Ampère's law for magnets

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

⃗ \nabla \times ⃗ 𝐇 = ⃗ 𝐉 𝑓 + 𝜕 ⃗ 𝐃 𝜕 𝑡

where $⃗ 𝐉 𝑓 = ⃗ 𝐉 - ⃗ 𝐉 𝑏 - 𝜕 ⃗ 𝐏 𝜕 𝑡$ is the free current density.

Derivation

Noting the expression for bound current density

⃗ \nabla \times ⃗ 𝐇 = 1 𝜇 0 ⃗ \nabla \times ⃗ 𝐁 - ⃗ \nabla \times ⃗ 𝐌 = ⃗ 𝐉 + 𝜖 0 𝜕 ⃗ 𝐄 𝜕 𝑡 - ⃗ 𝐉 𝑏 = ⃗ 𝐉 𝑓 + 𝜕 ⃗ 𝐏 𝜕 𝑡 + 𝜖 0 𝜕 ⃗ 𝐄 𝜕 𝑡 = ⃗ 𝐉 𝑓 + 𝜕 ⃗ 𝐃 𝜕 𝑡

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

⃗ 𝐉 = ⃗ 𝐉 𝑓 + ⃗ 𝐉 𝑏 + 𝜕 ⃗ 𝐏 𝜕 𝑡

as required.

or equivalently by Stokes's theorem

\oint 𝜕 Σ ⃗ 𝐇 \cdot 𝑑 ⃗ 𝐫 = (\iint Σ ⃗ 𝐉 𝑓 \cdot 𝑑 ⃗ 𝐚 + 𝑑 𝑑 𝑡 \iint Σ ⃗ 𝐃 \cdot 𝑑 ⃗ 𝐚) = 𝜇 0 (𝐼 𝑓, Σ + 𝜕 Φ 𝐷, Σ 𝜕 𝑡)

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