# 电极化

## 定义

${\displaystyle \mathbf {P} ={\langle \mathbf {p} \rangle \over V}}$

${\displaystyle \mathbf {P} ={d\mathbf {p} \over dV}}$

${\displaystyle \mathbf {p} =\iiint \mathbf {P} \,dV}$

${\displaystyle \mathbf {M} ={d\mathbf {m} \over dV},\quad \mathbf {m} =\iiint \mathbf {M} \,dV}$

## 束缚电荷

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

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

${\displaystyle \sigma _{bound}=\mathbf {P} \cdot {\hat {\mathbf {n} }}_{\mathrm {out} }}$

${\displaystyle \mathbf {J} _{p}={\frac {\partial \mathbf {P} }{\partial t}}}$

${\displaystyle \mathbf {J} _{total}=\mathbf {J} _{free}+\mathbf {J} _{bound}+\mathbf {J} _{p}=\mathbf {J} _{free}+\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}}$

“自由电流”是由外处进来的电流，不是由电介质的束缚电荷所构成的电流。“束缚电流”是由电介质束缚电荷产生的磁偶极子所构成的电流，一个原子尺寸的现象。

## 电极化强度与电场的关系

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

### 各向同性电介质

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

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

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

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

### 各向异性电介质

${\displaystyle P_{i}=\sum _{j}\epsilon _{0}\chi _{ij}E_{j}}$

${\displaystyle P_{i}/\epsilon _{0}=\sum _{j}\chi _{ij}^{(1)}E_{j}+\sum _{jk}\chi _{ijk}^{(2)}E_{j}E_{k}+\sum _{jk\ell }\chi _{ijk\ell }^{(3)}E_{j}E_{k}E_{\ell }+\cdots }$

## 参考文献

1. ^ McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3
2. ^ Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9
3. ^ Griffiths, David J., Introduction to Electrodynamics (3rd ed.), Prentice Hall: pp. 175, 179–184, 1998, ISBN 0-13-805326-X
4. ^ Jackson, John David, Classical Electrodynamic 3rd., USA: John Wiley & Sons, Inc.: pp. 151–154, 1999, ISBN 978-0-471-30932-1