Perfect electric dipole

A perfect electric dipole with electric dipole moment $⃗ 𝐩$ is an idealized charge distribution in electrostatics whose Multipole expansion of the electrostatic potential has all terms vanish except for the dipole term. Thus the potential due to a perfect dipole located at $⃗ 𝐫'$ is

𝑉 (⃗ 𝐫) = 1 4 𝜋 𝜖 0 ⃗ 𝐩 \cdot ˆ 𝖗 𝔯 2

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

𝑉 (⃗ 𝐫) = 1 4 𝜋 𝜖 0 ∭ Ω ⃗ 𝐏 (⃗ 𝐫') \cdot ˆ 𝖗 𝔯 2 𝑑 𝜏'

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

#state/tidy | #lang/en | #SemBr