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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 $q$:

$$\begin{aligned} \vb{F} = q \vb{E} \end{aligned}$$

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, $\vb{p}$ 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 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.

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