---
title: "Electric field"
firstLetter: "E"
publishDate: 2021-07-12
categories:
- Physics
- Electromagnetism

date: 2021-07-12T09:46:25+02:00
draft: false
markup: pandoc
---

## Electric 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
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 as follows,
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 **relative permittivity** $\varepsilon_r \equiv 1 + \chi_e^{(1)}$
and the **absolute permittivity** $\varepsilon \equiv \varepsilon_r \varepsilon_0$,
so that:

$$\begin{aligned}
    \vb{D}
    = \varepsilon_0 \vb{E} + \vb{P}
    = \varepsilon_0 \vb{E} + \varepsilon_0 \chi_e^{(1)} \vb{E}
    = \varepsilon_0 \varepsilon_r \vb{E}
    = \varepsilon \vb{E}
\end{aligned}$$

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 $\vb{E}(t)$:

$$\begin{aligned}
    \vb{P}(t)
    = (\chi_e^{(1)} * \vb{E})(t)
    = \int_{-\infty}^\infty \chi_e^{(1)}(t - \tau) \: \vb{E}(\tau) \:d\tau
\end{aligned}$$

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/).