Derive magnetic boundary conditions

Boundary conditions give the relations between the field components just on either side of the interface between two media. Take region 1 (μ₁) and region 2 (μ₂), with a_N pointing from region 2 into region 1. We resolve every field into a normal and a tangential part and use the two integral laws.

Normal component of B

Use a small cylindrical pillbox across the boundary, with top and bottom faces of area ΔS in each medium and height Δh → 0. The magnetic Gauss law is ∮ B·dS = 0:

B₁n ΔS − B₂n ΔS + (side contribution → 0) = 0
⇒ B₁n = B₂n
⇒ μ₁H₁n = μ₂H₂n

The normal component of B is continuous; the normal component of H is not (H₁n/H₂n = μ₂/μ₁).

Tangential component of H

Take a small rectangular loop abcda of width Δw straddling the boundary, with height Δh → 0. Let the surface current density at the boundary be K (A/m), perpendicular to the loop plane. Ampere's law ∮ H·dl = I_enc:

H₁t Δw − H₂t Δw + (sides → 0) = K Δw
⇒ H₁t − H₂t = K

In vector form: a_N12 × (H₁ − H₂) = K, where a_N12 is the unit normal from region 2 to region 1.

Corresponding statement for B: B₁t/μ₁ − B₂t/μ₂ = K.

Special case

If there is no surface current (K = 0, true for most real interfaces, since conductivities are finite):

H₁t = H₂t         (tangential H continuous)
B₁t / μ₁ = B₂t / μ₂

Summary

These conditions also give the refraction law of magnetic field lines: tan θ₁ / tan θ₂ = μ₁/μ₂.