# 磁偶极子与磁介质

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

# 磁偶极子

$\dif \bd{A}=\frac{\mu_0}{4\pi} \frac{I\dif \bd{l}'}{R}\\ （R为电流源到场点的距离）$

$2\dif A \cos \varphi = \frac{\mu_0 I}{4\pi R} a \dif \varphi \cdot 2\cos\varphi$

\begin{align} R&=[(r\cos\theta^2)^2 + (a\sin\varphi)^2+(r\sin\theta-a\cos\varphi)^2]^{1/2}\\ &=r(1+\frac{a^2}{r^2}-\frac{2a}{r}\sin\theta\cos\varphi)^{1/2} \end{align} \begin{align} &\because r \gg a\\ &\therefore \frac{1}{R}\approx\frac{1}{r} (1+\frac{a}{r}\sin\theta\cos\varphi) \end{align}

\begin{align} A_\varphi(r,\theta)&=\frac{\mu_0Ia}{2\pi r}\int_0^\pi (1+\frac{a}{r}\sin\theta\cos\varphi)\cos \varphi \dif \varphi\\ &=\frac{\mu_0Ia}{2\pi r} \cdot \frac{a}{r} \sin\theta \cdot \frac{\pi}{2}\\ &=\frac{\mu_0 IS \sin\theta}{4\pi r^2}\\ \bd{A}(r,\theta)&= \frac{\mu_0 IS \sin\theta}{4\pi r^2} \bd{a}_\varphi \end{align}

$\bd{B}=\nabla\times \bd{A} \approx \frac{\mu_0 SI}{4\pi r^3}(\bd{a}_r 2\cos\theta + \bd{a}_\theta \sin\theta)\\ \bd{E}\approx \frac{p}{4\pi\varepsilon_0 r^3 }(\bd{a}_r 2\cos\theta + \bd{a}_\theta \sin\theta)$

\begin{align} \bd{A}(r,\theta)&= \frac{\mu_0 IS \sin\theta}{4\pi r^2} \bd{a}_\varphi\\ &= \frac{\mu_0}{4\pi} \cdot IS \cdot \bd{a}_z \times \frac{\bd{r}}{r^3}\\ &=\frac{\mu_0}{4\pi} \bd{p}_m \times (\nabla\frac{1}{r}) \end{align}

$p_m=IS$ 称为磁偶极距。

# 磁化

• 抗磁性（水、铜、有机物）：产生的磁场与外加磁场相反（$\mu<\mu_0$）
• 顺磁性（锂、钠、铝、氧气）：产生的磁场与外加磁场相同，磁性较弱（$\mu>\mu_0$）
• 铁磁性（铁、钴、镍、钆）：产生的磁场与外加磁场相同，磁性很强，甚至会大于原有磁场，有磁畴
• 亚铁磁性物质

$\bd{M}=\lim_{\Delta V\rightarrow0} \frac{\sum_{i=1}^{n}\bd{p}_{mi}}{\Delta V}\;\rm{(A/m)}$

\begin{align} \bd{A}(\bd{r})&=\frac{\mu_0}{4\pi}\int_{V'} \frac{\bd{M}\times\bd{R}}{R^3} \dif V'\\ &=\frac{\mu_0}{4\pi}\int_{V'} \bd{M}\times\nabla' \left(\frac{\bd{R}}{R^3}\right) \dif V'\\ &=\frac{\mu_0}{4\pi}\int_{V'} \frac{\nabla'\times \bd{M}}{R}\dif V' - \frac{\mu_0}{4\pi}\int_{V'} \nabla \times \left( \frac{\bd{M}}{R} \right) \dif V'\\ &=\frac{\mu_0}{4\pi}\int_{V'} \frac{\nabla'\times \bd{M}}{R}\dif V' + \frac{\mu_0}{4\pi}\int_{S'} \frac{\bd{M}\times \bd{n}}{R} \dif S'\\ &=\frac{\mu_0}{4\pi}\int_{V'} \frac{\bd{J}_m}{R}\dif V' + \frac{\mu_0}{4\pi}\int_{S'} \frac{\bd{J}_{sm}}{R} \dif V' \end{align}

$\bd{J}_m=\nabla\times \bd{M}（省略'）\\ \bd{J}_{sm}=\bd{M}\times\bd{n}$

• 磁介质均匀且介质中无传导电流时，磁化体电流 $J_m=0$（想象内部电流环相互抵消）
• 磁介质表面总有磁化面电流 $J_{ms}\neq 0$
• 穿过整块介质的任意界面上的磁化电流总量等于0，$I_m+I_{sm}=0$（进去的电流等于出去的电流）

# 媒质中的安培环路定律

$\oint_c \bd{B}\cdot\dif l=\mu_0 (I+I_m)\\$

\begin{align} I_m&=\int_S J_m \cdot \dif \bd{S}\\ &=\int_S (\nabla\times\bd{M})\cdot \dif \bd{S}\\ &=\oint_c \bd{M} \cdot \dif l \end{align}

$\oint_c \bd{B}\cdot \dif \bd{l} =\mu_0 (I+\oint_c \bd{M} \cdot \dif l)\\ \oint_c \left( \frac{\bd{B}}{\mu_0}-\bd{M} \right) \cdot \dif \bd{l}=I$

$\bd{M}=\chi_m \bd{H}\\ 磁化率：\chi_m$

\begin{align} \bd{B}&=\mu_0(\bd{H}+\bd{M})\\ &=\mu_0(1+\chi_m) \bd{H}\\ &=\mu \bd{H} \end{align}\\ 磁导率：\mu

$积分：\oint_C \bd{H}\cdot \dif \bd{l}=I\\ 微分：\nabla\times\bd{H}=\bd{J}$