Multipole expansion of the electrostatic potential

The electrostatic potential due to sources localized to $Ω$ has the multipole expansion

𝑉 (⃗ 𝐫) = 1 4 𝜋 𝜖 0 \infty \sum 𝑛 = 0 1 𝑟 𝑛 + 1 ∭ Ω (𝑟') 𝑛 𝑃 𝑛 (c o s 𝛼) 𝜌 (⃗ 𝐫') 𝑑 𝜏' = 1 4 𝜋 𝜖 0 𝑄 𝑟 + 1 4 𝜋 𝜖 0 ⃗ 𝐩 \cdot ˆ 𝐫 𝑟 2 + O (𝑟 - 4)

where $𝑄$ is the total charge (i.e. monopole moment) of the distribution, $⃗ 𝐩$ is the Electric 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 the reciprocal to the electrostatic potential one gets the expression above.

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Multipole expansion of the electrostatic potential

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