# 磁通量

## 描述

${\displaystyle \displaystyle \Phi _{B}=\mathbf {B} \cdot \mathbf {a} =Ba\cos \theta }$

${\displaystyle \Phi _{B}=\iint \limits _{S}\mathbf {B} \cdot d\mathbf {S} }$

## 通过闭曲面的磁通量

${\displaystyle \Phi _{B}=\int \!\!\!\int \mathbf {B} \cdot d\mathbf {S} =0,}$

## 通过开曲面的磁通量

${\displaystyle {\mathcal {E}}=\oint _{\partial \Sigma (t)}\left(\mathbf {E} (\mathbf {r} ,\ t)+\mathbf {v\times B} (\mathbf {r} ,\ t)\right)\cdot d{\boldsymbol {\ell }}=-{d\Phi _{B} \over dt},}$

${\displaystyle {\mathcal {E}}}$电动势
${\displaystyle \Phi _{B}}$为通过开曲面的磁通量，这一开曲面的边界为${\displaystyle \partial \Sigma (t)}$
${\displaystyle \partial \Sigma (t)}$为一个随时间变化的闭曲线
${\displaystyle d{\boldsymbol {\ell }}}$是边界${\displaystyle \partial \Sigma (t)}$无穷小矢量元
${\displaystyle \mathbf {v} }$是线段${\displaystyle d{\boldsymbol {\ell }}}$的速度
${\displaystyle \mathbf {E} }$为电场
${\displaystyle \mathbf {B} }$磁场

## 与电通量的比较

${\displaystyle \Phi _{E}=\int \!\!\!\int _{S}\mathbf {E} \cdot d\mathbf {S} ={Q \over \epsilon _{0}},}$

${\displaystyle \mathbf {E} }$为电场
${\displaystyle S}$为任意闭曲面
${\displaystyle Q}$为曲面${\displaystyle S}$包围的电荷
${\displaystyle \epsilon _{0}}$真空电容率