Categories: Electromagnetism, Physics.

# 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 $q$:

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 $\vb{E}$):

Alternatively, for consistency with magnetic fields, $\vb{p}$ 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:

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 $\vb{E}$:

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:

I.e. $\varepsilon \equiv \varepsilon_r \varepsilon_0$,
where $\varepsilon_r \equiv 1 + \chi_e^{(1)}$ is
the **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.