PX157 - B12b - relative permittivity

{\vec{D}}_{r} = ϵ_{0} ϵ_{r} \vec{E}

where, $ϵ_{r} =$ relative permittivity

\begin{aligned} \vec{P} & = \vec{D} - ϵ_{0} \vec{E} \\ = ϵ_{0} ϵ_{r} \vec{E} - ϵ_{0} \vec{E} \\ = (ϵ_{r} - 1) ϵ_{0} \vec{E} \\ = χ_{e} ϵ_{0} \vec{E} \end{aligned}

where, $χ_{e} = (ϵ_{r} - 1)$

\subset \supset \iint_{S} \vec{P} \cdot d \vec{S} = - Q_{e n c l, b .}

\begin{aligned} ϵ_{0} \subset \supset \iint_{S} \vec{E} \cdot d \vec{S} & = Q_{e n c l} \\ = Q_{e n c l, f .} + Q_{e n c l, b .} \\ = Q_{e n c l, f .} - \subset \supset \iint_{S} \vec{P} \cdot d \vec{S} \end{aligned}

\subset \supset \iint_{S} (ϵ_{0} \vec{E} + \vec{P}) \cdot d \vec{S} =\subset \supset \iint_{S} \vec{D} \cdot d \vec{S} = Q_{e n c l, f .}

ϵ_{r} = \frac{C}{C_{0}}

where, $C$ and $C_{0}$ are the capacitances with and without the material

C_{0} = \frac{ϵ_{0} A}{d}, C = \frac{ϵ A}{d} = ϵ_{r} C_{0}