---
title: "Electric field"
sort_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/).