Polarization density ($\mathbf{P}$)

Electric polarization is electric dipole moment per unit volume. It produces bound volume and surface chargeAcross an interface, the net bound sheet charge is $({\mathbf{P}}_1-{\mathbf{P}}_2)\cdot \mathbf{n}$, where $\mathbf{n}$ points from medium one to medium two.

A tensor derivative of polarization with respect to a sublattice displacement at specified macroscopic electrical conditions. It includes electronic rearrangement and need not equal a formal ionic charge.

A phase without spontaneous switchable electric polarization, although an applied electric field can induce electric polarization. A material can be paraelectric above a transition to a ferroelectric phase.

electric polarization present in a polar material without an externally applied electric field.

Change of spontaneous electric polarization with temperature.

The temperature derivative of a component of spontaneous electric polarization under stated mechanical/electrical constraints. An electrode of area $A$ has ideal short-circuit response $I=pAdT/dt$. Mechanical constraints distinguish primary and secondary contributions.

A device that delivers electrical charge during changes of temperature of a polar material.

The piezoelectric contribution to temperature-dependent electric polarization caused by thermal strain.

The temperature dependence of electric polarization at fixed strain.

The electric field produced by the bound charge of a polarized body is its depolarization field. For a uniformly polarized ellipsoid in vacuum, the internal depolarization field is uniform and opposes the relevant component of the electric polarization: $E_{\mathrm{dep},i}=-N_i{P}_i/{ϵ}_0$. Its dimensionless geometric factors obey $N_x+N_y+N_z=1$.

For a homogeneous linear isotropic dielectric in vacuum, $P_i={ϵ}_0({ϵ}_r-1)E_{\mathrm{in},i}$. Combining this with the depolarization field and the imposed uniform electric field gives $E_{\mathrm{in},i}=E_{0,i}/[1+N_i({ϵ}_r-1)]$. A sphere has $N_i=1/3$. An infinite slab has $N=1$ normal to its faces and $N=0$ in its plane. A very long circular rod has $N=0$ along its axis and $N=1/2$ transverse to it. Slab and rod statements are limiting geometries, with end effects neglected; a finite cylinder does not generally have an exactly uniform internal electric field.

A sphere with constant electric polarization $\mathbf{P}=P\hat{\mathbf{z}}$ has zero volume bound charge and surface bound charge density ${\sigma }_b=P\cos \theta$. Integrating $\mathbf{r}{\sigma }_{b}dA$ gives the displayed electric dipole moment. The surface distribution is purely degree one in spherical harmonics, so its external static electric potential is exactly dipolar. Its interior field is finite and is not that of a point singularity.

The electric displacement is $\mathbf{D}={ϵ}_0\mathbf{E}+\mathbf{P}$ and obeys $\nabla \cdot \mathbf{D}={\rho }_{\mathrm{free}}$. In a linear isotropic dielectric, $\mathbf{D}=ϵ\mathbf{E}$.