Conductors and dielectrics in an electric field

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• Electric field strength near the surface of a conductor in vacuum: $$E_n = \frac{\sigma}{{\varepsilon}_0}$$
• Flux of polarization $\vec{P}$ across a closed surface: $$\oint \vec{P}.d\vec{S}=-q^\prime,$$ where $q^\prime$ is the algebraic sum of bound charges enclosed by this surface.
• Vector $\vec{D}$ and Gauss's theorem for it: $$\vec{D}=\varepsilon \vec{E}+\vec{P}, \oint \vec{D}.d\vec{S}=q,$$ where $q$ is the algebraic sum of extraneous charges inside a closed surface.
• Relations at the boundary between two dielectrics:$$P_{2n}-P_{1n}=-\sigma^\prime, D_{2n}-D_{1n}=\sigma, E_{2\tau}=E_{1\tau},$$ where $\sigma^\prime$ and $\sigma$ are the surface densities of bound and extraneous charges, and the unit vector $\vec{n}$ of the normal is directed from medium 1 to medium 2.
• In isotropic dielectrics:$$\vec{P}=\chi\varepsilon_0 \vec{E}, \vec{D}=\varepsilon \varepsilon_0 \vec{E}, \varepsilon =1+\chi.$$
• In the case of an isotropic uniform dielectric filling up all the space between the equipotential surfaces: $$\vec{E}=\frac{\vec{E}_0}{\varepsilon}.$$