# 磁标势

${\displaystyle \nabla \times \mathbf {H} =0,}$

${\displaystyle \mathbf {H} =-\nabla \psi .}$

${\displaystyle \nabla \cdot \mathbf {B} =\mu _{0}\nabla \cdot (\mathbf {H+M} )=0,}$

${\displaystyle \nabla ^{2}\psi =-\nabla \cdot \mathbf {H} =\nabla \cdot \mathbf {M} .}$

${\displaystyle \rho _{m}=-\nabla \cdot \mathbf {M} }$

## 利用磁标势求解磁場

${\displaystyle \nabla \times \mathbf {H} =\mathbf {J} }$
${\displaystyle \nabla \cdot \mathbf {B} =0}$

${\displaystyle \mathbf {H} =-\nabla \psi _{m}}$

${\displaystyle \nabla ^{2}\psi _{m}=0}$

### 鐵磁性物質的磁标势

${\displaystyle \mathbf {H} \ {\stackrel {def}{=}}\ {\frac {1}{\mu _{0}}}\mathbf {B} -\mathbf {M} }$

${\displaystyle \nabla \cdot \mathbf {B} =\mu _{0}\nabla \cdot ({\mathbf {H} +\mathbf {M} })=0}$

${\displaystyle \nabla ^{2}\psi _{m}=-\nabla \cdot \mathbf {H} =\nabla \cdot \mathbf {M} }$

${\displaystyle \nabla \cdot \mathbf {M} }$  可以視為磁場的源電流，就好似 ${\displaystyle \rho _{bound}=-\nabla \cdot \mathbf {P} }$ 靜電學束縛電荷一樣。這樣，類比束縛電荷，可以稱呼 ${\displaystyle \rho _{m}=-\nabla \cdot \mathbf {M} }$  為「束縛磁荷」。這樣，束縛磁荷的帕松方程式為

${\displaystyle \nabla ^{2}\psi _{m}=-\rho _{m}}$

${\displaystyle \psi _{m}(\mathbf {r} )=\ {\frac {1}{4\pi }}\int _{\mathbb {V} '}{\frac {\rho _{m}(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,d^{3}\mathbf {r} '}$

## 参考文献

• Duffin, W.J. Electricity and Magnetism, Fourth Edition. McGraw-Hill. 1990.
• Feynman, Richard P; Leighton, Robert B; Sands, Matthew. The Feynman Lectures on Physics Volume 2. Addison-Wesley. 1964. ISBN 020102117XP 请检查|isbn=值 (帮助).
• Jackson, John David. Classical Electrodynamics, Third Edition. John Wiley & Sons. 1998.
• Jackson, John Davd, Classical Electrodynamics 3rd, John-Wiley, 1999, ISBN 047130932X
• Kraus, John D., Electromagnetics 3rd, McGraw-Hill, 1984, ISBN 0070354235
• Ulaby, Fawwaz. Fundamentals of Applied Electromagnetics, Fifth Edition. Pearson Prentice Hall. 2007: 226–228. ISBN 0-13-241326-4.