# 电磁波本文重定向自 电磁波

## 概念

### 波动理论

${\displaystyle v=\nu \lambda \,\!}$

${\displaystyle u={\frac {1}{2\mu _{0}}}B^{2}+{\frac {\epsilon _{0}}{2}}E^{2}\,\!}$

### 传播速度

${\displaystyle n=c/v\,\!}$

## 从电磁理论推导

${\displaystyle \nabla \cdot \mathbf {E} =0\,\!}$（1）
${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}\,\!}$（2）
${\displaystyle \nabla \cdot \mathbf {B} =0\,\!}$（3）
${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\,\!}$（4）

${\displaystyle \nabla \times \left(\nabla \times \mathbf {E} \right)=\nabla \times \left(-{\frac {\partial \mathbf {B} }{\partial t}}\right)\,\!}$（5）

${\displaystyle \nabla \times \left(\nabla \times \mathbf {E} \right)=\nabla \left(\nabla \cdot \mathbf {E} \right)-\nabla ^{2}\mathbf {E} =-\nabla ^{2}\mathbf {E} \,\!}$（6）

${\displaystyle \nabla \times \left(-{\frac {\partial \mathbf {B} }{\partial t}}\right)=-{\frac {\partial }{\partial t}}\left(\nabla \times \mathbf {B} \right)=-\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}\,\!}$（7）

 ${\displaystyle \nabla ^{2}\mathbf {E} =\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}\,\!}$。

 ${\displaystyle \nabla ^{2}\mathbf {B} =\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}\,\!}$。

${\displaystyle \Box \mathbf {E} =0\,\!}$
${\displaystyle \Box \mathbf {B} =0\,\!}$

${\displaystyle \mathbf {E} =\mathbf {E} _{0}f\left(\mathbf {k} \cdot \mathbf {r} -\omega t\right)\,\!}$

${\displaystyle \nabla ^{2}f\left(\mathbf {k} \cdot \mathbf {r} -\omega t\right)={\frac {1}{{c_{0}}^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}f\left(\mathbf {k} \cdot \mathbf {r} -\omega t\right)\,\!}$

${\displaystyle \nabla \cdot \mathbf {E} =\mathbf {k} \cdot \mathbf {E} _{0}f'\left(\mathbf {k} \cdot \mathbf {r} -\omega t\right)=0\,\!}$

${\displaystyle \mathbf {E} \cdot \mathbf {k} =0\,\!}$

${\displaystyle \nabla \times \mathbf {E} ={\hat {\mathbf {k} }}\times \mathbf {E} _{0}f'\left(\mathbf {k} \cdot \mathbf {r} -\omega t\right)=-{\frac {\partial \mathbf {B} }{\partial t}}\,\!}$

${\displaystyle \mathbf {B} ={\frac {1}{\omega }}\mathbf {k} \times \mathbf {E} \,\!}$