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There is no such thing as a magnetic force. It is a useful fiction, part of what is actually the electromagnetic field.

What appears to be a magnetic field is really just electric fields changing in direction or intensity. If you analyze the situation using Special Relativity, you can see that it has exactly the effects we see experimentally.

Maxwell, a brilliant man, working with the knowledge we had then and experimental observations, came up with four equations

that exactly describe electric and magnetic fields. But no one had discovered relativity, just yet.

Picture two wires parallel to each other, a short distance apart. They have overall neutral charge, having as many electrons as protons.

Now say there is a current of electrons flowing through each wire, in opposite directions. Still overall neutral.

But look at it from the reference frame of an individual electron. All of its brethren moving with it appear normal. But the electrons moving in the opposite direction in the other wire appear closer together, due to the Lorentz Contraction

, which is length contraction due to special relativity. To those electrons, the electrons in this wire appear closer together. Well, the protons in each wire also appear closer together (from the viewpoint of the moving electrons) due to Lorentz Contraction, but at half the velocity (simplification).

So to the moving electrons, the other wire appears to have more negative charges than positive, and so are repelled. The positive nucleus are pulled along.

Notice that at no point have I invoked a magnetic field.

If I calculate the effects from this (many people smarter than me have done so), the repulsion between the wires agrees exactly with Maxwell's equations, and with experimental results to many, many decimal places

. If you redo this example with the electrons moving in the same direction, the wires attract as experimental results show.

You may be thinking, well, drift velocity of electrons is very slow even for very high currents. But there are MASSIVE numbers of them that are moving, 6.24x10^18 for every ampere of current. And the electric field is VERY powerful, so it all adds up. And in fact the magnetic fields created are relatively weak. We have to wind many turns around an electromagnet and put an iron core in it to boost the effects. Every turn around a coil is adding to the field, even though it is the same current. So two wires in a loop with 1A current flowing is the same "magnetic" field effect as one thicker wire with 2A current flowing.

And it isn’t uncommon to make an electromagnet with a thousand turns or more, and add an iron core to further multiply the effect.

TL;DR There is no magnetic force. It is moving electric charges and Special Relativity that make it appear as if there is something that we call a magnetic force. But doing the calculations with the older equations ignoring Special Relativity are much simpler, and understanding it as separate fields is simpler, so we continue to teach it that way. That, and inertia of minds.

This predicts radio waves, too, just as Maxwell's Equations did. Slosh electrons back and forth, as we do in antennas, and the changes in the angle of the electric field propagate away from the antenna at the speed of light. This moving electric field and Special Relativity cause that thing that appears to be a magnetic field at 90 degrees to the moving electric field.

More information:

Relativistic electromagnetism

See the top answer by Chris White:

How do moving charges produce magnetic fields?

For a much deeper understanding, I suggest this playlist of MIT lectures:

8.02 Physics II: Electricity and Magnetism, Spring 2002 (Complete Lectures by Walter Lewin)

Or just this one about magnetic fields and relativity:

So that is how an electric current creates a force that looks like a magnetic field. I don't know if anyone can answer "why", it depends on what you mean by "why".

Also posted as How do moving charges, such as electric current, produce a magnetic field?

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