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 (Cm) in SI units) to volume (in meters cubed).
Polarization density is denoted mathematically by ;
For a certain volume element <math>\Delta V</math> in the material, which carries a dipole moment <math>\Delta\mathbf p</math>, we define the polarization density :
<math display="block">\mathbf P = \frac{\Delta\mathbf p}{\Delta V}</math>
In general, the dipole moment <math>\Delta\mathbf p</math> changes from point to point within the dielectric. Hence, the polarization density of a dielectric inside an infinitesimal volume with an infinitesimal dipole moment is:
The net charge appearing as a result of polarization is called bound charge and denoted <math>Q_\text{b}</math>.
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 be isolated inside the dielectric. Due to polarization the positive bound charge <math>\mathrm d q_\text{b}^+</math> will be displaced a distance relative to the negative bound charge <math>\mathrm d q_\text{b}^-</math>, giving rise to a dipole moment <math> \mathrm d \mathbf p = \mathrm d q_\text{b}\mathbf d</math>. Substitution of this expression in yields
<math display="block">\mathbf P = {\mathrm d q_\text{b} \over \mathrm d V}\mathbf d </math>
Since the charge <math>\mathrm d q_\text{b}</math> bounded in the volume is equal to <math>\rho_\text{b} \mathrm d V</math> the equation for becomes: The field lines of the E-field are not shown: These point in the same directions, but many field lines start and end on the surface of the sphere, where there is bound charge. As a result, the density of E-field lines is lower inside the sphere than outside, which corresponds to the fact that the E-field is weaker inside the sphere than outside.]]
In a homogeneous, linear, non-dispersive and isotropic dielectric medium, the polarization is aligned with and proportional to the electric field :
<math display="block">\mathbf{P} = \chi\varepsilon_0 \mathbf E,</math>
where is the electric constant, and is the electric susceptibility of the medium. Note that in this case simplifies to a scalar, although more generally it is a tensor. This is a particular case due to the of the dielectric.
Taking into account this relation between and , equation () becomes:
<math display="block">\sigma_\text{b} = \mathbf{\hat{n_\text{out} \cdot \mathbf{P}</math>
where <math>\mathbf{\hat{n_\text{out}</math> is the normal vector to the surface pointing outwards. (see charge density for the rigorous proof)
Anisotropic dielectrics
The class of dielectrics where the polarization density and the electric field are not in the same direction are known as materials.
In such materials, the -th component of the polarization is related to the -th component of the electric field according to:
<math display="block">\mathbf{D} = \varepsilon_0\mathbf{E} + \mathbf{P}.</math>
This is known as the constitutive equation for electric fields. Here is the electric permittivity of empty space. In this equation, is the (negative of the) field induced in the material when the "fixed" charges, the dipoles, shift in response to the total underlying field , whereas is the field due to the remaining charges, known as "free" charges.
In general, varies as a function of depending on the medium, as described later in the article. In many problems, it is more convenient to work with and the free charges than with and the total charge. In other words, two people, Alice and Bob, looking at the same solid, may calculate different values of , and neither of them will be wrong. For example, if Alice chooses a unit cell with positive ions at the top and Bob chooses the unit cell with negative ions at the top, their computed vectors will have opposite directions. Alice and Bob will agree on the microscopic electric field in the solid, but disagree on the value of the displacement field <math>\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}</math>.
Even though the value of is not uniquely defined in a bulk solid, in uniquely defined.