Magnetic potential of a magnetized material

In Coulomb gauge, the potential due to a magnetized magnet $Ω$ is given by

⃗ 𝐀 (⃗ 𝐫) = 𝜇 0 4 𝜋 ∭ Ω ⃗ 𝐌 (⃗ 𝐫') \times ˆ 𝖗 𝔯 2 𝑑 𝜏' = 𝜇 0 4 𝜋 \subset \supset \iint 𝜕 Ω ⃗ 𝐊 𝑏 (⃗ 𝐫') 𝔯 𝑑 𝑎' + 𝜇 0 4 𝜋 ∭ Ω ⃗ 𝐉 𝑏 (⃗ 𝐫') 𝔯 𝑑 𝜏'

where $⃗ 𝐌$ is the magnetization, and $⃗ 𝐊 𝑏$ and $⃗ 𝐉 𝑏$ are the surface and volume bound current densities respectively given by

⃗ 𝐊 𝑏 (⃗ 𝐫') = ⃗ 𝐌 (⃗ 𝐫') \times ˆ 𝐧 ⃗ 𝐉 𝑏 (⃗ 𝐫') = ⃗ \nabla \times ⃗ 𝐌 (⃗ 𝐫')

If additional free current is present, the principle of superposition may be used to combine the potentials due to free current and due to magnetization (i.e. bound current).

Derivation

Applying ^GE5

⃗ 𝐀 (⃗ 𝐫) = 𝜇 0 4 𝜋 ∭ Ω ⃗ 𝐌 (⃗ 𝐫') \times ˆ 𝖗 𝔯 2 𝑑 𝜏' = 𝜇 0 4 𝜋 ∭ Ω ⃗ 𝐌 (⃗ 𝐫') \times ⃗ \nabla' [𝔯 - 1] 𝑑 𝜏' = 𝜇 0 4 𝜋 \subset \supset \iint Ω ⃗ 𝐌 (⃗ 𝐫') \times 𝑑 ⃗ 𝐚 𝔯 𝑑 𝜏' + 𝜇 0 4 𝜋 ∭ Ω ⃗ \nabla \times ⃗ 𝐌 (⃗ 𝐫') 𝔯 𝑑 𝜏'

as required.

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