Multipole expansion of the magnetic potential

In Coulomb gauge, the magnetic potential due to current $𝐼$ around a closed loop $𝜕 Σ$ has the multipole expansion

⃗ 𝐀 (⃗ 𝐫) = 𝜇 0 𝐼 4 𝜋 \infty \sum 𝑛 = 1 1 𝑟 𝑛 + 1 \oint 𝜕 Σ (𝑟') 𝑛 𝑃 𝑛 (c o s 𝛼) 𝑑 ⃗ 𝐥 = 𝜇 0 4 𝜋 ⃗ 𝐦 \times ˆ 𝐫 𝑟 2 + O (𝑟 - 4)

where $⃗ 𝐦$ is the magnetic dipole moment and $𝛼$ is the angle between $⃗ 𝐫$ and $⃗ 𝐫'$ so that $⃗ 𝐫 \cdot ⃗ 𝐫' = 𝑟 𝑟' c o s 𝛼$ . This may be combined with the principle of superposition.

Derivation

By applying the Multipole expansion of $1 / 𝔯$ to the magnetostatic potential on a closed loop $𝜕 Σ$ , and noting that the $𝑛 = 0$ term vanishes since there are no magnetic monopoles we get the above expression.

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