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 &epsilon;0 is the electric constant and &epsilon;r is the relative permittivity. In Gaussian units &epsilon;0 is not defined and may put equal to unity. In vacuum the dimensionless quantity &epsilon;r = 1 (both for SI and Gaussian units) and D is simply related, or equal, to E.

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

\boldsymbol{\nabla} \cdot \mathbf{D}(\mathbf{r}) = \rho(\mathbf{r}), $$ where the symbol &nabla;&sdot; gives the divergence of D(r) and &rho;(r) is the charge density at the point r.

As defined here, D and E are proportional, i.e., &epsilon;r is a number (a scalar). For a non-isotropic dielectric &epsilon;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

\mathbf{E}(\omega) = \int_{-\infty}^\infty \mathbf{E}(t) e^{- i\omega t} d\omega $$

(To be continued)