Electric displacement

In physics, electric displacement, usually denoted by D, is a vector field in a non-conducting medium, a dielectric, that is proportional to the electric field E. In SI units,

$\mathbf {D} (\mathbf {r} )=\epsilon _{0}\epsilon _{r}\mathbf {E} (\mathbf {r} ),$

where ε₀ is the electric constant and ε_r is the relative permittivity. In Gaussian units ε₀ is not defined and may put equal to unity. In vacuum the dimensionless quantity ε_r = 1 (both for SI and Gaussian units) and D is simply related, or equal, to E. Often D is termed an auxiliary field with the principal field being E. An alternative auxiliary field is the electric polarization P of the dielectric,

${\begin{aligned}\mathbf {P} &\equiv \mathbf {D} -\epsilon _{0}\mathbf {E} &\mathrm {(SI)} \\\mathbf {P} &\equiv (\mathbf {D} -\mathbf {E} )/4\pi &\mathrm {(Gaussian)} \\\end{aligned}}$

The vector field P describes the polarization (displacement of charges) occurring in a dielectric when it is inserted between the charged plates of a parallel-plate capacitor. Clearly, the fact that for any insulator ε_r > 1 (i.e., that D is not simply equal to ε_rE) has the same physical origin.

The electric displacement appears in one of the macroscopic Maxwell equations,

${\boldsymbol {\nabla }}\cdot \mathbf {D} (\mathbf {r} )=\rho (\mathbf {r} ),$

where the symbol ∇⋅ gives the divergence of D(r) and ρ(r) is the charge density at the point r.

As defined here, D and E are proportional, i.e., ε_r is a number (a scalar). For a non-isotropic dielectric ε_r may be a second rank tensor,

$D_{i}(\mathbf {r} )=\epsilon _{0}\sum _{j=1}^{3}(\epsilon _{r})_{ij}E_{j}(\mathbf {r} ),\quad i,j=1,2,3=x,y,z.$

In the case of a non-stationary (time-dependent) electric field, we have the Fourier transform of the electric field

${\mathcal {F}}(\mathbf {E} )(\omega )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\mathbf {E} (t)e^{-i\omega t}dt$

(To be continued)