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Magnetic fields are represented by field lines and it is stated that these lines are closed lines, going through the source (often shown as lines between the two poles of the source).

Approaching two sources with the same poles, the magnetic fields are represented by field lines, which deform each other.

But what happens in the case of the approach of opposite poles? In which point of space the two fields start to build the common field?

Answering, please consider, that for case two, you may rotate one of the magnets from case one.

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  • $\begingroup$ Try rephrasing the question with more clarity. $\endgroup$ – TechDroid Mar 2 at 9:55
  • $\begingroup$ @John Magnetic fields extended to infinity, but the still could be represented there as closed lines. How they break to build a common line? $\endgroup$ – HolgerFiedler Mar 2 at 10:03
  • $\begingroup$ Magnetic field lines are closed lines only in very particular cases. In general, a magnetic line can rotate indefinitely without being closed. $\endgroup$ – Vincent Fraticelli Mar 2 at 11:14
  • $\begingroup$ Magnetic field lines are just a visualisation of the magnetic field. The magnetic field itself is the superposition of the fields of the individual magnets. Note that that the magnets may polarise each other, so the total field is not the superposition of the fields of the isolated magnets. $\endgroup$ – my2cts Mar 2 at 11:47
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I believe that what you are asking is answered by the following image (from https://phys.libretexts.org) enter image description here

Your case 2 would be image (b), and case 1 would be image (c). These field lines can be found numerically using Maxwell's equation $\text{div} \vec{B} = 0$.

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  • $\begingroup$ As drawn the fields due each of the facing poles increase as the poles get closer.. So do the magnets poles get stronger when they get closer together? $\endgroup$ – Farcher Mar 2 at 11:59
  • $\begingroup$ Roman, try to rotate from (c) to (b) and draw a situation in between $\endgroup$ – HolgerFiedler Mar 2 at 13:29
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It is useful to divide the problem up.

How do the fields vary?

First understand that each configuration of currents (be they classical or quantum generates a particular contribution to the magnetic field. Call such a contribution a "source field".

In your case each magnet generates a source field that is roughly an extended dipole (and the lines do run as loops: the figure has simply neglects to show the lines inside the magnets). These sources field move and re-orient as you move the magnet but they are unaffected by other magnets (up to a point, anyway).

The magnetic field is the sum of all source fields.1

Magnetic fields are vectors2 so the word "sum" means "vector sum".

How do the field lines look?

Field lines are a visualization of fields.

They are drawn so that:

  • At every point on a drawn line the field is tangent to the line.
  • In any given region of space the strength of the electric field is proportional to the local density of field lines3

This representation can be quite lossy if you use too few lines (and most figures use too few to show much detail). Drawing a representative set of field lines from an arbitrary field is a bit of an art.

But if you have lost information in draw the lines, then you can't properly reconstruct the sum staring with only the line drawings and can't "add" the field line drawing directly.

How to construct a visualization of the sum?

The proper way to construct a detailed and accurate field-line maps for interactions between the magnets is to start by measuring or approximating the source fields,4 then add them in vector arithmetic, then draw a suitable set of field lines of the sum.

1 Usually we restrict the sum to strong, nearby sources for practical reasons, but in principle you should be summing over the retarded contribution of all sources in the historical light-cone of your event of interest. Luckily distant contributions tend to be small enough that we really don't care.

2 In the introductory treatment, anyway. A move complete treatment makes them a skew-symmetric portion of the electromagnetic field tensor, but the ideas here-in can be extended to that treatment in a straightforward way.

3 I believe that to get the density bit quantitatively right you actually have to draw your lines in three-dimensions, but we just fudge it.

4 With a sufficiently dense drawing of field line you can make a good approximation of the field from the line-representation. Thus the weasel words about not being able to add field-line drawing directly.

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  • $\begingroup$ In which point of space the two fields start to build the common field? Where the loop get reassembled? $\endgroup$ – HolgerFiedler Mar 3 at 5:29
  • $\begingroup$ At every point in space all the time (though again, we often ignore small enough contributions for reasons of convenience). And there are no physical loops that have to get re-built: there are just vector fields and a visualization of a vector field. $\endgroup$ – dmckee♦ Mar 3 at 15:47
  • $\begingroup$ Ok, in case (c) of Romans answer both vector fields are limited by a infinite line between them. How this line “collapses” during the rotation of one of the magnets? $\endgroup$ – HolgerFiedler Mar 4 at 11:15

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