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 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, \(\vb{p}\) 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.


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