Perfect magnetic dipole

A perfect magnetic dipole with magnetic dipole moment $⃗ 𝐦$ is an idealized current distribution in magnetostatics whose Multipole expansion of the magnetostatic potential has all terms vanish except for the dipole term. Thus the potential in Coulomb gauge due to a perfect dipole located at $⃗ 𝐫'$ is

⃗ 𝐀 = 𝜇 0 4 𝜋 ⃗ 𝐦 \times ˆ 𝖗 𝔯 2

and given a continuous distribution of perfect dipoles $⃗ 𝐌$ localized to $Ω$

⃗ 𝐀 = 𝜇 0 4 𝜋 ∭ Ω ⃗ 𝐌 (⃗ 𝐫') \times ˆ 𝖗 𝔯 2 𝑑 𝜏'

A perfect dipole arises by considering a physical dipole $⃗ 𝐦 = 𝐼 ⃗ 𝐚$ in the limit as $⃗ 𝐚 \to ⃗ 𝟎$ and $𝑞 \to \infty$ so that $⃗ 𝐦$ remains fixed.