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.

Relation of D to surface charge density σ

Two capacitors at same voltage V. On the left vacuum between the plates and on the right a dielectric with relative permittivity ε_r. Absolute values of surface charge densities are indicated by σ.

The electric displacement D (is the magnitude of vector D, which points from − to + charge) is equal to the true surface charge density σ_true indicated in the figure, where two parallel-plate capacitors are shown that are completely identical except for the matter between the plates. In particular the plates have the same voltage difference V.

To explain this interpretation of D, we recall that the relative permittivity may be defined as the ratio of two capacitances, C ≡ Q / V, namely the ratio of the capacitance C of the capacitor filled with dielectric to the capacitance C_vac of a capacitor in vacuum,

$\epsilon _{\mathrm {r} }\equiv {\frac {C}{C_{\mathrm {vac} }}}={\frac {Q_{\mathrm {true} }}{V}}\left[{\frac {Q_{\mathrm {free} }}{V}}\right]^{-1}={\frac {Q_{\mathrm {true} }}{Q_{\mathrm {free} }}}={\frac {\sigma _{\mathrm {true} }}{\sigma _{\mathrm {free} }}},$

where we used that Q is σ × A, with A the area of the plates.

Tensor character of D

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,

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

(To be continued)