What is the relation between bound charge density ρ_b and polarization P?

• A. ρ_b = −∇ · P; σ_b = P · n̂.
• B. ρ_b = ∇ · P; σ_b = −P · n̂.
• C. ρ_b = ∇ × P; σ_b = P × n̂.
• D. ρ_b = −∇ × P; σ_b = −P · n̂.

Answer: (A)

Bound charges come from how the polarization field P varies and ends at surfaces. The volume bound charge density is given by the negative divergence of P, ρ_b = -∇·P. This means where P lines spread out (diverge), you get negative bound charge, and where they converge, you get positive bound charge. At the boundaries of a dielectric, the normal component of polarization cannot vanish abruptly, producing a surface bound charge density equal to the normal component of P, σ_b = P · n̂, where n̂ is the outward normal. So if P has a component pointing out of the surface, you get positive bound charge there; if it points inward, you get negative. This combination—ρ_b = -∇·P for the bulk and σ_b = P · n̂ at surfaces—summarizes how polarization translates into bound charges.