Bound surface charge density on a boundary with polarization P is given by which expression?

Bound surface charge density on a boundary with polarization P is given by which expression?

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

Answer: (A)

Bound charges arise where the polarization ends at a boundary. The amount of bound charge per unit area on that surface is the component of the polarization that points outward through the surface, written as σ_b = P · n̂, with n̂ the outward normal. This convention matches the standard relations ρ_b = −∇ · P for volume bound charge and σ_b = P · n̂ for surface bound charge, which can be seen by imagining a small pillbox straddling the boundary and applying the divergence theorem. If the polarization stops at the boundary (P goes to zero outside), the surface term is the projection of P along the outward normal, hence the positive sign in σ_b = P · n̂. The other options don’t describe a surface bound charge density: ∇ · P gives the volume bound charge density, and P × n̂ is not a charge density (it relates to other quantities like bound currents in different contexts).

Bound charges arise where the polarization ends at a boundary. The amount of bound charge per unit area on that surface is the component of the polarization that points outward through the surface, written as σ_b = P · n̂, with n̂ the outward normal. This convention matches the standard relations ρ_b = −∇ · P for volume bound charge and σ_b = P · n̂ for surface bound charge, which can be seen by imagining a small pillbox straddling the boundary and applying the divergence theorem. If the polarization stops at the boundary (P goes to zero outside), the surface term is the projection of P along the outward normal, hence the positive sign in σ_b = P · n̂. The other options don’t describe a surface bound charge density: ∇ · P gives the volume bound charge density, and P × n̂ is not a charge density (it relates to other quantities like bound currents in different contexts).