# 磁感应强度

\begin{align*} \newcommand{\dif}{\mathop{}\!\mathrm{d}}\\ \newcommand{\p}{\partial}\\ \newcommand{\bd}{\boldsymbol} \end{align*}

# 磁感应强度

## 安培力定律

$\dif \bd{F}_{21}=\frac{\mu_0}{4\pi}\frac{I_2\dif \bd{l}_2 \ \times I_1\dif \bd{I}_1}{R_{21}^3}\times \bd{R}_{21}$

$\bd{F}_{21}=\frac{\mu_0}{4\pi}\oint_{c_2}{\oint_{c_1}} \frac{I_2\dif \bd{l}_2 \times I_1 \dif \bd{l}_1 \times \bd{a}_R}{R^2}$

## 比奥-萨伐尔定律

$\bd{F}_{21}=\oint_{c_2} I_2\dif \bd{l}_2 \times \left( \frac{\mu_0}{4\pi}{\oint_{c_1}} \frac{ I_1 \dif \bd{l}_1 \times \bd{a}_R}{R^2} \right)$

$\bd{B}(\bd{r})=\frac{\mu_0}{4\pi}{\oint_{c’}} \frac{ I \dif \bd{l}' \times \bd{a}_R}{R^2}$

$体电流 \bd{B}(\bd{r})=\frac{\mu_0}{4\pi}{\int_{V'}} \frac{ \bd{J}(\bd{r}') \times \bd{a}_R}{R^2}\dif V'\\ 面电流 \bd{B}(\bd{r})=\frac{\mu_0}{4\pi}{\oint_{S'}} \frac{ \bd{J}_S(\bd{r}') \times \bd{a}_R}{R^2}\dif S'\\ 线电流直接用电流表示即可$

$$\bd{B}=\frac{\mu_0}{4\pi}\int_{c'} \frac{I\dif \bd{l}'\times \bd{a}_R}{R^2}\\$$ 根据几何关系，我们可以写出：
$$\begin{cases} \dif \bd{l}'=\dif \bd{z}'\\ z'=z-\frac{r}{\tan\theta}\\ \dif z'= r\csc^2 \theta\dif \theta\\ \dif \bd{z}'=\bd{a}_z r\csc^2 \theta\dif \theta\\ \bd{a}_R=\bd{a}_r \sin\theta+\bd{a}_z\cos\theta\\ \dif z'\times \bd{a}_R=\bd{a}_\varphi r \csc^2\theta\sin\theta \dif \theta\\ R=r\csc\theta \end{cases}$$ 将上面最后两项代入比奥-萨伐尔定律：
\begin{align} \bd{B}&=\frac{\mu_0}{4\pi}\int_{c'} \frac{I\dif \bd{l}'\times \bd{a}_R}{R^2}\\ &=\frac{\mu_0 I}{4\pi} \int_{\theta_1}^{\theta_2} \frac{r\csc^2\theta\sin\theta\dif \theta}{r^2\csc^2 \theta}\\ &=\frac{\mu_0 I}{4\pi r} \int_{\theta_1}^{\theta_2} \sin\theta\dif\theta\\ &=\frac{\mu_0 I}{4\pi r} (\cos\theta_1 - \cos \theta_2) \end{align} 由上可以推出无限长直导线的磁场强度为 $\bd{B}=\frac{\mu_0 I}{2\pi r} \bd{a}_\varphi$

# 洛伦兹力

\begin{align*} \dif \bd{F} &= I\dif \bd{I}\times\bd{B}=\bd{J}\dif V\times\bd{B}=\rho \bd{v} \dif V \times \bd{B}\\ &=\dif q \bd{v}\times \bd{B} \end{align*}

$洛伦兹力\;\bd{F}=q\bd{v}\times\bd{B}$