Categories: Electromagnetism, Physics.

Electric field

The electric field E\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 qq:

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

This definition implies that the direction of E\vb{E} is from positive to negative charges, since opposite charges attracts and like charges repel.

If two opposite point charges with magnitude qq are observed from far away, they can be treated as a single object called a dipole, which has an electric dipole moment p\vb{p} defined like so, where d\vb{d} is the vector going from the negative to the positive charge (opposite direction of E\vb{E}):

p=qd\begin{aligned} \vb{p} = q \vb{d} \end{aligned}

Alternatively, for consistency with magnetic fields, p\vb{p} can be defined from the aligning torque τ\vb{\tau} experienced by the dipole when placed in an E\vb{E}-field. In other words, p\vb{p} satisfies:

τ=p×E\begin{aligned} \vb{\tau} = \vb{p} \times \vb{E} \end{aligned}

Where p\vb{p} has units of Cm\mathrm{C m}. The polarization density P\vb{P} is defined from p\vb{p}, and roughly speaking represents the moments per unit volume:

PdpdV    p=VPdV\begin{aligned} \vb{P} \equiv \dv{\vb{p}}{V} \:\:\iff\:\: \vb{p} = \int_V \vb{P} \dd{V} \end{aligned}

If P\vb{P} has the same magnitude and direction throughout the body, then this becomes p=PV\vb{p} = \vb{P} V, where VV is the volume. Therefore, P\vb{P} has units of C/m2\mathrm{C / m^2}.

A nonzero P\vb{P} complicates things, since it contributes to the field and hence modifies E\vb{E}. We thus define the “free” displacement field D\vb{D} from the “bound” field P\vb{P} and the “net” field E\vb{E}:

Dε0E+P    E=1ε0(DP)\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 ε0\varepsilon_0 is a known constant. It is important to point out some inconsistencies here: D\vb{D} and P\vb{P} contain a factor of ε0\varepsilon_0, and therefore measure flux density, while E\vb{E} does not contain ε0\varepsilon_0, and thus measures field intensity. Note that this convention is the opposite of the magnetic analogues B\vb{B}, H\vb{H} and M\vb{M}, and that M\vb{M} has the opposite sign of P\vb{P}.

The polarization P\vb{P} is a function of E\vb{E}. In addition to the inherent polarity of the material P0\vb{P}_0 (zero in most cases), there is a (possibly nonlinear) response to the applied E\vb{E}-field:

P=P0+ε0χe(1)E+ε0χe(2)EE+ε0χe(3)E2E+...\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 χe(n)\chi_e^{(n)} are the electric susceptibilities of the medium. For simplicity, we often assume that only the n ⁣= ⁣1n\!=\!1 term is nonzero, which is the linear response to E\vb{E}. In that case, we define the absolute permittivity ε\varepsilon so that:

D=ε0E+P=ε0E+ε0χe(1)E=ε0εrE=εE\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. εεrε0\varepsilon \equiv \varepsilon_r \varepsilon_0, where εr1+χe(1)\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 E\vb{E}, meaning that χe(1)\chi_e^{(1)} is a function of time, and that P\vb{P} is the convolution of χe(1)(t)\chi_e^{(1)}(t) and E(t)\vb{E}(t):

P(t)=ε0(χe(1)E)(t)=ε0χe(1)(tτ)E(τ)dτ\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 χe(1)(t)=0\chi_e^{(1)}(t) = 0 for t<0t < 0 in order to ensure causality, which leads to the Kramers-Kronig relations.