If we have an resting iron ball, and we put near a magnet it would be move and accelerate. I'm understanding that the magnetic field is a relativistic effect of electric field, and if I consider magnetic field as just electric, it would be ok. But what about classic physic? How does it explain it?
[Physics] Why do magnets attract of repel objects, if magnetic field only chages the direction of particles
electromagnetismmagnetic fieldsspecial-relativity
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There is a difference between your two cases. When you are talking about a charge passing between magnets you are thinking of it as a uniform magnetic field. But it is not uniform, it gets stronger as you approach the magnets. if it were a uniform field the magnetic dipole you put in the middle would not feel a force along the line of the magnets, although it would still experience a torque.
To see where the force along the magnets comes from, imagine a single charge moving in a circle with velocity always perpendicular to the line of the magnets. If it were a uniform magnetic field the Lorentz force would act perpendicular to the line of the magnets producing an outward radial force. This is what you realized.
But really the field is getting weaker as you go away from the magnets, and since the magnetic field has no divergence that means the field lines must expand outward as you go away from the magnets. You see this on any diagram of the magnetic field of a dipole. So if you look at what the Lorentz force on the charge moving in a circle is now the the magnetic field has a (possibly small) component outwards you will see the charge picks up a force along the line of the magnets (in addition to the original force outward).
The "lines" you see when viewing iron filings around a magnet have more to do with the fact that they are tiny slivers of iron, and less to do with magnetic field lines as one normally talks about them.
Also, over the length scale of one of these slivers, the magnetic field is largely constant, and a ferromagnet (or magnetic dipole) placed in a constant magnetic field will not accelerate (it will, however, align itself with the field). Once two slivers line themselves up head to tail, the field they create around them makes it more favorable for other slivers to join the chain rather than to lie haphazard, because the filings distort the field around them. So it is simply energetically preferred for these slivers to line up head to tail and form longer chains, but if you look closely, the chains break and merge.
Magnetic field lines are just a way of visualizing magnetic fields, in the same way that electric field lines are used to visualize electric fields (lines of force). There are no "gaps" between true magnetic field lines -- they occupy all space. We just draw them that way to convey a sense of their intensity.
I also don't quite agree with the statement that friction prevents them from clustering on the magnet. It's a bit more complicated than that, and, indeed, you can watch the same behaviour in air by suspending a magnet above the filings and allowing them to lift up. Once the filings start attaching themselves to the magnet, a magnetic circuit is created which changes how the field looks.
Best Answer
First, relativity isn't needed to understand macroscopic magnetic effects such as the attraction of large magnetic objects. That was explained and understood well before Einstein's work; the explanation culminated in Maxwell's equations, but the magneto-static parts were well understood earlier in the 19th century. Relativity changed none of that (it was mechanics that had problems, not E&M)
It's true that there wasn't an agreed-upon microscopic explanation of why bulk iron would "magnetize", but the effect was well understood in terms of behavior: both the math and material-specific data were in common use for engineering in the 1880's.
The misconception comes from a statement that you'll occasionally see in a text books now, usually right after the Lorentz force ($\vec{F} = q \vec{v} \times \vec{B}$) is introduced: "Because $\vec{F}$ is always perpendicular to $\vec{v}$, the Lorentz force can do no work". That's true for the simple case of an individual electron moving (slowly) in a completely static magnetic field, but it's misleading for the macroscopic case: it's far from the situation of two bulk magnets attracting each other.
The force between two magnets involves a lot of electrons in different places, not just one. The combination of the Lorentz forces on those results in a net force on the magnet which can do work. The traditional diagram for that is:
To say it another way, if the electrons move only under a Lorentz force, no work is done. But if they are also subject to mechanical forces which move them, then that motion may have a component along the Lorentz force, and work can be done.
But you don’t need to know electrons to understand this. A 19th century engineer or natural scientist ("physicist" wasn't in common use until very late) would understand attraction in a couple of ways:
There's more energy in the magnetic field(s) of two magnets oriented N-S when they're farther apart than when they're closer together: there's a mechanical force that turns that difference into work as the magnets move
A magnetic dipole in a non-uniform magnetic field feels a force. Each bit of the iron ball is a separate bit of magnet with its own dipole. Add those up, in the local field, to get the entire force.
Pre-relativity E&M worked with currents and fields ($\vec{E}$) and ($\vec{B}$) that, within the macroscopic domain, worked quite well. They didn't really know about the electron (1897) or the proton/nucleus(1911), let alone special relativity (1907), but they were able to explain and use magnetism quite well: The transformer dates back to the 1830's, for example.