Electric potential of a polarized material

The potential due to a polarized diëlectric $Ω$ is given by

𝑉 (⃗ 𝐫) = 1 4 𝜋 𝜖 0 ∭ Ω ⃗ 𝐏 (⃗ 𝐫') \cdot ˆ 𝖗 𝔯 2 𝑑 𝜏' = 1 4 𝜋 𝜖 0 \subset \supset \iint 𝜕 Ω 𝜎 𝑏 (⃗ 𝐫') 𝔯 𝑑 𝑎' + 1 4 𝜋 𝜖 0 ∭ Ω 𝜌 𝑏 (⃗ 𝐫') 𝔯 𝑑 𝜏'

where $⃗ 𝐏$ is the electric polarization density, and $𝜎 𝑏$ and $𝜌 𝑏$ are the surface and volume bound charge densities respectively given by

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

If additional free charge is present, the principle of superposition may be used to combine the potentials due to free charge and due to polarization (i.e. bound charge).

Derivation

Applying ^GE1

𝑉 (⃗ 𝐫) = 1 4 𝜋 𝜖 0 ∭ Ω ⃗ 𝐏 (⃗ 𝐫) \cdot ˆ 𝖗 𝔯 2 𝑑 𝜏' = 1 4 𝜋 𝜖 0 ∭ Ω ⃗ 𝐏 (⃗ 𝐫) \cdot ⃗ \nabla' [𝔯 - 1] 𝑑 𝜏' = 1 4 𝜋 𝜖 0 \subset \supset \iint 𝜕 Ω ⃗ 𝐏 (⃗ 𝐫') \cdot 𝑑 ⃗ 𝐚 𝔯 - 1 4 𝜋 𝜖 0 ∭ Ω ⃗ \nabla' \cdot ⃗ 𝐏 (⃗ 𝐫') 𝔯 𝑑 𝜏'

as required.

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