Electrodynamics MOC

Magnetostatics MOC

Magnetostatics is a special case of electrodynamics where the Magnetic field is time-independent and there is no electric field, i.e.

⃗𝐄=0𝜕⃗𝐁𝜕𝑡=⃗𝟎

Magnetostatics was established empicircally by the Biot-Savart Law, but of course it is fully encoded in Maxwell’s equations whose differential form become

0=𝜌𝑞⃗∇⋅⃗𝐁=0𝜕⃗𝐁𝜕𝑡=0⃗∇×⃗𝐁=𝜇0⃗𝐉

whence it immediately follows that 𝜕⃗𝐉𝜕𝑡 =⃗𝟎 and the charge continuity equation becomes

⃗∇⋅⃗𝐉=0

Since the Poynting vector and thus momentum density vanishes, no energy nor momentum is transported by the fields and no momentum is stored by the fields.

Potential

A magnetostatic system is completely described by its magnetic potential

⃗𝐁=⃗∇×⃗𝐀⃗∇(⃗∇⋅⃗𝐀)−∇2⃗𝐀=𝜇0⃗𝐉

where the Coulomb gauge ⃗∇ ⋅⃗𝐀 =0 further reduces the above to Poisson’s equation

⃗𝐁=⃗∇×⃗𝐀∇2⃗𝐀=−𝜇0⃗𝐉

so for localized sources the solution is

⃗𝐀(⃗𝐫)=𝜇04𝜋∭Ω⃗𝐉(⃗𝐫′)𝔯𝑑𝜏′⃗𝐀(⃗𝐫)=𝜇04𝜋∬Σ⃗𝐊(⃗𝐫′)𝔯𝑑𝑎′⃗𝐀(⃗𝐫)=𝜇04𝜋∫Π⃗𝐈(⃗𝐫′)𝔯𝑑𝑙′

for volume, surface, and line current density respectively. See also Multipole expansion of the magnetostatic potential.

Further properties

• Magnetostatic energy density
• Magnetostatic boundary conditions at a surface

Applications

• Magnet
  • Magnetic dipole moment
  • Perfect magnetic dipole
  • Magnetic potential of a magnetized material
  • Linear magnet
• H-field
  • Ampère’s law for magnets

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