# 洛伦兹-亥维赛单位制

## 有理化

{\displaystyle {\begin{aligned}\nabla \cdot \mathbf {D} &=\rho /\beta ,\\\quad \nabla \cdot \mathbf {B} &=0,\\\quad \kappa \nabla \times \mathbf {E} &=-{\frac {\partial \mathbf {B} }{\partial t}},\\\quad \kappa \nabla \times \mathbf {H} &={\frac {\partial \mathbf {D} }{\partial t}}+\mathbf {J} /\beta ,\end{aligned}}}

• 高斯单位制β = 1/4πκ = c
• 洛伦兹-亥维赛单位制β = 1κ = c
• 国际单位制β = 1κ = 1

## 有源麦克斯韦方程组

${\displaystyle \nabla \cdot \mathbf {E} =\rho \,}$
${\displaystyle \nabla \cdot \mathbf {B} =0\,}$
${\displaystyle \nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}\,}$
${\displaystyle \nabla \times \mathbf {B} ={\frac {1}{c}}{\frac {\partial \mathbf {E} }{\partial t}}+{\frac {1}{c}}\mathbf {J} \,}$

${\displaystyle \mathbf {F} _{q}=q\left(\mathbf {E} +{\frac {\mathbf {v} _{q}}{c}}\times \mathbf {B} \right)\,}$

${\displaystyle q_{\mathrm {LH} }\ =\ {\sqrt {4\pi }}\ q_{\mathrm {G} }}$
${\displaystyle \mathbf {E} _{\mathrm {LH} }\ =\ {\mathbf {E} _{\mathrm {G} } \over {\sqrt {4\pi }}}}$
${\displaystyle \mathbf {B} _{\mathrm {LH} }\ =\ {\mathbf {B} _{\mathrm {G} } \over {\sqrt {4\pi }}}}$.

## 几种单位制下电磁学方程的形式比较

### 麦克斯韦方程组

（宏观）
${\displaystyle \nabla \cdot \mathbf {D} =\rho _{\text{f}}}$ ${\displaystyle \nabla \cdot \mathbf {D} =4\pi \rho _{\text{f}}}$ ${\displaystyle \nabla \cdot \mathbf {D} =\rho _{\text{f}}}$

（微观）
${\displaystyle \nabla \cdot \mathbf {E} =\rho }$ ${\displaystyle \nabla \cdot \mathbf {E} =4\pi \rho }$ ${\displaystyle \nabla \cdot \mathbf {E} =\rho /\epsilon _{0}}$

（宏观）
${\displaystyle \nabla \times \mathbf {H} ={\frac {1}{c}}\mathbf {J} _{\text{f}}+{\frac {1}{c}}{\frac {\partial \mathbf {D} }{\partial t}}}$ ${\displaystyle \nabla \times \mathbf {H} ={\frac {4\pi }{c}}\mathbf {J} _{\text{f}}+{\frac {1}{c}}{\frac {\partial \mathbf {D} }{\partial t}}}$ ${\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{\text{f}}+{\frac {\partial \mathbf {D} }{\partial t}}}$

（微观）
${\displaystyle \nabla \times \mathbf {B} ={\frac {1}{c}}\mathbf {J} +{\frac {1}{c}}{\frac {\partial \mathbf {E} }{\partial t}}}$ ${\displaystyle \nabla \times \mathbf {B} ={\frac {4\pi }{c}}\mathbf {J} +{\frac {1}{c}}{\frac {\partial \mathbf {E} }{\partial t}}}$ ${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}}$

### 电介质与磁介质

${\displaystyle \mathbf {D} =\mathbf {E} +\mathbf {P} }$ ${\displaystyle \mathbf {D} =\mathbf {E} +4\pi \mathbf {P} }$ ${\displaystyle \mathbf {D} =\epsilon _{0}\mathbf {E} +\mathbf {P} }$
${\displaystyle \mathbf {P} =\chi _{\text{e}}\mathbf {E} }$ ${\displaystyle \mathbf {P} =\chi _{\text{e}}\mathbf {E} }$ ${\displaystyle \mathbf {P} =\chi _{\text{e}}\epsilon _{0}\mathbf {E} }$
${\displaystyle \mathbf {D} =\epsilon \mathbf {E} }$ ${\displaystyle \mathbf {D} =\epsilon \mathbf {E} }$ ${\displaystyle \mathbf {D} =\epsilon \mathbf {E} }$
${\displaystyle \epsilon =1+\chi _{\text{e}}}$ ${\displaystyle \epsilon =1+4\pi \chi _{\text{e}}}$ ${\displaystyle \epsilon /\epsilon _{0}=1+\chi _{\text{e}}}$

${\displaystyle \chi _{\text{e}}^{\text{SI}}=\chi _{\text{e}}^{\text{LH}}=4\pi \chi _{\text{e}}^{\text{G}}}$

${\displaystyle \mathbf {B} =\mathbf {H} +\mathbf {M} }$ ${\displaystyle \mathbf {B} =\mathbf {H} +4\pi \mathbf {M} }$ ${\displaystyle \mathbf {B} =\mu _{0}(\mathbf {H} +\mathbf {M} )}$
${\displaystyle \mathbf {M} =\chi _{\text{m}}\mathbf {H} }$ ${\displaystyle \mathbf {M} =\chi _{\text{m}}\mathbf {H} }$ ${\displaystyle \mathbf {M} =\chi _{\text{m}}\mathbf {H} }$
${\displaystyle \mathbf {B} =\mu \mathbf {H} }$ ${\displaystyle \mathbf {B} =\mu \mathbf {H} }$ ${\displaystyle \mathbf {B} =\mu \mathbf {H} }$
${\displaystyle \mu =1+\chi _{\text{m}}}$ ${\displaystyle \mu =1+4\pi \chi _{\text{m}}}$ ${\displaystyle \mu /\mu _{0}=1+\chi _{\text{m}}}$

• BH分别为磁感应强度磁场强度
• M磁化强度
• ${\displaystyle \mu }$磁导率
• ${\displaystyle \mu _{0}}$为真空磁导率；
• ${\displaystyle \chi _{\text{m}}}$磁化率

${\displaystyle \chi _{\text{m}}^{\text{SI}}=\chi _{\text{m}}^{\text{LH}}=4\pi \chi _{\text{m}}^{\text{G}}}$

### 矢势与标势

${\displaystyle \mathbf {E} =-\nabla \phi }$ ${\displaystyle \mathbf {E} =-\nabla \phi }$ ${\displaystyle \mathbf {E} =-\nabla \phi }$

（通常形式）
${\displaystyle \mathbf {E} =-\nabla \phi -{\frac {1}{c}}{\frac {\partial \mathbf {A} }{\partial t}}}$ ${\displaystyle \mathbf {E} =-\nabla \phi -{\frac {1}{c}}{\frac {\partial \mathbf {A} }{\partial t}}}$ ${\displaystyle \mathbf {E} =-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}}$

## 形式变换规律

${\displaystyle \left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)}$ ${\displaystyle {\sqrt {4\pi }}\left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)}$ ${\displaystyle {\frac {1}{\sqrt {\epsilon _{0}}}}\left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)}$

${\displaystyle \left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)}$ ${\displaystyle {\frac {1}{\sqrt {4\pi }}}\left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)}$ ${\displaystyle {\frac {1}{\sqrt {\mu _{0}}}}\left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)}$

${\displaystyle \left(\epsilon ,\mu \right)}$ ${\displaystyle \left(\epsilon ,\mu \right)}$ ${\displaystyle \left({\frac {\epsilon }{\epsilon _{0}}},{\frac {\mu }{\mu _{0}}}\right)}$

${\displaystyle \left(\chi _{\text{e}},\chi _{\text{m}}\right)}$ ${\displaystyle 4\pi \left(\chi _{\text{e}},\chi _{\text{m}}\right)}$ ${\displaystyle \left(\chi _{\text{e}},\chi _{\text{m}}\right)}$

## 自然单位制

${\displaystyle \nabla \cdot \mathbf {E} =\rho \,}$
${\displaystyle \nabla \cdot \mathbf {B} =0\,}$
${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}\,}$
${\displaystyle \nabla \times \mathbf {B} ={\frac {\partial \mathbf {E} }{\partial t}}+\mathbf {J} \,}$
${\displaystyle \mathbf {F} _{q}=q(\mathbf {E} +\mathbf {v} _{q}\times \mathbf {B} )\,}$

## 注释

1. ^ 这个量会在国际单位制中用到。但在高斯单位制与洛伦兹-亥维赛单位制中，由于其数值为1，因而常常被忽略。
2. ^ 与真空电容率类似，真空磁导率会在国际单位制中用到，但在高斯单位制与洛伦兹-亥维赛单位制中则常常被忽略。