Diëlectric

Electric potential of a polarized material

The potential due to a polarized diëlectric Ω is given by

𝑉(⃗𝐫)=14𝜋𝜖0∭Ω⃗𝐏(⃗𝐫′)⋅ˆ𝖗𝔯2𝑑𝜏′=14𝜋𝜖0⊂⊃∬𝜕Ω𝜎𝑏(⃗𝐫′)𝔯𝑑𝑎′+14𝜋𝜖0∭Ω𝜌𝑏(⃗𝐫′)𝔯𝑑𝜏′

where ⃗𝐏 is the electric polarization density, and 𝜎𝑏 and 𝜌𝑏 are the surface and volume bound charge densities respectively given by

𝜎𝑏(⃗𝐫′)=⃗𝐏(⃗𝐫′)⋅ˆ𝐧𝜌𝑏(⃗𝐫′)=−⃗∇⋅⃗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)=14𝜋𝜖0∭Ω⃗𝐏(⃗𝐫)⋅ˆ𝖗𝔯2𝑑𝜏′=14𝜋𝜖0∭Ω⃗𝐏(⃗𝐫)⋅⃗∇′[𝔯−1]𝑑𝜏′=14𝜋𝜖0⊂⊃∬𝜕Ω⃗𝐏(⃗𝐫′)⋅𝑑⃗𝐚𝔯−14𝜋𝜖0∭Ω⃗∇′⋅⃗𝐏(⃗𝐫′)𝔯𝑑𝜏′

as required.

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