Polarization density

In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the volumetric density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its atoms or molecules gain electric dipole moment and the dielectric is said to be polarized. Electric polarization of a given dielectric material sample is defined as the quotient of electric dipole moment (a vector quantity, expressed as coulombs-meters (C⋅m) in SI units) to volume (in meters cubed). Polarization density is denoted mathematically by P; in SI units, it is expressed in coulombs per square meter (C/m2). Polarization density also describes how a material responds to an applied electric field as well as the way the material changes the electric field, and can be used to calculate the forces that result from those interactions. It can be compared to magnetization, which is the measure of the corresponding response of a material to a magnetic field in magnetism. Similar to ferromagnets, which have a non-zero permanent magnetization even if no external magnetic field is applied, ferroelectric materials have a non-zero polarization in the absence of external electric field.

Definition An external electric field that is applied to a dielectric material, causes a displacement of bound charged elements. A bound charge is a charge that is associated with an atom or molecule within a material. It is called "bound" because it is not free to move within the material like free charges. Positive charged elements are displaced in the direction of the field, and negative charged elements are displaced opposite to the direction of the field. The molecules may remain neutral in charge, yet an electric dipole moment forms. For a certain volume element Δ V {\displaystyle \Delta V} in the material, which carries a dipole moment Δ p {\displaystyle \Delta \mathbf {p} } , we define the polarization density P:

P = Δ p Δ V {\displaystyle \mathbf {P} ={\frac {\Delta \mathbf {p} }{\Delta V}}}

In general, the dipole moment Δ p {\displaystyle \Delta \mathbf {p} } changes from point to point within the dielectric. Hence, the polarization density P of a dielectric inside an infinitesimal volume dV with an infinitesimal dipole moment dp is:

The net charge appearing as a result of polarization is called bound charge and denoted Q b {\displaystyle Q_{\text{b}}} . This definition of polarization density as a "dipole moment per unit volume" is widely adopted, though in some cases it can lead to ambiguities and paradoxes.

Other expressions Let a volume dV be isolated inside the dielectric. Due to polarization the positive bound charge d q b + {\displaystyle \mathrm {d} q_{\text{b}}^{+}} will be displaced a distance d relative to the negative bound charge d q b − {\displaystyle \mathrm {d} q_{\text{b}}^{-}} , giving rise to a dipole moment d p = d q b d {\displaystyle \mathrm {d} \mathbf {p} =\mathrm {d} q_{\text{b}}\mathbf {d} } . Substitution of this expression in (1) yields

P = d q b d V d {\displaystyle \mathbf {P} ={\mathrm {d} q_{\text{b}} \over \mathrm {d} V}\mathbf {d} }

Since the charge d q b {\displaystyle \mathrm {d} q_{\text{b}}} bounded in the volume dV is equal to ρ b d V {\displaystyle \rho _{\text{b}}\mathrm {d} V} the equation for P becomes:

where ρ b {\displaystyle \rho _{\text{b}}} is the density of the bound charge in the volume under consideration. It is clear from the definition above that the dipoles are overall neutral and thus ρ b {\displaystyle \rho _{\text{b}}} is balanced by an equal density of opposite charges within the volume. Charges that are not balanced are part of the free charge discussed below.

Gauss's law for the field of P For a given volume V enclosed by a surface S, the bound charge Q b {\displaystyle Q_{\text{b}}} inside it is equal to the flux of P through S taken with the negative sign, or

Differential form By the divergence theorem, Gauss's law for the field P can be stated in differential form as:

− ρ b = ∇ ⋅ P , {\displaystyle -\rho _{\text{b}}=\nabla \cdot \mathbf {P} ,}

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