Bound surface charge density on a dielectric surface is related to polarization by which expression?

Bound surface charge density on a dielectric surface is related to polarization by which expression?

• A. σ_b = P · n̂
• B. σ_b = E · n̂
• C. σ_b = P × n̂
• D. σ_b = 0

Answer: (A)

Bound charges on a dielectric surface come from where the polarization ends at the boundary. Polarization P is the dipole moment per unit volume, and at the surface the portion of P that points outward ends up as a bound surface charge density equal to the projection of P along the outward normal: σ_b = P · n̂. This result can be seen by considering a tiny pillbox straddling the surface; for a uniform P the bound volume charge ρ_b = -∇·P vanishes inside the bulk, and the net charge on the surface equals the normal component of P times the area, i.e., σ_b = P · n̂. If there is no normal component of P, there is no bound surface charge; if P has a normal component in the outward direction, σ_b takes the corresponding sign. The other options don’t describe bound surface charge: the normal component of E is not the bound charge, a cross product would yield a vector rather than a scalar charge density, and zero is not generally true.

Bound charges on a dielectric surface come from where the polarization ends at the boundary. Polarization P is the dipole moment per unit volume, and at the surface the portion of P that points outward ends up as a bound surface charge density equal to the projection of P along the outward normal: σ_b = P · n̂. This result can be seen by considering a tiny pillbox straddling the surface; for a uniform P the bound volume charge ρ_b = -∇·P vanishes inside the bulk, and the net charge on the surface equals the normal component of P times the area, i.e., σ_b = P · n̂. If there is no normal component of P, there is no bound surface charge; if P has a normal component in the outward direction, σ_b takes the corresponding sign. The other options don’t describe bound surface charge: the normal component of E is not the bound charge, a cross product would yield a vector rather than a scalar charge density, and zero is not generally true.