3 恒定磁场

3.1 恒定磁场的基本定律

3.1.1 回路的安培力

• 即 $C_1$$C_1$ 对 $C_2$$C_2$ 的作用力，其中 ${\mathit{a}}_R={\mathit{r}}_2-{\mathit{r}}_1$$\bm a _R = \bm r_2 - \bm r_1$.

• 真空中的磁导率 ${\mu }_0=4\pi \times {10}^{-7}\mathrm{H}/\mathrm{m}$$\mu_0 = 4\pi \cp10^{-7}\ \mathrm{H / m}$.

3.1.2 比奥萨瓦定律

• 磁感应强度 (磁通量密度) $\mathit{B}$$\bm B$

  • 单位为：$T=\mathrm{Wb}/{\mathrm{m}}^2={10}^4\mathrm{Gs}$$T = \mathrm{Wb/m^2} = 10^4\ \mathrm{Gs}$.

  • 线电流：${\displaystyle \mathit{B}={\displaystyle \frac{{\mu }_0}{4\pi }}{\oint }_{C^{\prime }}{\displaystyle \frac{I\mathrm{d}{\mathit{l}}^{\prime }\times {\mathit{a}}_R}{R^2}}}$$\d {\bm B} = \dfrac{\mu_0}{4\pi} \oint_{C'} \dfrac{I\dd{\bm l' \cp \bm a_R}}{R^2}$.

  • 面电流：$\mathit{B}={\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle {\iint }_{S^{\prime }}{\displaystyle \frac{{\mathit{J}}_{\text{s}}({\mathit{r}}^{\prime })\times {\mathit{a}}_R}{R^2}}\mathrm{d}{S}^{\prime }}$$\bm B = \dfrac{\mu_0}{4\pi} \d\iint_{S'} \dfrac{\bm J_\text s(\bm r') \cp \bm a_R}{R^2} \dd{S'}$.

  • 体电流：$\mathit{B}={\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle {\iiint }_{V^{\prime }}{\displaystyle \frac{\mathit{J}({\mathit{r}}^{\prime })\times {\mathit{a}}_R}{R^2}}\mathrm{d}{V}^{\prime }}$$\bm B = \dfrac{\mu_0}{4\pi} \d\iiint_{V'} \dfrac{\bm J(\bm r') \cp \bm a_R}{R^2} \dd{V'}$.

• 例子

  • 有限长直导线：$\mathit{B}={\displaystyle \frac{{\mu }_{0}I}{4\pi r}}(\cos {\theta }_1-\cos {\theta }_2){\mathit{a}}_{\phi }$$\bm B = \dfrac{\mu_0 I}{4\pi r} (\cos\theta_1 - \cos\theta_2) \bm a_\varphi$.

  • 无穷长直导线：$\mathit{B}={\displaystyle \frac{{\mu }_{0}I}{2\pi r}}{\mathit{a}}_{\phi }$$\bm B = \dfrac{\mu_0 I}{2\pi r} \bm a_\varphi$.

  • 电流环的轴线：$\mathit{B}={\displaystyle \frac{{\mu }_{0}I{r}^2}{2(z^2+r^2{)}^{3/2}}}{\mathit{a}}_z$$\bm B = \dfrac{\mu_0 I r^2}{2(z^2 + r^2)^{3/2}} \bm a_z$.

  • 电流环的圆心：$\mathit{B}={\displaystyle \frac{{\mu }_{0}I}{2r}}{\mathit{a}}_z$$\bm B = \dfrac{\mu_0 I}{2r} \bm a_z$.

  • 无限大面电流：$B={\displaystyle \frac{{\mu }_0{J}_{\text{s}}}{2}}$$B = \dfrac{\mu_0 J_\text s}{2}$.

3.1.3 洛伦兹力方程

• 宏观回路：${\displaystyle \mathit{F}={\oint }_{l}I\mathrm{d}\mathit{l}\times \mathit{B}}$$\d \bm F = \oint_l I \dd{\bm l} \cp \bm B$.

• 微观粒子：$\mathit{F}=q\mathit{v}\times \mathit{B}$$\bm F = q \bm v \cp \bm B$.

• 洛伦兹力方程：$\mathit{F}=q(\mathit{v}\times \mathit{B}+\mathit{E})$$\bm F = q (\bm v \cp \bm B + \bm E)$.

3.2 真空中恒定磁场方程

3.2.1 散度与磁通连续性原理

• 旋度场与无源性

  • $\mathit{B}={\displaystyle \frac{{\mu }_0}{4\pi }}\mathbf{\nabla }\times {\displaystyle {\int }_{V^{\prime }}{\displaystyle \frac{\mathit{J}(r^{\prime })}{R}}\mathrm{d}{V}^{\prime }}$$\bm B = \dfrac{\mu_0}{4\pi} \curl \dint_{V'} \dfrac{\bm J(r')}{R} \dd{V'}$.

  • $\mathbf{\nabla }\mathbf{\cdot }\mathit{B}=0$$\div \bm B = 0$.

• 磁通量与连续性

  • 磁通 ${\mathrm{\Phi }}_{\text{m}}={\displaystyle {\int }_S\mathit{B}\mathbf{\cdot }\mathrm{d}\mathit{S}}$$\Phi_\text m = \dint_S \bm B \bm\cdot \dd{\bm S}$ 单位为韦伯 $\mathrm{Wb}$$\rm{Wb}$.

  • ${\displaystyle {\oint }_S\mathit{B}\mathbf{\cdot }\mathrm{d}\mathit{S}={\int }_V\mathbf{\nabla }\mathbf{\cdot }\mathit{B}\mathrm{d}V=0}$$\d \oint_S \bm B \bm\cdot \dd{\bm S} = \int_V \div \bm B \dd{V} = 0$.

3.2.2 真空中的安培环路定理

• 积分形式：${\displaystyle {\oint }_l\mathit{B}\mathbf{\cdot }\mathrm{d}\mathit{l}={\mu }_{0}I}$$\d\oint_l \bm B \bm\cdot \dd{\bm l} = \mu_0 I$.

• 微分形式：$\mathbf{\nabla }\times \mathit{B}={\mu }_0\mathit{J}$$\curl \bm B = \mu_0 \bm J$.

• 其中 $I$$I$ 为真实电流（自由电流），包括传导电流和运流电流.

3.2.3 位函数与矢量泊松方程

• 矢量磁位 $\mathit{A}$$\bm A$.

  • $\mathit{B}\equiv \mathbf{\nabla }\times \mathit{A}$$\bm B \equiv \curl \bm A$.

  • $\mathit{A}$$\bm A$ 不唯一. 若 $\mathit{A}$$\bm A$ 满足上式，则 $\mathit{A}+\mathbf{\nabla }\varphi$$\bm A + \grad \phi$ 也满足.

  • 库仑规范：规定 $\mathbf{\nabla }\mathbf{\cdot }\mathit{A}=0$$\div \bm A = 0$.

• 矢量泊松方程：$\mathrm{\Delta }\mathit{A}=\mathbf{\nabla }(\mathbf{\nabla }\mathbf{\cdot }\mathit{A})-\mathbf{\nabla }\times (\mathbf{\nabla }\times \mathit{A})=-{\mu }_0\mathit{J}$$\Delta \bm A = \grad (\div \bm A) - \curl (\curl \bm A) = -\mu_0 \bm J$.

3.2.4 矢量磁位的计算与应用

• 矢量磁位的计算

  可由毕奥–萨瓦定律或矢量泊松方程推得.

  • 线电流：$\mathit{A}={\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle {\int }_{V^{\prime }}{\displaystyle \frac{\mathit{J}(r^{\prime })}{R}}\mathrm{d}{V}^{\prime }}$$\bm A = \dfrac{\mu_0}{4\pi} \dint_{V'} \dfrac{\bm J(r')}{R} \dd{V'}$.

  • 面电流：${\displaystyle \mathit{A}={\displaystyle \frac{{\mu }_0}{4\pi }}{\int }_{S^{\prime }}{\displaystyle \frac{{\mathit{J}}_s}{R}}\mathrm{d}{S}^{\prime }}$$\d \bm A = \dfrac{\mu_0}{4\pi} \int_{S'} \dfrac{\bm J_s}{R} \dd{S'}$.

  • 体电流：$\mathit{A}={\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle {\int }_{l^{\prime }}{\displaystyle \frac{\mathit{I}}{R}}\mathrm{d}{l}^{\prime }}$$\bm A = \dfrac{\mu_0}{4\pi} \dint_{l'} \dfrac{\bm I}{R} \dd{l'}$.

  • 或由 $\mathrm{\Delta }\mathit{A}=-{\mu }_0\mathit{J}$$\Delta \bm A = -\mu_0 \bm J$ 与边界条件求解.

• 矢量磁位的应用

  • 求磁感应强度：$\mathit{B}=\mathbf{\nabla }\times \mathit{A}$$\bm B = \curl \bm A$.

  • 求磁通量：${\mathrm{\Phi }}_{\text{m}}={\displaystyle {\int }_S\mathit{B}\mathbf{\cdot }\mathrm{d}\mathit{S}={\oint }_C\mathit{A}\mathbf{\cdot }\mathrm{d}\mathit{l}}$$\Phi_\text m = \dint_S \bm B \vdot \dd{\bm S} = \oint_C \bm A \vdot \dd{\bm l}$.

3.3 磁偶极子与介质磁化

3.3.1 磁偶极子及其矢量磁位

• 磁偶极子

  • 一个小载流圆环，右手螺旋方向.

  • 磁偶极矩 (磁矩)：${\mathit{p}}_{\text{m}}=I\mathit{S}=IS{\mathit{a}}_z$$\bm p_\text m = I \bm S = IS \bm a_z$.

• 矢量磁位

  • 矢量磁位：$\mathit{A}={\displaystyle \frac{{\mu }_0}{4\pi r^2}}{\mathit{p}}_{\text{m}}\times {\mathit{a}}_r={\displaystyle \frac{{\mu }_0}{4\pi r^3}}{\mathit{p}}_{\text{m}}\times \mathit{r}$$\bm A = \dfrac{\mu_0}{4\pi r^2} \bm p_\text m \cp \bm a_r = \dfrac{\mu_0}{4\pi r^3} \bm p_\text m \cp \bm r$.

  • 磁感应强度：$\mathit{B}={\displaystyle \frac{{\mu }_{0}IS}{4\pi r^3}}({\mathit{a}}_{r}2\cos \theta +{\mathit{a}}_{\theta }\sin \theta )$$\bm B = \dfrac{\mu_0 I S}{4\pi r^3} (\bm a_r 2\cos\theta + \bm a_\theta \sin\theta)$.

3.3.2 磁化介质及其电流密度

• 介质的磁化

  • 介质磁化后：束缚电流 (磁化电流)，附加磁场.

  • 介质的分类：抗磁质，顺磁质，铁磁质.

  • 铁磁质：剩磁、磁滞现象.

• 磁化强度矢量 $\mathit{M}=\underset{\mathrm{\Delta }V\to 0}{lim}{\displaystyle \frac{\sum {\mathit{p}}_{\text{m}}}{\mathrm{\Delta }V}}$$\bm M = \liml_{\Delta V \to 0} \dfrac{\sum \bm p_\text m}{\Delta V} %\dsum_{k=1}^N \dfrac{\bm m_k}{\Delta V}$.

  • 均匀磁化 / 非均匀磁化.

  • 磁化体电流密度：${\mathit{J}}_{\text{m}}=\mathbf{\nabla }\times \mathit{M}$$\bm J_\text{m} = \curl \bm M$.

  • 磁化面电流密度：${\mathit{J}}_{\text{sm}}=\mathit{M}\times \mathit{n}$$\bm J_{\text{sm}} = \bm M \cp \bm n$.

3.3.3 介质中的恒定磁场方程

• 磁场强度 $\mathit{H}$$\bm H$.

  • 考虑 $\mathbf{\nabla }\times \mathit{B}={\mu }_0(\mathit{J}+{\mathit{J}}_{\text{m}})={\mu }_0(\mathit{J}+\mathbf{\nabla }\times \mathit{M})$$\curl \bm B = \mu_0(\bm J + \bm J_\text m) = \mu_0 (\bm J + \curl \bm M)$.

  • 于是定义磁场强度 $\mathit{H}:={\displaystyle \frac{\mathit{B}}{{\mu }_0}}-\mathit{M}$$\bm H := \dfrac{\bm B}{\mu_0} - \bm M$，单位为 $\mathrm{A}/\mathrm{m}$$\mathrm{A/m}$.

• 安培环路定律

  • 微分形式：$\mathbf{\nabla }\times \mathit{H}=\mathit{J}$$\curl \bm H = \bm J$.

  • 积分形式：${\displaystyle {\oint }_l\mathit{H}\mathbf{\cdot }\mathrm{d}\mathit{l}=I}$$\d\oint_l \bm H \vdot \dd{\bm l} = I$.

• 磁通连续性原理

  • 微分形式：$\mathbf{\nabla }\mathbf{\cdot }\mathit{B}=0$$\div \bm B = 0$.

  • 积分形式：${\displaystyle {\oint }_S\mathit{B}\mathbf{\cdot }\mathrm{d}\mathit{S}=0}$$\d\oint_S \bm B \vdot \dd{\bm S} = 0$.

• 矢量电位函数

  • 泊松方程：$\mathrm{\Delta }\mathit{A}=-\mathbf{\nabla }\times (\mathbf{\nabla }\times \mathit{A})=-\mu \mathit{J}$$\Delta \bm A = -\curl (\curl \bm A) = - \mu \bm J$.

  • 对于无源区域，拉普拉斯方程：$\mathrm{\Delta }\mathit{A}=\mathbf{0}$$\Delta \bm A = \bm 0$.

3.3.4 磁化强度与相对磁导率

• 磁化强度：$\mathit{M}\equiv {\chi }_{\text{m}}\mathit{H}$$\bm M \equiv \chi_\text m \bm H$.

  • 介质的磁化率 ${\chi }_{\text{m}}$$\chi_\text m$ 无量纲.

  • 在真空中，${\chi }_{\text{m}}=0$$\chi_\text m = 0$.

  • 顺磁质 ${\chi }_{\text{m}}>0$$\chi_\text m > 0$，抗磁质 ${\chi }_{\text{m}}<0$$\chi_\text m < 0$.

  • 铁磁质 ${\chi }_{\text{m}}\gg 1$$\chi_\text m \gg 1$，非铁磁质 ${\chi }_{\text{m}}\approx 1$$\chi_\text m \approx 1$.

• 磁感应强度：$\mathit{B}={\mu }_0(\mathit{H}+\mathit{M})={\mu }_0(1+{\chi }_{\text{m}})\mathit{H}={\mu }_0{\mu }_{\text{r}}\mathit{H}=\mu \mathit{H}$$\bm B = \mu_0 (\bm H + \bm M) = \mu_0 (1 + \chi_\text m) \bm H = \mu_0 \mu_\text r \bm H = \mu \bm H$.

  • 相对磁导率 ${\mu }_{\text{r}}$$\mu_\text r$ 无量纲.

  • 磁导率 $\mu ={\mu }_0{\mu }_{\text{r}}$$\mu = \mu_0 \mu_\text r$ 单位为 $\mathrm{H}/\mathrm{m}$$\rm{H/m}$.

  • 顺磁质和抗磁质的 ${\mu }_{\text{r}}\approx 1$$\mu_\text r \approx 1$.

  • 铁磁质的 ${\mu }_{\text{r}}\gg 1$$\mu_\text r \gg 1$ 且非常数.

3.3.5 磁化强度与电流的计算

• 磁化强度和磁化电流的计算

  • $\mathit{M}={\chi }_{\text{m}}\mathit{H}=({\mu }_{\text{r}}-1)\mathit{H}={\displaystyle \frac{{\mu }_{\text{r}}-1}{\mu }}\mathit{B}$$\bm M = \chi_\text m \bm H = (\mu_\text r - 1) \bm H = \dfrac{\mu_\text r - 1}{\mu} \bm B$.

  • ${\mathit{J}}_{\text{m}}=\mathbf{\nabla }\times \mathit{M}={\displaystyle \frac{{\mu }_{\text{r}}-1}{{\mu }_{\text{r}}}}\mathit{J}+\mathbf{\nabla }{\displaystyle \frac{{\mu }_{\text{r}}-1}{\mu }}\times \mathit{B}$$\bm J_\text m = \curl \bm M = \dfrac{\mu_\text r - 1}{\mu_\text r} \bm J + \grad \dfrac{\mu_\text r - 1}{\mu} \cp \bm B$.

  • ${\mathit{J}}_{\text{sm}}=\mathit{M}\times {\mathit{e}}_{\text{n}}={\displaystyle \frac{{\mu }_{\text{r}}-1}{\mu }}\mathit{B}\times {\mathit{e}}_{\text{n}}$$\bm J_\text{sm} = \bm M \cp \bm e_\text n = \dfrac{\mu_\text r - 1}{\mu} \bm B \cp \bm e_\text n$.

• 对于线性各向同性介质

  • ${\mathit{J}}_{\text{m}}={\displaystyle \frac{{\mu }_{\text{r}}-1}{{\mu }_{\text{r}}}}\mathit{J}$$\bm J_\text m = \dfrac{\mu_\text r - 1}{\mu_\text r} \bm J$.

  • $\mathbf{\nabla }\mathbf{\cdot }\mathit{M}=-\mathbf{\nabla }\mathbf{\cdot }\mathit{H}=0$$\div \bm M = -\div \bm H = 0$.

• 媒质的本构方程

  • $\mathit{D}=\epsilon \mathit{E}$$\bm D = \ve \bm E$.

  • ${\mathit{J}}_{\text{c}}=\sigma \mathit{E}$$\bm J_\text c = \sigma \bm E$.

  • $\mathit{B}=\mu \mathit{H}$$\bm B = \mu \bm H$.

3.4 恒定磁场的边界条件

3.4.1 磁感应强度边界条件

• $\mathit{n}\cdot ({\mathit{B}}_1-{\mathit{B}}_2)=0$$\bm n \cdot (\bm B_1 - \bm B_2) = 0$.

• $B_{\text{1n}}=B_{\text{2n}}={\mu }_1{H}_{\text{1n}}={\mu }_2{H}_{\text{2n}}$$B_\text{1n} = B_\text{2n} = \mu_1 H_\text{1n} = \mu_2 H_\text{2n}$.

• 若理想导体内部没有电磁场也没有电流，则 $B_{\text{n}}=0$$B_\text n = 0$.

3.4.2 磁场强度的边界条件

• $\mathit{n}\times ({\mathit{H}}_1-{\mathit{H}}_2)={\mathit{J}}_{\text{s}}$$\bm n \cp (\bm H_1 - \bm H_2) = \bm J_\text s$.

• $H_{\text{1t}}-H_{\text{2t}}={\displaystyle \frac{B_{\text{1t}}}{{\mu }_1}}-{\displaystyle \frac{B_{\text{2t}}}{{\mu }_2}}=J_{\text{s}}$$H_\text{1t} - H_\text{2t} = \dfrac{B_\text{1t}}{\mu_1} - \dfrac{B_\text{2t}}{\mu_2} = J_\text s$.

• 若媒质 2 为导体，则 $H_{\text{t}}=J_{\text{s}}$$H_\text t = J_\text s$.

• 若导体表面有自由面电流，则将产生切向磁场强度分量.

3.4.3 电位函数的边界条件

• 矢量磁位

  • ${\mathit{A}}_1={\mathit{A}}_2$$\bm A_1 = \bm A_2$.

  • $\mathit{n}\times ({\displaystyle \frac{\mathbf{\nabla }\times {\mathit{A}}_1}{{\mu }_1}}-{\displaystyle \frac{\mathbf{\nabla }\times {\mathit{A}}_2}{{\mu }_2}})={\mathit{J}}_{\text{s}}$$\bm n \cp \pqty{\dfrac{\curl \bm A_1}{\mu_1} - \dfrac{\curl \bm A_2}{\mu_2}} = \bm J_\text s$.

• 标量磁位

  • ${\varphi }_{\text{m1}}={\varphi }_{\text{m2}}$$\phi_\text{m1} = \phi_\text{m2}$.

  • ${\mu }_1{\displaystyle \frac{\partial {\varphi }_{\text{m1}}}{\partial n}={\mu }_2\frac{\partial {\varphi }_{\text{m2}}}{\partial n}}$$\mu_1 \d\pdv{\phi_\text{m1}}{n} = \mu_2 \pdv{\phi_\text{m2}}{n}$.

3.4.4 分界面上的磁场方向

• 若 ${\mathit{J}}_{\text{s}}=\mathbf{0}$$\bm J_\text s = \bm 0$，则

  • ${\displaystyle \frac{\tan {\theta }_1}{\tan {\theta }_2}}={\displaystyle \frac{{\mu }_1}{{\mu }_2}}$$\dfrac{\tan\theta_1}{\tan\theta_2} = \dfrac{\mu_1}{\mu_2}$.

  • ${\displaystyle \frac{B_1}{B_2}}={\displaystyle \frac{\cos {\theta }_2}{\cos {\theta }_1}}$$\dfrac{B_1}{B_2} = \dfrac{\cos\theta_2}{\cos\theta_1}$.

• 铁磁质的 $\mu \gg 1$$\mu \gg 1$，因此空气中的磁场近似垂直于纯铁表面.

3.4.5 电流密度的计算方法

• 自由电流

  • 自由电流体密度：$\mathit{J}=\mathbf{\nabla }\times \mathit{H}$$\bm J = \curl \bm H$.

  • 自由电流面密度：${\mathit{J}}_{\text{s}}=\mathit{n}\times ({\mathit{H}}_1-{\mathit{H}}_2)$$\bm J_\text s = \bm n \cp (\bm H_1 - \bm H_2)$.

• 磁化电流

  • 磁化体电流密度：${\mathit{J}}_{\text{m}}=\mathbf{\nabla }\times \mathit{M}$$\bm J_\text{m} = \curl \bm M$.

  • 磁化面电流密度：${\mathit{J}}_{\text{sm}}={\mathit{M}}_1\times {\mathit{n}}_1+{\mathit{M}}_2\times {\mathit{n}}_2$$\bm J_{\text{sm}} = \bm M_1 \cp \bm n_1 + \bm M_2 \cp \bm n_2$.

3.5 无源磁场的标量磁位

3.5.1 标量磁位与磁压定义

• 对于无源区域 $\mathit{J}=0$$\bm J = 0$，

  • 矢量磁位：${\mathbf{\nabla }}^2\mathit{A}=-\mathbf{\nabla }\times \mathit{H}=\mathbf{0}$$\grad^2 \bm A = -\curl \bm H = \bm 0$.

  • 标量磁位：$\mathit{H}\equiv -\mathbf{\nabla }{\varphi }_{\text{m}}$$\bm H \equiv -\grad \phi_\text{m}$.

  • 标量磁位 ${\varphi }_{\text{m}}$$\phi_\text m$ 的单位为安培.

  • 磁压：$U_{\text{mAB}}={\displaystyle {\int }_A^B\mathit{H}\mathbf{\cdot }\mathrm{d}\mathit{l}={\varphi }_{\text{mA}}-{\varphi }_{\text{mB}}}$$U_\text{mAB} = \dint_A^B \bm H \vdot \dd{\bm l} = \phi_\text{mA} - \phi_\text{mB}$.

• 多值性

  • 标量磁位与磁压都具有多值性.

  • 产生原因：${\displaystyle {\oint }_C\mathit{H}\mathbf{\cdot }\mathrm{d}\mathit{l}=I\ne 0}$$\d\oint_C \bm H \vdot \dd{\bm l} = I \ne 0$.

  • 消除方法：规定积分路径不穿过电流回路围成的区域 (磁障碍面).

3.5.2 拉氏方程与泊松方程

• $0=\mathbf{\nabla }\mathbf{\cdot }\mathit{B}={\mu }_0\mathbf{\nabla }\mathbf{\cdot }(\mathit{H}+\mathit{M})\Rightarrow \mathbf{\nabla }\mathbf{\cdot }\mathit{H}=-\mathbf{\nabla }\mathbf{\cdot }\mathit{M}$$0 = \div \bm B = \mu_0 \div (\bm H + \bm M) \RA \div \bm H = -\div \bm M$.

• 均匀磁化：${\mathbf{\nabla }}^2{\varphi }_{\text{m}}=-\mathbf{\nabla }\mathbf{\cdot }\mathit{H}=-\mathbf{\nabla }\mathbf{\cdot }{\displaystyle \frac{\mathit{B}}{\mu }}=0$$\grad^2 \phi_\text m = -\div \bm H = -\div \dfrac{\bm B}{\mu} = 0$.

• 非均匀磁化：${\mathbf{\nabla }}^2{\varphi }_{\text{m}}=\mathbf{\nabla }\mathbf{\cdot }\mathit{M}\equiv -{\rho }_{\text{m}}$$\grad^2 \phi_\text m = \div \bm M \equiv -\rho_\text m$.

3.5.3 磁荷密度与边界条件

• 磁荷密度

  • 磁荷体密度 ${\rho }_{\text{m}}=-\mathbf{\nabla }\mathbf{\cdot }\mathit{M}$$\rho_\text m = -\div \bm M %= \div \bm H$.

  • 磁荷面密度 ${\rho }_{\text{sm}}=\mathit{M}\mathbf{\cdot }\mathit{n}$$\rho_\text{sm} = \bm M \vdot \bm n$.

• 磁场强度的边界条件

  • $\mathit{n}\mathbf{\cdot }({\mathit{H}}_1-{\mathit{H}}_2)={\rho }_{\text{sm}}$$\bm n \vdot (\bm H_1 - \bm H_2) = \rho_\text{sm}$.

  • $H_{1\text{n}}-H_{\text{2n}}={\rho }_{\text{sm}}$$H_{1\text n} - H_\text{2n} = \rho_\text{sm}$.

3.5.4 电位函数的边界条件

• 矢量磁位的边界条件

  • ${\mathit{A}}_1={\mathit{A}}_2$$\bm A_1 = \bm A_2$.

  • $\mathit{n}\times ({\displaystyle \frac{\mathbf{\nabla }\times {\mathit{A}}_1}{{\mu }_1}}-{\displaystyle \frac{\mathbf{\nabla }\times {\mathit{A}}_2}{{\mu }_2}})={\mathit{J}}_{\text{s}}$$\bm n \cp \pqty{\dfrac{\curl \bm A_1}{\mu_1} - \dfrac{\curl \bm A_2}{\mu_2}} = \bm J_\text s$.

• 标量磁位的边界条件

  • ${\varphi }_{\text{m1}}={\varphi }_{\text{m2}}$$\phi_\text{m1} = \phi_\text{m2}$.

  • ${\mu }_1{\displaystyle \frac{\partial {\varphi }_{\text{m1}}}{\partial n}={\mu }_2\frac{\partial {\varphi }_{\text{m2}}}{\partial n}}$$\mu_1 \d\pdv{\phi_\text{m1}}{n} = \mu_2 \pdv{\phi_\text{m2}}{n}$.

3.5.5 磁感应强度计算方法

• 利用标量磁位

  • ${\rho }_{\text{m}}=-\mathbf{\nabla }\mathbf{\cdot }\mathit{M},{\rho }_{\text{sm}}=\mathit{M}\mathbf{\cdot }\mathit{n}$$\rho_\text m = -\div \bm M, \rho_\text{sm} = \bm M \vdot \bm n$.

  • ${\varphi }_{\text{m}}={\displaystyle {\int }_{V^{\prime }}{\displaystyle \frac{{\rho }_{\text{m}}}{4\pi R}}\mathrm{d}{V}^{\prime }+{\displaystyle {\int }_S{\displaystyle \frac{{\rho }_{\text{sm}}}{4\pi R}}\mathrm{d}{S}^{\prime }}}$$\phi_\text m = \dint_{V'} \dfrac{\rho_\text m}{4\pi R} \dd{V'} + \dint_S \dfrac{\rho_\text{sm}}{4\pi R} \dd{S'}$.

  • $\mathit{H}=-\mathbf{\nabla }{\varphi }_{\text{m}},\mathit{B}={\mu }_0(\mathit{H}+\mathit{M})$$\bm H = -\grad \phi_\text m, \bm B = \mu_0 (\bm H + \bm M)$.

• 利用矢量磁位

  • ${\mathit{J}}_{\text{m}}=\mathbf{\nabla }\times \mathit{M},{\mathit{J}}_{\text{sm}}=\mathit{M}\times \mathit{n}$$\bm J_\text m = \curl \bm M, \bm J_\text{sm} = \bm M \cp \bm n$.

  • $\mathit{B}={\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle {\int }_{V^{\prime }}{\displaystyle \frac{{\mathit{J}}_{\text{m}}\times {\mathit{a}}_R}{R^2}}\mathrm{d}{V}^{\prime }+{\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle {\int }_{S^{\prime }}{\displaystyle \frac{{\mathit{J}}_{\text{s}}\times {\mathit{a}}_R}{R^2}}\mathrm{d}{S}^{\prime }}}$$\bm B = \dfrac{\mu_0}{4\pi} \dint_{V'} \dfrac{\bm J_\text{m} \cp \bm a_R}{R^2} \dd{V'} + \dfrac{\mu_0}{4\pi} \dint_{S'} \dfrac{\bm J_\text s \cp \bm a_R}{R^2} \dd{S'}$.

  • $\mathit{H}={\displaystyle \frac{\mathit{B}}{{\mu }_0}}-\mathit{M}$$\bm H = \dfrac{\bm B}{\mu_0} - \bm M$.

3.6 恒定磁场的其它问题

3.6.1 自电感与互电感

• 自电感 $L$$L$.

  • 磁通：$\mathrm{\Phi }={\displaystyle {\int }_S\mathit{B}\mathbf{\cdot }\mathrm{d}\mathit{S}}$$\Phi = \dint_S \bm B \vdot \dd{\bm S}$.

  • 磁链：$\mathrm{\Psi }=n\mathrm{\Phi }$$\Psi = n \Phi$.

  • 自感：$L={\displaystyle \frac{\mathrm{\Psi }}{I}}$$L = \dfrac{\Psi}{I}$.

• 互电感 $M$$M$.

  • 互磁链：${\mathrm{\Psi }}_{12}$$\Psi_{12}$.

  • $M_{12}={\displaystyle \frac{{\mathrm{\Psi }}_{12}}{I_2}}$$M_{12} = \dfrac{\Psi_{12}}{I_2}$.

  • $M_{12}=M_{21}=M$$M_{12} = M_{21} = M$.

• 自感大于零，互感可正可负.

3.6.2 内自感与外自感

• 磁链：$\mathrm{\Psi }=n\mathrm{\Phi }={\mathrm{\Psi }}_{\mathrm{i}}+{\mathrm{\Psi }}_{\mathrm{o}}$$\Psi = n \Phi = \Psi_\i + \Psi_\o$.

  • 内磁链：${\mathrm{\Psi }}_{\mathrm{i}}$$\Psi_\i$ 为与部分电流交链的磁链.

  • 外磁链：${\mathrm{\Psi }}_{\mathrm{o}}$$\Psi_\o$ 为与全部电流交链的磁链.

• 电感：$L={\displaystyle \frac{\mathrm{\Psi }}{I}}=L_{\mathrm{i}}+L_{\mathrm{o}}$$L = \dfrac{\Psi}{I} = L_\i + L_\o$.

  • 内自感：$L_{\mathrm{i}}={\displaystyle \frac{{\psi }_{\mathrm{i}}}{I}}$$L_\i = \dfrac{\psi_\i}{I}$.

  • 外自感：$L_{\mathrm{o}}={\displaystyle \frac{{\psi }_{\mathrm{o}}}{I}}$$L_\o = \dfrac{\psi_\o}{I}$.

• 长直导体的内自感

  • 注意 $\mathit{B}$$\bm B$ 和 $\mathit{S}$$\bm S$ 的方向. $B_{\mathrm{i}}={\displaystyle \frac{{\mu }_0}{2\pi r}}{\displaystyle \frac{r^{2}I}{a^2}}={\displaystyle \frac{{\mu }_{0}Ir}{2\pi a^2}}$$B_\i = \dfrac{\mu_0}{2\pi r} \dfrac{r^2 I}{a^2} = \dfrac{\mu_0 I r}{2\pi a^2}$,

  • $L_{\mathrm{i}}={\displaystyle \frac{\mathrm{\Psi }}{I}}{\displaystyle {\int }_0^a{\displaystyle \frac{r^2}{a^2}}{\mathit{B}}_{\mathrm{i}}\mathbf{\cdot }\mathrm{d}\mathit{S}={\displaystyle \frac{1}{I}}{\int }_0^a{\displaystyle \frac{r^2}{a^2}}{\displaystyle \frac{{\mu }_{0}Ir}{2\pi a^2}}l\mathrm{d}r={\displaystyle \frac{{\mu }_{0}l}{8\pi }}}$$L_\i = \dfrac{\Psi}{I} \dint_0^a \dfrac{r^2}{a^2} \bm B_\i \vdot \dd{\bm S} = \dfrac{1}{I} \int_0^a \dfrac{r^2}{a^2} \dfrac{\mu_0 I r}{2\pi a^2} l \dr = \dfrac{\mu_0 l}{8\pi}$.

3.6.3 自感与互感计算

• 互感

  • 矢量磁位：${\mathit{A}}_{21}={\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle {\oint }_{l_1}{\displaystyle \frac{I_1\mathrm{d}{\mathit{l}}_1}{R}}}$$\bm A_{21} = \dfrac{\mu_0}{4\pi} \d\oint_{l_1} \dfrac{I_1 \dd {\bm l_1}}{R}$.

  • 磁通：${\mathrm{\Phi }}_{21}={\displaystyle {\int }_S\mathit{B}\mathbf{\cdot }\mathrm{d}\mathit{S}={\oint }_{l_2}{\mathit{A}}_{21}\mathbf{\cdot }\mathrm{d}{\mathit{l}}_2}$$\Phi_{21} = \dint_S \bm B \vdot \dd{\bm S} = \oint_{l_2} \bm A_{21} \vdot \dd{\bm l_2}$.

  • 诺依曼公式：$M={\displaystyle \frac{N_1{N}_2{\mu }_0}{4\pi }}{\displaystyle {\oint }_{l_1}{\oint }_{l_2}{\displaystyle \frac{\mathrm{d}{\mathit{l}}_2\cdot \mathrm{d}{\mathit{l}}_1}{R}}}$$M = \dfrac{N_1 N_2 \mu_0}{4 \pi} \d\oint_{l_1} \oint_{l_2} \dfrac{\dd{\bm l_2} \cdot \dd{\bm l_1}}{R}$.

• 自感

  • 外自感：$L_{\mathrm{o}}={\displaystyle \frac{N^2{\mu }_0}{4\pi }}{\displaystyle {\oint }_{l_1}{\oint }_{l_2}{\displaystyle \frac{\mathrm{d}{\mathit{l}}_2\cdot \mathrm{d}{\mathit{l}}_1}{R}}}$$L_\o = \dfrac{N^2 \mu_0}{4 \pi} \d\oint_{l_1} \oint_{l_2} \dfrac{\dd{\bm l_2} \cdot \dd{\bm l_1}}{R}$.

  • 内自感：$L_{\mathrm{i}}={\displaystyle \frac{{\mu }_{0}l}{8\pi }}$$L_\i = \dfrac{\mu_0 l}{8 \pi}$.

  • 自电感：$L=L_{\mathrm{i}}+L_{\mathrm{o}}$$L = L_\i + L_\o$.

3.6.4 能量与能量密度

• $n$$n$ 回路电流系统

  • $W_{\text{m}}={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n{I}_i{\mathrm{\Psi }}_i={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i,j}{M}_{ij}{I}_i{I}_j}}$$W_\text m = \dfrac{1}{2} \insum I_i \Psi_i = \dfrac{1}{2} \dsum_{i, j} M_{ij} I_i I_j$.

  • $W_{\text{m}}={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n{L}_i{I}_i^2+{\displaystyle \frac{1}{2}}\sum _{i\ne j}{M}_{ij}{I}_i{I}_j}$$W_\text m = \dfrac{1}{2} \insum L_i I_i^2 + \dfrac{1}{2} \sum_{i \ne j} M_{ij} I_i I_j$.

• 两个回路

  • 互感系数：$M\le \sqrt{L_1{L}_2}$$M \le \sqrt{L_1 L_2}$.

  • 耦合系数：$k={\displaystyle \frac{M}{\sqrt{L_1{L}_2}}}$$k = \dfrac{M}{\sqrt{L_1 L_2}}$.

• 磁场能量

  • ${\omega }_{\text{m}}={\displaystyle \frac{1}{2}}\mathit{B}\cdot \mathit{H}={\displaystyle \frac{1}{2}}\mu H^2={\displaystyle \frac{B^2}{2\mu }}$$\omega_\text m = \dfrac{1}{2} \bm B \cdot \bm H = \dfrac{1}{2} \mu H^2 = \dfrac{B^2}{2 \mu}$.

  • $W_{\text{m}}={\displaystyle \frac{1}{2}}{\displaystyle {\int }_V\mathit{H}\mathbf{\cdot }\mathit{B}\mathrm{d}V}$$W_\text m = \dfrac{1}{2} \dint_V \bm H \vdot \bm B \dd{V}$.

3.6.5 磁场力虚位移法

对于复杂系统，利用 $\mathrm{d}W=\mathrm{d}{W}_{\text{m}}+F_g\mathrm{d}g$$\dd{W} = \dd{W_\text m} + F_g \dd{g}$.

• 若回路电流不变

  • $\mathrm{d}W={\displaystyle \sum _{k=1}^N{I}_k\mathrm{d}{\mathrm{\Psi }}_k}$$\dd{W} = \dsum_{k=1}^N I_k \dd{\Psi_k}$.

  • $\mathrm{d}{W}_{\text{m}}={\displaystyle \frac{1}{2}}{\displaystyle \sum _{k=1}^N{I}_k\mathrm{d}{\mathrm{\Psi }}_k}$$\dd{W_\text m} = \dfrac{1}{2} \dsum_{k=1}^N I_k \dd{\Psi_k}$.

  • $F_g={{\displaystyle \frac{\partial W_{\text{m}}}{\partial g}\phantom{\int }}|}_{I_k=\text{常}\text{数}}$$F_g = \eval{\d\pdv{W_\text m}{g}}_{I_k = 常数}$.

• 若回路磁链不变

  • $\mathrm{d}W=0$$\dd{W} = 0$.

  • $F_g=-{{\displaystyle \frac{\partial W_{\text{m}}}{\partial g}\phantom{\int }}|}_{{\mathrm{\Psi }}_k=\text{常}\text{数}}$$F_g = -\eval{\dpdv{W_\text m}{g}}_{\Psi_k = 常数}$.