Electric displacement

In physics, the electric displacement, usually denoted by D, is a vector field in a non-conducting medium, a dielectric. The displacement D is proportional to the electric field E in which the dielectric is placed. 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 does not occur 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 (SI), or equal (Gaussian), to E. Often D is termed an auxiliary field with E the principal field. An other 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 chunk of dielectric when it is brought into an electric field. The fact that for any insulator ε_r > 1 (i.e., that D is not simply equal to ε₀E) has as its cause the polarization of the dielectric.

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.

Tensor character of relative permittivity

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

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

so that D and E are not necessarily parallel.