{C m}$.
-The **polarization density** $\vb{P}$ is defined from $\vb{p}$,
+Where $$\vb{p}$$ has units of $$\mathrm{C m}$$.
+The **polarization density** $$\vb{P}$$ is defined from $$\vb{p}$$,
and roughly speaking represents the moments per unit volume:
$$\begin{aligned}
@@ -53,16 +53,16 @@ $$\begin{aligned}
\vb{p} = \int_V \vb{P} \dd{V}
\end{aligned}$$
-If $\vb{P}$ has the same magnitude and direction throughout the body,
-then this becomes $\vb{p} = \vb{P} V$, where $V$ is the volume.
-Therefore, $\vb{P}$ has units of $\mathrm{C / m^2}$.
+If $$\vb{P}$$ has the same magnitude and direction throughout the body,
+then this becomes $$\vb{p} = \vb{P} V$$, where $$V$$ is the volume.
+Therefore, $$\vb{P}$$ has units of $$\mathrm{C / m^2}$$.
-A nonzero $\vb{P}$ complicates things,
-since it contributes to the field and hence modifies $\vb{E}$.
+A nonzero $$\vb{P}$$ complicates things,
+since it contributes to the field and hence modifies $$\vb{E}$$.
We thus define
-the "free" **displacement field** $\vb{D}$
-from the "bound" field $\vb{P}$
-and the "net" field $\vb{E}$:
+the "free" **displacement field** $$\vb{D}$$
+from the "bound" field $$\vb{P}$$
+and the "net" field $$\vb{E}$$:
$$\begin{aligned}
\vb{D} \equiv \varepsilon_0 \vb{E} + \vb{P}
@@ -70,21 +70,21 @@ $$\begin{aligned}
\vb{E} = \frac{1}{\varepsilon_0} (\vb{D} - \vb{P})
\end{aligned}$$
-Where the **electric permittivity of free space** $\varepsilon_0$ is a known constant.
+Where the **electric permittivity of free space** $$\varepsilon_0$$ is a known constant.
It is important to point out some inconsistencies here:
-$\vb{D}$ and $\vb{P}$ contain a factor of $\varepsilon_0$,
+$$\vb{D}$$ and $$\vb{P}$$ contain a factor of $$\varepsilon_0$$,
and therefore measure **flux density**,
-while $\vb{E}$ does not contain $\varepsilon_0$,
+while $$\vb{E}$$ does not contain $$\varepsilon_0$$,
and thus measures **field intensity**.
Note that this convention is the opposite
-of the magnetic analogues $\vb{B}$, $\vb{H}$ and $\vb{M}$,
-and that $\vb{M}$ has the opposite sign of $\vb{P}$.
+of the magnetic analogues $$\vb{B}$$, $$\vb{H}$$ and $$\vb{M}$$,
+and that $$\vb{M}$$ has the opposite sign of $$\vb{P}$$.
-The polarization $\vb{P}$ is a function of $\vb{E}$.
+The polarization $$\vb{P}$$ is a function of $$\vb{E}$$.
In addition to the inherent polarity
-of the material $\vb{P}_0$ (zero in most cases),
+of the material $$\vb{P}_0$$ (zero in most cases),
there is a (possibly nonlinear) response
-to the applied $\vb{E}$-field:
+to the applied $$\vb{E}$$-field:
$$\begin{aligned}
\vb{P} =
@@ -93,10 +93,10 @@ $$\begin{aligned}
+ \varepsilon_0 \chi_e^{(3)} |\vb{E}|^2 \: \vb{E} + ...
\end{aligned}$$
-Where the $\chi_e^{(n)}$ are the **electric susceptibilities** of the medium.
-For simplicity, we often assume that only the $n\!=\!1$ term is nonzero,
-which is the linear response to $\vb{E}$.
-In that case, we define the **absolute permittivity** $\varepsilon$ so that:
+Where the $$\chi_e^{(n)}$$ are the **electric susceptibilities** of the medium.
+For simplicity, we often assume that only the $$n\!=\!1$$ term is nonzero,
+which is the linear response to $$\vb{E}$$.
+In that case, we define the **absolute permittivity** $$\varepsilon$$ so that:
$$\begin{aligned}
\vb{D}
@@ -106,15 +106,15 @@ $$\begin{aligned}
= \varepsilon \vb{E}
\end{aligned}$$
-I.e. $\varepsilon \equiv \varepsilon_r \varepsilon_0$,
-where $\varepsilon_r \equiv 1 + \chi_e^{(1)}$ is
+I.e. $$\varepsilon \equiv \varepsilon_r \varepsilon_0$$,
+where $$\varepsilon_r \equiv 1 + \chi_e^{(1)}$$ is
the [**dielectric function**](/know/concept/dielectric-function/)
or **relative permittivity**,
whose calculation is of great interest in physics.
-In reality, a material cannot respond instantly to $\vb{E}$,
-meaning that $\chi_e^{(1)}$ is a function of time,
-and that $\vb{P}$ is the convolution of $\chi_e^{(1)}(t)$ and $\vb{E}(t)$:
+In reality, a material cannot respond instantly to $$\vb{E}$$,
+meaning that $$\chi_e^{(1)}$$ is a function of time,
+and that $$\vb{P}$$ is the convolution of $$\chi_e^{(1)}(t)$$ and $$\vb{E}(t)$$:
$$\begin{aligned}
\vb{P}(t)
@@ -122,6 +122,6 @@ $$\begin{aligned}
= \varepsilon_0 \int_{-\infty}^\infty \chi_e^{(1)}(t - \tau) \: \vb{E}(\tau) \:d\tau
\end{aligned}$$
-Note that this definition requires $\chi_e^{(1)}(t) = 0$ for $t < 0$
+Note that this definition requires $$\chi_e^{(1)}(t) = 0$$ for $$t < 0$$
in order to ensure causality,
which leads to the [Kramers-Kronig relations](/know/concept/kramers-kronig-relations/).