# 電位移

${\displaystyle \mathbf {D} \ {\stackrel {\mathrm {def} }{=}}\ \varepsilon _{0}\mathbf {E} +\mathbf {P} }$

## 概述

${\displaystyle \nabla \cdot \mathbf {E} =\rho _{total}/\varepsilon _{0}}$

${\displaystyle \nabla \cdot \mathbf {P} =-\rho _{bound}}$

${\displaystyle \rho _{total}=\rho _{free}+\rho _{bound}}$

${\displaystyle \nabla \cdot \mathbf {D} =\rho _{free}}$

${\displaystyle \nabla \times \mathbf {D} =\varepsilon _{0}(\nabla \times \mathbf {E} )+(\nabla \times \mathbf {P} )}$

${\displaystyle \nabla \times \mathbf {D} =\nabla \times \mathbf {P} }$

## 線性電介質

「線性電介質」，對於外電場的施加，會產生線性響應。例如，鐵電材料是非線性電介質。假設線性電介質具有各向同性，則其電場與電極化強度的關係式為[2]

${\displaystyle \mathbf {P} =\chi _{e}\varepsilon _{0}\mathbf {E} }$

${\displaystyle \mathbf {D} =(1+\chi _{e})\varepsilon _{0}\mathbf {E} =\varepsilon \mathbf {E} }$

${\displaystyle \nabla \cdot (\varepsilon \mathbf {E} )=\rho _{free}}$

${\displaystyle \nabla \cdot \mathbf {E} =\rho _{free}/\varepsilon }$

${\displaystyle \varepsilon _{r}\ {\stackrel {\mathrm {def} }{=}}\ \varepsilon /\varepsilon _{0}}$

${\displaystyle \varepsilon _{r}=1+\chi _{e}}$

## 應用範例

${\displaystyle \oint _{\mathbb {S} }\mathbf {D} _{+}\cdot \mathrm {d} \mathbf {a} =Q}$

${\displaystyle 2D_{+}A=Q}$ ;

${\displaystyle D_{+}=Q/2A}$

${\displaystyle D_{-}=Q/2A}$

${\displaystyle E=D/\varepsilon =Q/\varepsilon A}$

${\displaystyle V=Ed=Qd/\varepsilon A}$

${\displaystyle C=Q/V=\varepsilon A/d}$

## 參考文獻

1. ^ Griffiths, David J., Introduction to Electrodynamics (3rd ed.), Prentice Hall: pp. 175, 179–184, 1998, ISBN 0-13-805326-X
2. ^ Jackson, John David, Classical Electrodynamic 3rd., USA: John Wiley & Sons, Inc.: pp. 151–154, 1999, ISBN 978-0-471-30932-1