Electric Displacement

$\vec{D}\equiv {\epsilon }_0\vec{E}+\vec{P}={\epsilon }_0(1+{\chi }_e)\vec{E}=\epsilon \vec{E}$

• Electric displacement for a linear media
• $\epsilon$ is the permittivity of the substance
• $\vec{P}$ is the polarization of an atom
• ${\chi }_e$ is the electric susceptibility of the substance

$\nabla \cdot \vec{D}={\rho }_f$

• Gauss’s law for electric displacement
• ${\rho }_f$ is the free charge density

$\oint D\cdot d\vec{a}=Q_{f_{\text{enc}}}$

• $Q_{f_{\text{enc}}}$ is the total enclosed free charge in the volume