A twist on Maxwell's equations boundary conditions

[Ahmad Kishki]

we have that Ht1 (x,y,z) - Ht2 (x,y,z) = Js and for the special case Ht1 (x,y,z) - Ht2 (x,y,z) = 0 where there is no surface current. At a boundary with Js =0, which for simplicity let's asume is at at x = a, then knowing that Ht1 and Ht2 are the magnetic fields to the left and right of the boundary respectively, we can then re write the boundary condition as lim e->0 Ht1 (a-e,y,z) - Ht2 (a+e,y,z) = 0 from which. Ht'(a,y,z) = 0 would follow if Ht1 (x,y,z) = Ht2 (x,y,z)?

 

[EM_Guy]

Ahmad, I'm not sure that I follow you. Are you asking whether the partial derivative of the tangential H field with respect to x is equal to zero? If so, I see no reason that it needs to be.

What is the greater context of your question? You are talking about boundary conditions of magnetic fields. In general, the difference between the tangential magnetic fields on either side of the boundary is equal to the current density on the boundary - as you have already pointed out. When there are no sources or charges, the difference between the tangential magnetic fields on either side of the boundary is equal to zero. If the material on one side of the boundary is a perfect electric conductor (PEC), then the tangential magnetic field at the boundary is equal to the surface current density on the conductor (noting that no field can exist inside of a PEC).