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+---
+title: "Electric field"
+date: 2021-07-12
+categories:
+- Physics
+- Electromagnetism
+layout: "concept"
+---
+
+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 $q$:
+
+$$\begin{aligned}
+    \vb{F}
+    = q \vb{E}
+\end{aligned}$$
+
+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 $q$
+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 $\vb{E}$):
+
+$$\begin{aligned}
+    \vb{p} = q \vb{d}
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    \vb{\tau} = \vb{p} \times \vb{E}
+\end{aligned}$$
+
+Where $\vb{p}$ has units of $\mathrm{C m}$.
+The **polarization density** $\vb{P}$ is defined from $\vb{p}$,
+and roughly speaking represents the moments per unit volume:
+
+$$\begin{aligned}
+    \vb{P} \equiv \dv{\vb{p}}{V}
+    \:\:\iff\:\:
+    \vb{p} = \int_V \vb{P} \dd{V}
+\end{aligned}$$
+
+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 $\mathrm{C / m^2}$.
+
+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 $\vb{E}$:
+
+$$\begin{aligned}
+    \vb{D} \equiv \varepsilon_0 \vb{E} + \vb{P}
+    \:\:\iff\:\:
+    \vb{E} = \frac{1}{\varepsilon_0} (\vb{D} - \vb{P})
+\end{aligned}$$
+
+Where the **electric permittivity of free space** $\varepsilon_0$ is a known constant.
+It is important to point out some inconsistencies here:
+$\vb{D}$ and $\vb{P}$ contain a factor of $\varepsilon_0$,
+and therefore measure **flux density**,
+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 $\vb{P}$.
+
+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),
+there is a (possibly nonlinear) response
+to the applied $\vb{E}$-field:
+
+$$\begin{aligned}
+    \vb{P} =
+    \vb{P}_0 + \varepsilon_0 \chi_e^{(1)} \vb{E}
+    + \varepsilon_0 \chi_e^{(2)} |\vb{E}| \: \vb{E}
+    + \varepsilon_0 \chi_e^{(3)} |\vb{E}|^2 \: \vb{E} + ...
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    \vb{D}
+    = \varepsilon_0 \vb{E} + \vb{P}
+    = \varepsilon_0 \vb{E} + \varepsilon_0 \chi_e^{(1)} \vb{E}
+    = \varepsilon_0 \varepsilon_r \vb{E}
+    = \varepsilon \vb{E}
+\end{aligned}$$
+
+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 $\vb{E}(t)$:
+
+$$\begin{aligned}
+    \vb{P}(t)
+    = \varepsilon_0 \big(\chi_e^{(1)} * \vb{E}\big)(t)
+    = \varepsilon_0 \int_{-\infty}^\infty \chi_e^{(1)}(t - \tau) \: \vb{E}(\tau) \:d\tau
+\end{aligned}$$
+
+Note that this definition requires $\chi_e^{(1)}(t) = 0$ for $t < 0$
+in order to ensure causality,
+which leads to the [Kramers-Kronig relations](/know/concept/kramers-kronig-relations/).
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