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# 磁场

## 定义

${\displaystyle \mathbf {F} =q\mathbf {v} \times \mathbf {B} }$

${\displaystyle \mathbf {F} }$是作用力，以牛顿为测量单位，
${\displaystyle q}$电荷量，以库仑为测量单位，
${\displaystyle \mathbf {v} }$是电荷${\displaystyle q}$速度，以米／秒为测量单位。

${\displaystyle {\boldsymbol {\tau ))={\boldsymbol {\mu ))\times \mathbf {B} }$

${\displaystyle {\boldsymbol {\tau ))}$是力矩，以牛顿·米为测量单位，
${\displaystyle {\boldsymbol {\mu ))}$是磁偶极子的磁偶极矩，以米2·安培为测量单位。

## 历史

1825年，安培又发表了安培定律。这定律也能够描述载流导线产生的磁场。更重要的，这定律帮助建立整个电磁理论的基础。于1831年，麦可·法拉第证实，随着时间而变化的磁场会生成电场。这实验结果展示出电与磁之间更密切的关系。

## B场与H场

B场的各种不同命名

H场的各种不同命名

H场以方程定义为[6]

${\displaystyle \mathbf {H} \ {\stackrel {def}{=))\ {\frac {\mathbf {B} }{\mu _{0))}-\mathbf {M} }$

${\displaystyle \mathbf {H} ={\frac {\mathbf {B} }{\mu '))}$

${\displaystyle \mathbf {H} ={\frac {\mathbf {B} }{\mu _{0))))$

## 磁场线

### 磁场散度为0

${\displaystyle \oint _{\mathbb {S} }\mathbf {B} \cdot \mathrm {d} \mathbf {a} =0}$

### H场线开始与终结于磁极

${\displaystyle \mathbf {H} \ {\stackrel {def}{=))\ {\frac {\mathbf {B} }{\mu _{0))}-\mathbf {M} }$

${\displaystyle J_{b}\ {\stackrel {def}{=))\ \nabla \times \mathbf {M} ={\boldsymbol {0))}$
${\displaystyle K_{b}\ {\stackrel {def}{=))\ \mathbf {M} \times {\hat {n))=M\sin \theta }$

${\displaystyle \mathbf {B} ={\hat {z))2\mu _{0}M/3}$

${\displaystyle \mathbf {H} =-{\hat {z))M/3}$

## 磁偶极子

${\displaystyle {\boldsymbol {\mu ))=I\mathbf {a} }$

### 磁偶极子的磁场

${\displaystyle \mathbf {A} (\mathbf {r} )={\frac {\mu _{0)){4\pi r^{2))}({\boldsymbol {\mu ))\times {\hat {\mathbf {r} )))}$

${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$

${\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0)){4\pi r^{3))}\left(3({\boldsymbol {\mu ))\cdot {\hat {\mathbf {r} ))){\hat {\mathbf {r} ))-{\boldsymbol {\mu ))\right)+{\frac {2\mu _{0}{\boldsymbol {\mu ))}{3))\delta ^{3}(\mathbf {r} )}$

### 磁偶极子感受到的磁力矩

${\displaystyle {\boldsymbol {\tau ))=\left(IwB_{0}{\frac {w\sin {\theta )){2))+IwB_{0}{\frac {w\sin {\theta )){2))\right){\hat {\mathbf {y} ))=Iw^{2}B_{0}\sin {\theta }{\hat {\mathbf {y} ))}$

${\displaystyle {\boldsymbol {\tau ))={\boldsymbol {\mu ))\times \mathbf {B} }$

### 磁偶极子的势能

${\displaystyle W=-\int _{\theta _{1))^{\theta _{2))\tau \ \mathrm {d} \theta =-\int _{\theta _{1))^{\theta _{2))\mu B_{0}\sin {\theta }\ \mathrm {d} \theta =\mu B_{0}(\cos {\theta _{2))-\cos {\theta _{1)))}$

${\displaystyle W=\mu B_{0}\cos {\theta _{2))={\boldsymbol {\mu ))\cdot \mathbf {B} }$

${\displaystyle W_{a}=-W=-{\boldsymbol {\mu ))\cdot \mathbf {B} }$

${\displaystyle U=-{\boldsymbol {\mu ))\cdot \mathbf {B} }$

## 永久磁铁

### 永久磁铁的磁场

${\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0)){4\pi r^{3))}\left[3({\boldsymbol {\mu ))\cdot {\hat {\mathbf {r} ))){\hat {\mathbf {r} ))-{\boldsymbol {\mu ))\right]}$

### 外磁场作用于磁铁的力矩

${\displaystyle {\boldsymbol {\tau ))={\boldsymbol {\mu ))\times \mathbf {B} }$

### 外磁场作用于磁铁的磁力

${\displaystyle \mathbf {F} =\mathbf {\nabla } \left({\boldsymbol {\mu ))\cdot \mathbf {B} \right)}$

## 电流与磁场

### 运动中的带电粒子所生成的磁场

${\displaystyle \mathbf {E} ={\frac {q}{4\pi \epsilon _{0))}{\frac {1-v^{2}/c^{2)){(1-v^{2}\sin ^{2}\theta /c^{2})^{3/2))}{\frac {\mathbf {r} -\mathbf {w} }{|\mathbf {r} -\mathbf {w} |^{3))))$
${\displaystyle \mathbf {B} =\mathbf {v} \times {\frac {1}{c^{2))}\mathbf {E} }$

${\displaystyle \mathbf {E} ={\frac {q}{4\pi \epsilon _{0))}\ {\frac {\mathbf {r} -\mathbf {w} }{|\mathbf {r} -\mathbf {w} |^{3))))$
${\displaystyle \mathbf {B} ={\frac {\mu _{0}q\mathbf {v} }{4\pi ))\times {\frac {\mathbf {r} -\mathbf {w} }{|\mathbf {r} -\mathbf {w} |^{3))))$

### 电流产生的磁场

${\displaystyle \mathrm {d} \mathbf {B} ={\frac {\mu _{0}I}{4\pi ))\mathrm {d} {\boldsymbol {\ell ))^{\ \prime }\times {\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{3))))$

${\displaystyle \oint _{\mathbb {C} }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell ))=\mu _{0}I_{\mathrm {enc} ))$
${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} }$

${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t))}$

### 电流与磁力

#### 移动中的带电粒子所感受到的磁力

${\displaystyle \mathbf {F} =q\mathbf {v} \times \mathbf {B} }$

#### 载流导线所感受到的磁力

${\displaystyle \mathbf {F} =q\mathbf {v} \times \mathbf {B} }$

${\displaystyle \mathbf {F} =\int _{\mathbb {C} }\mathbf {v} \times \mathbf {B} \ \mathrm {d} q=\int _{\mathbb {C} }\mathbf {v} \times \mathbf {B} \lambda \ \mathrm {d} \ell =\int _{\mathbb {C} }\mathbf {I} \times \mathbf {B} \ \mathrm {d} \ell }$

${\displaystyle \mathbf {F} =I\int _{\mathbb {C} }\mathrm {d} {\boldsymbol {\ell ))\times \mathbf {B} }$

${\displaystyle F=ILB}$

## 磁性物质内外的H场与B场

### 磁化强度

${\displaystyle \mathbf {M} \ {\stackrel {def}{=))\ n{\boldsymbol {\mu ))}$

${\displaystyle J_{b}\ {\stackrel {def}{=))\ \nabla \times \mathbf {M} }$
${\displaystyle K_{b}\ {\stackrel {def}{=))\ \mathbf {M} \times {\hat {n))}$

### H场与磁性物质

H场${\displaystyle \mathbf {H} }$定义为

${\displaystyle \mathbf {H} \ {\stackrel {def}{=))\ {\frac {\mathbf {B} }{\mu _{0))}-\mathbf {M} }$

${\displaystyle \oint _{\mathbb {C} }\mathbf {H} \cdot \mathrm {d} {\boldsymbol {\ell ))=\oint _{\mathbb {C} }\left({\frac {\mathbf {B} }{\mu _{0))}-\mathbf {M} \right)\cdot \mathrm {d} {\boldsymbol {\ell ))=I_{\mathrm {tot} }-I_{\mathrm {b} }=I_{\mathrm {f} ))$

${\displaystyle \oint _{\mathbb {S} }\mathbf {H} \cdot \mathrm {d} \mathbf {a} =\oint _{\mathbb {S} }\left({\frac {\mathbf {B} }{\mu _{0))}-\mathbf {M} \right)\cdot \mathrm {d} \mathbf {a} =(0+q_{M})=q_{M))$

### 磁性物质

• 抗磁性物质响应出的磁化强度与外磁场呈相反方向，会趋于朝着磁场较弱的区域移动，即被磁场排斥[18]
• 顺磁性物质响应出的磁化强度与外磁场呈相同方向，会趋于朝着磁场较强的区域移动，即被磁场吸引[18]
• 铁磁性物质内部有很多未配对电子。由于交换作用（exchange interaction），这些电子的自旋趋于与相邻未配对电子的自旋呈相同方向。由于铁磁性物质内部又分为很多磁畴，虽然磁畴内部所有电子的自旋会单向排列，造成“饱和磁矩”，磁畴与磁畴之间，磁矩的方向与大小都不相同。所以，未被磁化的铁磁性物质，其净磁矩与磁化矢量都等于零。假设施加外磁场，这些磁畴的磁矩还趋于与外磁场呈相同方向，从而形成有可能相当强烈的磁化矢量与其感应磁场。随着外磁场的增高，磁化强度也会增高，直到“饱和点”，净磁矩等于饱合磁矩。这时，再增高外磁场也不会改变磁化强度。假设现在撤除外磁场，则铁磁性物质仍能保存一些磁化的状态，净磁矩与磁化矢量不等于零。所以，经过磁化处理后的铁磁性物质具有“自发磁矩”。
• 反铁磁性物质内部的相邻价电子的自旋趋于相反方向。这种物质的净磁矩与磁化强度都等于零。大多数反铁磁性物质只存在于低温状况。假设温度超过奈尔温度，则通常会变为顺磁性物质。
• 亚铁磁性物质内部是由两种以上原子组成，不同次晶格的不同原子，其磁矩的方向相反，数值大小不相等，所以，净磁矩与磁化强度都不等于零，具有较微弱的铁磁性。
• 超导体（和铁磁超导体[19][20]：当温度低于某临界温度，磁场小于某临界磁场时，这些物质会特征地变成完美导电体电导率变得无穷大，磁性也变得非常显著，当磁场小于某更小的临界磁场时，这物质会成为完美抗磁性物质。超导体常常会在某宽广的温度和磁场值域内（称为“混合态”），展现出磁化强度对于磁场的复杂磁滞依赖关系。

${\displaystyle \mathbf {M} =\chi _{m}\mathbf {H} }$

${\displaystyle \mathbf {B} =\mu '\mathbf {H} }$

## 电磁学：电场与磁场之间的关系

### 法拉第电磁感应定律：含时磁场生成的电场

${\displaystyle \Phi _{B}(t)\ {\stackrel {def}{=))\ \int _{\mathbb {S} }\mathbf {B} (\mathbf {r} ,\,t)\cdot \mathrm {d} \mathbf {a} }$

${\displaystyle {\mathcal {E))=-{\frac {\mathrm {d} \Phi _{B)){\mathrm {d} t))}$

${\displaystyle {\mathcal {E))=-{\frac {\partial \Phi _{B)){\partial t))=-\int _{\mathbb {S} }{\frac {\partial \mathbf {B} }{\partial t))\cdot \mathrm {d} \mathbf {a} }$

${\displaystyle {\mathcal {E))=\oint _{\mathbb {C} }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell ))}$

${\displaystyle \oint _{\mathbb {C} }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell ))=-\int _{\mathbb {S} }{\frac {\partial \mathbf {B} }{\partial t))\cdot \mathrm {d} \mathbf {a} }$

${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t))}$

### 麦克斯韦-安培方程：含时电场生成的磁场

1861年，麦克斯韦将安培定律方程重新推导一遍，使得符合电动力学条件，并且发表结果于论文《论物理力线》内。麦克斯韦认为，含时电场会生成磁场，假若电场含时间，则前述安培定律方程不成立，必须加以修正。经过修正后，新的方程称为麦克斯韦-安培方程，是麦克斯韦方程组中的一个方程，以积分形式表示为

${\displaystyle \oint _{\mathbb {C} }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell ))=\mu _{0}\int _{\mathbb {S} }\left(\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t))\right)\cdot \mathrm {d} \mathbf {a} }$

${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t))}$

${\displaystyle \mathbf {J} _{D}\ {\stackrel {def}{=))\ {\frac {\partial \mathbf {D} }{\partial t))}$

${\displaystyle \mathbf {J} _{D}=\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t))+{\frac {\partial \mathbf {P} }{\partial t))}$

### 磁矢势

${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$
${\displaystyle \mathbf {E} =-\nabla \varphi -{\frac {\partial \mathbf {A} }{\partial t))}$

## 重要应用领域

### 磁路

H场在磁路学领域很有用处。在一个线性物质内部，

${\displaystyle \mathbf {B} =\mu '\mathbf {H} }$

${\displaystyle \mathbf {J} =\sigma \mathbf {E} }$

${\displaystyle \Phi ={\frac {F}{R))_{m))$

## 注释

1. ^ 更精确地分类，磁场是一种赝矢量力矩角速度也是准矢量。当坐标反演时，准矢量会保持不变。
2. ^ 基本粒子，像电子正子等等，会产生自己内有的磁场，这是一种相对论性效应，并不是因为粒子运动而产生的。但是，对于大多数状况，这磁场可以模想为是由粒子所载有的电荷因为旋转运动而产生的。因此，这相对论性效应称为自旋。磁铁产生的磁场主要是由内部未配对电子的自旋形成的。
3. ^ 他的论文《Epistola Petri Peregrini de Maricourt ad Sygerum de Foucaucourt Militem de Magnete》，简称为《Epistola de magnete（磁石书信）》，发表日期认定为公元1269年。
4. ^ 按照磁化强度的定义，在这模型里，类比弹簧的物理，为了要将磁化强度增加${\displaystyle \delta \mathbf {M} }$，拉扯与扭转磁荷之间的相互作用，所需的机械功每单位体积为${\displaystyle \delta W=\mathbf {H} \cdot \mu _{0}\delta \mathbf {M} }$。由于磁场为${\displaystyle \mathbf {B} =\mu _{0}(\mathbf {H} +\mathbf {M} )}$，包括了制备外磁场于真空所需的H场项目。所以，增加磁场所需的机械功每单位体积为${\displaystyle \delta W=\mathbf {H} \cdot \mu _{0}\delta \mathbf {B} }$。这是正确的结果。但是，用来推导的模型并不正确。
5. ^ ，回顾沉思，这模型的成功，大多是因为，在磁性物质外部，电偶极子的电场跟磁偶极子的磁场有相同的样式。只有在磁性物质内部，简单的磁荷模型无法解释磁场的物理行为。
6. ^ Edward Purcell, in Electricity and Magnetism, McGraw-Hill, 1963, writes, Even some modern writers who treat B as the primary field feel obliged to call it the magnetic induction because the name magnetic field was historically preempted by H. This seems clumsy and pedantic. If you go into the laboratory and ask a physicist what causes the pion trajectories in his bubble chamber to curve, he'll probably answer "magnetic field", not "magnetic induction." You will seldom hear a geophysicist refer to the Earth's magnetic induction, or an astrophysicist talk about the magnetic induction of the galaxy. We propose to keep on calling B the magnetic field. As for H, although other names have been invented for it, we shall call it "the field H" or even "the magnetic field H."
David Griffith, in Introduction to Electrodynamics, Prentice-Hall, Inc, 1999, page 271, writes, Many author call H, not B, the "magnetic field." Then, they have to invent a new word for B? the "flux density," or magnetic "induction"(an absurd choice, since that term already has at least two other meanings in electrodynamics). Anyway, B is indisputably the fundamental quantity, so I shall continue to call it the "magnetic field," as everyone does in the spoken language. H has no sensibile name: just call it "H".
Andrew Zangwill, in Modern Electrodynamics, Cambridge University Press, 2013, page 44, writes, In this book, only the "electric field" E and the "magnetic field" B are fundamental. We give no special names to the auxiliary fields D and H.
7. ^ 。试想将一个指南针置入磁铁内部，指南针的指北极会指向磁铁的指北极。又将两块条形磁铁，端点连着端点，排列于一直线，假设相连结的两个端点是磁异性，则两块条形磁铁会相吸引，否则，两块条形磁铁会相排斥。
8. ^ 如同前面所述，磁场线的密度与磁场大小有关。
9. ^ 曾经有几个实验制备出最初被认为可能是磁单极子的事件，但后来都无法获得学术界确认。详尽细节，请参阅磁单极子
10. ^ 这里，微小的意思是，观测者离磁铁足够的遥远，这样，磁铁的尺寸可以被设定为无穷小；当尺寸驱向无穷小极限时，磁铁可以理想化成为磁偶极子。较大尺寸的磁铁的磁场必须包括更多项目，不只是与磁矩有关，还会与磁铁的几何形状有关。
11. ^ 根据狭义相对论，移动中的带电粒子会产生电场和磁场，而对于处于与粒子同速度的参考系的观察者，粒子是固定不动的，所以，只可能会产生电场。同样的物理原则应该可以应用于各个不同的参考系，这意味着电场和磁场是同样物理现象的两面。更详尽细节，请参阅条目经典电磁学与狭义相对论（classical electromagnetism and special relativity）。

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