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## Main Question or Discussion Point

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

[tex]\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}[/tex].

Somehow according to the paper, this reduces to [tex]\mathbf{F} = \nabla (\mathbf{m} \cdot \mathbf{B} ) = \mathbf{m} \times (\nabla \times \mathbf{B} ) + (\mathbf{m} \cdot \nabla) \mathbf{B}[/tex] 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 [tex]\nabla \times \mathbf{B} = \mathbf{0}[/tex] 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-x1MLny6y8C&pg=PA102&lpg=PA102&dq=curl+B+non-zero&source=bl&ots=ykJ9TvbyB4&sig=o2aArVM1JK_dcGa9LXWJ7CyF0Gg&hl=en&ei=RMMaSsiXF4SVkAW32PT4DA&sa=X&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.

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

[tex]\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}[/tex].

Somehow according to the paper, this reduces to [tex]\mathbf{F} = \nabla (\mathbf{m} \cdot \mathbf{B} ) = \mathbf{m} \times (\nabla \times \mathbf{B} ) + (\mathbf{m} \cdot \nabla) \mathbf{B}[/tex] 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 [tex]\nabla \times \mathbf{B} = \mathbf{0}[/tex] 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-x1MLny6y8C&pg=PA102&lpg=PA102&dq=curl+B+non-zero&source=bl&ots=ykJ9TvbyB4&sig=o2aArVM1JK_dcGa9LXWJ7CyF0Gg&hl=en&ei=RMMaSsiXF4SVkAW32PT4DA&sa=X&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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