Maxwell’s equations in materials

Ampère’s law for magnets

It follows from Ampère’s circuital law that the auxiliary H-field satisfies

⃗∇×⃗𝐇=⃗𝐉𝑓+𝜕⃗𝐃𝜕𝑡

where ⃗𝐉𝑓 =⃗𝐉 −⃗𝐉𝑏 −𝜕⃗𝐏𝜕𝑡 is the free current density.

Derivation

Noting the expression for ^bcd

⃗∇×⃗𝐇=1𝜇0⃗∇×⃗𝐁−⃗∇×⃗𝐌=⃗𝐉+𝜖0𝜕⃗𝐄𝜕𝑡−⃗𝐉𝑏=⃗𝐉𝑓+𝜕⃗𝐏𝜕𝑡+𝜖0𝜕⃗𝐄𝜕𝑡=⃗𝐉𝑓+𝜕⃗𝐃𝜕𝑡

where we used the fact that changing electric polarization density is a source of current

⃗𝐉=⃗𝐉𝑓+⃗𝐉𝑏+𝜕⃗𝐏𝜕𝑡

as required.

or equivalently by Stokes’s theorem

∮𝜕Σ⃗𝐇⋅𝑑⃗𝐫=(∬Σ⃗𝐉𝑓⋅𝑑⃗𝐚+𝑑𝑑𝑡∬Σ⃗𝐃⋅𝑑⃗𝐚)=𝜇0(𝐼𝑓,Σ+𝜕Φ𝐷,Σ𝜕𝑡)

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