# Small question about maxwell's equation curl of H

The first page of this short pdf from MIT sums the starting point to formulate my question:
https://ocw.mit.edu/courses/physics...gnetism-fall-2006/lecture-notes/lecture29.pdf

We can see that
∇xH = Jfree
and
∇xB o (Jfree +∇xM)
∇xB o (Jfree +JB)

And now my question is, if KB ≠ 0 , how I can't see its contribution to B in the last equation?
I have been solving problems in class and in them it appears a contribution to B , after using the curl theorem, like this
∫KB·dl

jtbell
Mentor
I assume that by ##\vec K_B## you mean the bound surface currents on a magnetized material. These are a subset of the bound current density ##\vec J_B##, namely the bound currents that lie on the surface of the material (idealized as an infinitesimally thin sheet) rather than being distributed through the volume of its interior.

Mathematically I think you might incorporate ##\vec K_B## into the last equation by multiplying it by a Dirac delta function that represents the surface, thereby converting it to a volume current density.

• Homework Helper
Gold Member
The evaluation of ## \nabla \times \vec{M} ## includes surface currents, and, as by Stokes theorem, the discontinuity in ## \vec{M} ## at the boundary results in a surface current per unit length ## \vec{K}_m=\vec{M} \times \hat{n} ##, that comes from ## \nabla \times \vec{M}=\vec{J}_m ##. ## \\ ## Basically, this derivation, (of ## \nabla \times \vec{H}=\vec{J}_{free} ##), starts with ## \vec{B}=\mu_o (\vec{H}+\vec{M}) ##, and you take the curl of both sides. The derivation of the equation ## \vec{B}=\mu_o (\vec{H}+\vec{M}) ## can be somewhat lengthy, and the derivation is often omitted in many E&M textbooks. They like to use the analogous electrostatic equation ## \vec{D}=\epsilon_o \vec{E} +\vec{P} ##, to justify it.

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Thank's both, and sorry for replying late. I thing I understood the general idea of your replies but for the moment Im going to look up in the bibliography the said discontinuity at the boundary.

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