Electric dipole moment

The electric dipole moment $⃗ 𝐩$ measures the separation of positive and negative charges within a system, determining the dipole term in the Multipole expansion of the electrostatic potential. In general, for a charge distribution $𝜌$ localized to $Ω$ the electric dipole moment is given by

⃗ 𝐩 (⃗ 𝐫) = ∭ Ω ⃗ 𝖗 𝜌 (⃗ 𝐫') 𝑑 𝜏'

In the case of a point charge $𝑄$ at position $⃗ 𝐫$ , we have $⃗ 𝐩 = 𝑄 ⃗ 𝐫$ . It is useful to introduce the concept of electric polarization density $⃗ 𝐏$ , so that the dipole moment due to a volume $Ω$ is given by

⃗ 𝐩 = ∭ Ω ⃗ 𝐏 (⃗ 𝐫') 𝑑 𝜏'

Analysis of the Electric potential of a polarized material motivates the surface and volume bound charge densities

𝜎 𝑏 (⃗ 𝐫') = ⃗ 𝐏 (⃗ 𝐫') \cdot ˆ 𝐧 𝜌 𝑏 (⃗ 𝐫') = - ⃗ \nabla \cdot ⃗ 𝐏 (⃗ 𝐫')

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Electric dipole moment

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