What is the boundary condition for the normal component of the electric displacement D across a boundary with surface free charge σ_free?

What is the boundary condition for the normal component of the electric displacement D across a boundary with surface free charge σ_free?

• A. D2_n − D1_n = σ_free
• B. D2_t − D1_t = σ_free
• C. E2_n − E1_n = σ_free/ε0
• D. D2_n + D1_n = σ_free

Answer: (A)

The normal component of the electric displacement D has a jump equal to the surface free charge density at a boundary. This comes from Gauss’s law in differential form, ∇·D = ρ_free. Imagine a tiny Gaussian pillbox straddling the boundary, with its faces parallel to the boundary and area A. The flux through the pillbox is D2_n A on the side in medium 2 minus D1_n A on the side in medium 1, and by Gauss’s law this must equal the free charge on the surface, σ_free A. Dividing by A gives D2_n − D1_n = σ_free. Interpretation: the discontinuity in the normal component of D is sourced only by free charges on the boundary; bound charges (related to polarization) are already included in how E behaves through the relation D = ε0E + P. If there are no free charges on the surface, the normal component of D is continuous across the boundary.

The normal component of the electric displacement D has a jump equal to the surface free charge density at a boundary. This comes from Gauss’s law in differential form, ∇·D = ρ_free. Imagine a tiny Gaussian pillbox straddling the boundary, with its faces parallel to the boundary and area A. The flux through the pillbox is D2_n A on the side in medium 2 minus D1_n A on the side in medium 1, and by Gauss’s law this must equal the free charge on the surface, σ_free A. Dividing by A gives D2_n − D1_n = σ_free.

Interpretation: the discontinuity in the normal component of D is sourced only by free charges on the boundary; bound charges (related to polarization) are already included in how E behaves through the relation D = ε0E + P. If there are no free charges on the surface, the normal component of D is continuous across the boundary.