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The current associated with electric polarization and magnetization in matter. If the magnetic response is neglected, it is ${\mathbf{J}}_b={\partial }_t\mathbf{P}$. It is distinguished from a free conduction current in the Maxwell equations in matter.

Ignoring magnetization, electric polarization charge transport gives bound current ${\mathbf{J}}_{\mathrm{b}}={\partial }_t\mathbf{P}$. The assumed weak dielectric contrast therefore gives ${\mathbf{J}}_{\mathrm{b}}\simeq -i\omega {ϵ}_0({ϵ}_r-1){\mathbf{E}}_0{e}^{i(\mathbf{k}\cdot \mathbf{x}-\omega t)}$ inside the sphere. This is the first Born approximation: the incident field supplies the electric polarization to leading order in ${ϵ}_r-1$.

For $\mathbf{x}=r\mathbf{n}$, the far-distance expansions are $|\mathbf{x}-{\mathbf{x}}^{\prime }|=r-\mathbf{n}\cdot {\mathbf{x}}^{\prime }+O(a^2/r)$ and $|\mathbf{x}-{\mathbf{x}}^{\prime }{|}^{-1}\simeq r^{-1}$. Substitute the retarded time into the phase in the Lorenz gauge vector potential. The phase becomes $kr-\omega t+(\mathbf{k}-k\mathbf{n})\cdot {\mathbf{x}}^{\prime }$, so, using ${\mu }_0{ϵ}_0=c^{-2}$,The assumptions $r\gg a$ and $k{a}^2/r\ll 2\pi$ control amplitude and phase errors respectively. In the radiation zone the supplied electric and magnetic fields satisfy ${\mathbf{B}}_{\mathrm{scatt}}=c^{-1}\mathbf{n}\times {\mathbf{E}}_{\mathrm{scatt}}$. Complex phasors therefore give the time-averaged radial Poynting flux $|{\mathbf{E}}_{\mathrm{scatt}}{|}^2/(2{\mu }_{0}c)$. The incident flux is $|{\mathbf{E}}_0{|}^2/(2{\mu }_{0}c)$, soOrient spherical coordinates along $\mathbf{q}$ to evaluate the form factor:ConsequentlyAt $q=0$ the bracket is interpreted by its limit $a^3/3$, not a singularity. The transverse factor accounts for polarization of an electromagnetic wave; for complex polarization of an electromagnetic wave its squared norm means the Hermitian norm.

The electric polarization $P$ is electric dipole moment per unit volume. A polarized body acts as the bound volume and surface charge densityIndeed, summing the scalar potential of its dipoles and integrating by parts givesThis is exactly the potential of ${\rho }_{\mathrm{b}}$ and ${\sigma }_{\mathrm{b}}$.

For the central free charge, Gauss's law for the electric displacement field givesThus on the sphere,

A dielectric is an electrically insulating material whose bound positive and negative charges can be displaced slightly by an electric field, producing electric polarization without sustained conduction current. Its macroscopic electric response is described by its permittivity.

The electric polarization $\mathbf{P}$ is electric dipole moment per unit volume. It producesThe electric displacement field $\mathbf{D}={ϵ}_0\mathbf{E}+\mathbf{P}$ is useful becauseso it separates free charge from the bound charge represented by $\mathbf{P}$. In a linear isotropic dielectric, $\mathbf{D}=ϵ\mathbf{E}$.

Spherical symmetry and Gauss's law for $\mathbf{D}$ give, on both sides of the interface,ThusWith $\mathbf{n}={\mathbf{e}}_r$ pointing from medium one to medium two, the net bound sheet charge at $r=R$ is the polarization jumpIf one assigns surface charge separately to the inner body and outer medium, their contributions are ${\mathbf{P}}_1\cdot \mathbf{n}$ and $-{\mathbf{P}}_2\cdot \mathbf{n}$; their sum is the displayed physical interface charge.