# Force on point magnetic dipole

Homework Helper
Hi guys, I'm involved in a research project regarding discrete bit patterned media and I was tasked to figure out how a magnetic force microscope works for imaging magnetic islands on thin magnetic films coated on silicon substrate, as part of a preparatory literature review.

So I pulled out a few books from the library (and a paper Boyer, 1987) and came across this which was derived for an loop current model and which they attempted to show was conceptually equivalent to the one derived for magnetic dipole under the assumption of no-currents
$$\mathbf{F} = \nabla (\mathbf{m} \cdot \mathbf{B} ) = \mathbf{m} \times (\nabla \times \mathbf{B} ) + \mathbf{B} \times (\nabla \times \mathbf{m}) + (\mathbf{m} \cdot \nabla) \mathbf{B} + (\mathbf{B} \cdot \nabla)\mathbf{m}$$.

Somehow according to the paper, this reduces to $$\mathbf{F} = \nabla (\mathbf{m} \cdot \mathbf{B} ) = \mathbf{m} \times (\nabla \times \mathbf{B} ) + (\mathbf{m} \cdot \nabla) \mathbf{B}$$ with the other two terms disappearing because, as the paper says that m doesn't depend on coordinates. What does that mean and why? I understand m is always perpendicular to the current loop (IdS, in fact). Further, a later assumption made was that $$\nabla \times \mathbf{B} = \mathbf{0}$$ but I don't see why we should assume that bound and free current is 0. When is this valid and why?

I've done only a second-year EE E&M course so far where magnetic dipoles was omitted so the lecturer could start on transmission lines. So please do point out where I can read up on this. Thanks a lot.

P.S. I've seen https://www.physicsforums.com/showthread.php?t=210771" as well, where pam says that curl B is only non-zero on the surface of a permanent magnet but I don't see why.

http://books.google.com/books?id=I-...&oi=book_result&ct=result&resnum=2#PPA102,M1"also seems to say curl B is nonzero at the point where the electric field is changing or alternatively when there is current or displacement current.

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its been a while but:

the curl of B? thats just current.

magnetization can be thought of as infinitesimal current loops (think 'tiny squares').
if the magnetization within the body of the permanent magnet (think 'grid of tiny squares') is constant (I'm guessing thats what it means by 'm doesn't depend on coordinates') then the currents cancel out everywhere except at the surface so that the magnet field of the permanent magnet can be though of as being due entirely to surface currents. hence curl is nonzero only at the surface.

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Homework Helper
Hi granpa. Yeah it's been I think about a semester since that I took that class. Amazingly short memory I have.

That explains a lot, so thanks for that. But it is assumed that the bound current (and hence the infinitesimal current loops) is 0, so how does that current model apply? We also assume that m is a constant vector (each magnetic dipole moment m). By definition of m, we assume that each m is due only to the bound current encircling a closed planar loop, so there's no reason to assume that m changes since neither I_b nor dS changes? But don't the dipoles interact with each other (just as you said currents cancel out everywhere) ? Doens't this cause I_b to change?

the bound current is zero in the interior because the magnetization is constant. the current in adjacent loops cancels out (if they are equal). it is nonzero on the surface.

picture 2 square loops adjacent to one another. imagine that each has a clockwise current. the current in the wire that they share is the sum of the 2 currents. the 2 currents are equal but in oppsite direction. so they cancel out completely.

the rest of your post I cant make much sense out of

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Further, a later assumption made was that $$\nabla \times \mathbf{B} = \mathbf{0}$$ but I don't see why we should assume that bound and free current is 0. When is this valid and why?
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I'm confused. wouldnt that just be the curl of the applied magnetic field. unless current is actually passing through the magnetic material then the curl will certainly be zero there.

crickets chirping.