Maxwell's equations in materials

Gauß's law for diëlectrics

It follows from Gauß's law that the auxiliary field of Electric displacement ⃗𝐃 satisfies

⃗∇⋅⃗𝐃=𝜌𝑓

where 𝜌𝑓 is the free charge density.

Derivation

Noting the expression for bound charge density,

⃗∇⋅⃗𝐃=𝜖0⃗∇⋅⃗𝐄+⃗∇⋅⃗𝐏=𝜌−𝜌𝑏=𝜌𝑓

as claimed.

Using Острогра́дский's divergence theorem we see

⊂⊃∬𝜕Ω⃗𝐃⋅𝑑⃗𝐚=∭Ω𝜌𝑓𝑑𝜏′

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