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Does a magnetic field have concentrations of magnetic force lines as seen when putting iron filings over top a bar magnet or are these imaginary? I.e. are they just an artifact of the iron being a 'conductor' of the magnetic field lines making them look like they are concentrated along these path lines but are really continuum of strength around the bar magnet when there are no filings are present?

Also I learned that field lines do not cross, yet there are magnet configurations who's forces are explained as the magnetic force vectors are indeed crossing and are additive such as a Halbach Magnet Array. So what is actually happening here?

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Field lines are a visualization tool for the magnetic field (and for other fields, too). They are a way of representing the field (which is a physical---if intangible---thing) in a drawing. They are not unique in this: there are other ways of representing vector fields, but they have a long history.

Now, magnetic fields are vector fields subject to superposition, which means that the total field can be given by adding up the contributions for multiple fields. If you visualize each on the contributions separately then the field lines from one may cross the field lines from another, but it remains true that

  • No single contribution will be represented with lines that cross
  • The net (total) field will not be represented with lines that cross
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    $\begingroup$ What are the two other ways of representing vector fields? $\endgroup$ – Adrian Dec 15 '14 at 4:07

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