Firstly, please note that I am talking about the period BEFORE electricity and magnetism were unified. So I am NOT seeking for answers based on Ampere atomic current model of magnets.

I have read the following statement about the property of magnets at two different places. One from here:

Definition of magnetic poles. If the magnet be a long thin cylindrical bar, uniformly magnetized in such a way that the magnetic axis is parallel to the length of the bar, the points in which the main axis cuts the ends of the bar are the magnetic poles. The end of the bar which tends to point north, when the magnet is freely suspended, is the north, or positive pole; the other is the south or negative pole. Such a magnet is called solenoidal, and behaves to other magnets as if the poles were centres of force, the rest of the magnet being devoid of magnetic action. In all actual magnets the magnetisation differs from uniformity. No two single points can strictly be taken as centres of force completely representing the action of the magnet. For many practical purposes, however, a well-made bar magnet may be treated as solenoidal with sufficient accuracy; that is to say, its action may be regarded as due to two poles or centres of force, on near each end of the magnet.

and the other from Maxwell's treatise Vol II, Article 373:

phenomena can be accounted for by supposing that like ends of the magnets repel each other, that unlike ends attract each other, and that the intermediate parts of the magnets have no sensible mutual action.

The ends of a long thin magnet are commonly called its Poles. In the case of an indefinitely thin magnet, uniformly magnetized throughout its length, the extremities act as centres of force and the rest of the magnet appears devoid of magnetic action. In actual magnets the magnetization deviates from uniformity, so that no single points can be taken as the poles. Coulomb, however, by using long thin rods magnetized with care, succeeded in establishing the law of force between two like magnetic poles*.

The repulsion between two like magnetic poles is in the straight line joining them, and is numerically equal to the product of the strengths of the poles divided by the square of the distance between them.

374.] This law, of course, assumes that the strength of each pole is measured in terms of a certain unit, the magnitude of which may be deduced from the terms of the law. The unit-pole is a pole which points north, and is such that, when placed at unit distance in air from another unit-pole, it repels it with unit of force, the unit of force being defined as in Art. 6. A pole which points south is reckoned negative.

If $m_1$ and $m_2$ are the strengths of two magnetic poles $l$ the distance between them, and $f$ the force of repulsion, all expressed numerically, then $$f = \frac{m_1m_2}{l^2}.$$

But if $[m]$, $[L]$ and $[F]$ be the concrete units of magnetic pole, length and force, then $$ f[F] = \left[ \frac{m}{L} \right] ^2\frac{m_1m_2}{l^2} , $$ whence it follows that $$ [m^2] = [L^2F] = [L^2 \frac{ML}{T^2}] $$ or $$ [m] = [L^\frac{3}{2} T^{-1} M^\frac{1}{2}]. $$ The dimensions of the unit pole are therefore $\frac{3}{2}$ as regards length, $(— 1)$ as regards time, and $\frac12$ as regards mass. These dimensions are the same as those of the electrostatic unit of electricity, which is specified in exactly the same way in Arts. 41, 42.

* His experiments on magnetism with the Torsion Balance are contained in the Memoirs of the Academy of Paris, 1780-9, and in Biot's Traité de Physique, tom. iii.

It says that in the magnetic pole model, only the surfaces at the extremities (poles) act as centers of force and the rest of the magnet appears free from magnetic action.

How can we ensure this? For example, why cannot we have half of the magnet's volume being north pole and the other half volume being south pole? Why do we necessarily have the surfaces at the extremities as centers of force instead of volumes (analogous to the charge model)?

  • $\begingroup$ Do u=you mean before Maxwell, or perhaps Ampere ? Coulomb was born in 1736 and I don't think you want to go back that far. $\endgroup$ – my2cts Sep 9 '18 at 11:10
  • $\begingroup$ What I really mean is before the unification of E&M by Ampere in 1820s. I used the term "before Coulomb" so as to avoid the Ampere/Maxwell confusion. $\endgroup$ – Joe Sep 9 '18 at 11:41
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    $\begingroup$ Please do not post images of texts you want to quote, but type it out instead so it is readable for all users and so that it can be indexed by search engines. For formulae, use MathJax instead. $\endgroup$ – Emilio Pisanty Sep 10 '18 at 14:56
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    $\begingroup$ I'm voting to close this question as off-topic because, as per the site guidelines, this question belongs on History of Science and Mathematics. $\endgroup$ – Emilio Pisanty Sep 10 '18 at 15:25
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    $\begingroup$ @Joe That's not the way closing questions works. The idea is that if the question in it's present state isn't appropriate, it's closed so that no new answers can be added. $\endgroup$ – user191954 Sep 11 '18 at 3:13

"Why do we necessarily have the surfaces at the extremities as centers of force instead of volumes (analogous to the charge model)?"

On your pre-Ampère model we must think of a magnet as being made of very many short magnets, end-to-end, that is with North pole of one next to the South pole of its neighbour. So, except at the ends of the magnet, the poles cancel with each other. [The non-existence of isolated poles was known at least since the time of Gilbert (c1600).]

This model is consistent with the appearance of 'new poles' when you snap a magnet in two; each 'half' has a North pole at one end and a South at the other.

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  • $\begingroup$ Mathematically speaking, does this mean magnetic poles have infinitesimal (very small) volume or just areas? Please also give a reason. $\endgroup$ – Joe Sep 9 '18 at 9:36
  • $\begingroup$ A pole is just some surface through which magnetic flux is going. North means the lines are emerging and south means that the lines are going in. This means that inside a bar magnet there are infinitely many north and south poles, all of them overlapping. Any surface through which nearly all of the flux passes can be considered a north or a south pole , depending from which side you are approaching it. The whole confusion is caused by the underlying desire to define magnetic charges, which do not exist. $\endgroup$ – my2cts Sep 9 '18 at 12:19
  • $\begingroup$ I wrote about magnetic poles quite recently, in response to the question "Definition of a magnetic field". Maybe the question and answers will be of some interest. $\endgroup$ – Philip Wood Sep 9 '18 at 12:30
  • $\begingroup$ That was me :):):) $\endgroup$ – Joe Sep 9 '18 at 12:31
  • $\begingroup$ Ah. It occurred to me that it might have been you after I'd written the comment! $\endgroup$ – Philip Wood Sep 9 '18 at 12:40

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