Polarization density

In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is said to be polarized. The electric dipole moment induced per unit volume of the dielectric material is called the electric polarization of the dielectric.It follows that the negative bound charge d q b − = ρ b −   d V 1 = ρ b − d 1   d A {displaystyle mathrm {d} q_{b}^{-}= ho _{b}^{-} mathrm {d} V_{1}= ho _{b}^{-}d_{1} mathrm {d} A} moved from the outer part of the surface dA inwards, while the positive bound charge d q b + = ρ b   d V 2 = ρ b d 2   d A {displaystyle mathrm {d} q_{b}^{+}= ho _{b} mathrm {d} V_{2}= ho _{b}d_{2} mathrm {d} A} moved from the inner part of the surface outwards.for the volume V containing the bound charge Q b {displaystyle Q_{b}} . And since Q b {displaystyle Q_{b}} is the integral of the bound charge density ρ b {displaystyle ho _{b}} taken over the entire volume V enclosed by S, the above equation yields In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is said to be polarized. The electric dipole moment induced per unit volume of the dielectric material is called the electric polarization of the dielectric. 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. The SI unit of measure is coulombs per square meter, and polarization density is represented by a vector P. An external electric field that is applied to a dielectric material, causes a displacement of bound charged elements. These are elements which are bound to molecules and are not free to move around the material. 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: 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_{b}} . This definition of polarization as a 'dipole moment per unit volume' is widely adopted,though in some cases it can lead to ambiguities and paradoxes. Let a volume dV be isolated inside the dielectric. Due to polarization the positive bound charge d q b + {displaystyle mathrm {d} q_{b}^{+}} will be displaced a distance d {displaystyle mathbf {d} } relative to the negative bound charge d q b − {displaystyle mathrm {d} q_{b}^{-}} , giving rise to a dipole moment d p = d q b d {displaystyle mathrm {d} mathbf {p} =mathrm {d} q_{b}mathbf {d} } . Substitution of this expression in (1) yields Since the charge d q b {displaystyle mathrm {d} q_{b}} bounded in the volume dV is equal to ρ b d V {displaystyle ho _{b}mathrm {d} V} the equation for P becomes: