What is the boundary condition for the tangential component of the magnetic field H across a boundary with a surface current K?

What is the boundary condition for the tangential component of the magnetic field H across a boundary with a surface current K?

• A. n × (H2 − H1) = K_free
• B. n · (H2 − H1) = K_free
• C. H2_t − H1_t = K_free
• D. n × (E2 − E1) = K_free

Answer: (A)

The tangential component of the magnetic field can jump across a boundary if there is a surface current flowing along that boundary. This comes from Ampere-Maxwell’s law when you look at a tiny loop that straddles the boundary: the line integral of H around the loop equals the free current crossing the loop. As the loop height shrinks, only the sides parallel to the boundary contribute, giving a difference in the tangential H components on the two sides that equals the surface current density flowing along the boundary. Mathematically, this is written as n × (H2 − H1) = K_free, where n is the unit normal to the boundary pointing from the first medium to the second, H1 and H2 are the magnetic fields just on each side, and K_free is the surface current density. So, this relation correctly captures why the tangential part of H can have a discontinuity equal to the surface current. The other forms mix normal components, treat the difference as a scalar, or bring in E-field terms, which do not describe the boundary behavior of H with a surface current.

The tangential component of the magnetic field can jump across a boundary if there is a surface current flowing along that boundary. This comes from Ampere-Maxwell’s law when you look at a tiny loop that straddles the boundary: the line integral of H around the loop equals the free current crossing the loop. As the loop height shrinks, only the sides parallel to the boundary contribute, giving a difference in the tangential H components on the two sides that equals the surface current density flowing along the boundary. Mathematically, this is written as n × (H2 − H1) = K_free, where n is the unit normal to the boundary pointing from the first medium to the second, H1 and H2 are the magnetic fields just on each side, and K_free is the surface current density.

So, this relation correctly captures why the tangential part of H can have a discontinuity equal to the surface current. The other forms mix normal components, treat the difference as a scalar, or bring in E-field terms, which do not describe the boundary behavior of H with a surface current.