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Several people have told me that a current-carrying wire usually (or always) repels a nearby permanent magnet....

Most recently, I saw this Veritasium video on YouTube: 'How Special Relativity Makes Magnets Work'. At 2:40 to 2:45 of the video, the host says that, 'A wire with current in it deflects nearby magnets'...

But why? Why would a wire with current always, or even usually, deflect another magnet? Rather than attract it?

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"Deflects" here is not the same as "repel" and is not the opposite of "attract". The orientation of a magnetic needle is what may change in the presence of a current. This was what Oersted obsrved in 1820. If there is a net force it will depend on the gradient of the field. So, I think you missinterpret what the video intends to say when you take it to mean repulsion only.

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It's the Lorentz force at work $\vec{F} = q\vec{v} \times \vec{B}$, not magnetic attraction or repulsion per se as you might imagine between permanent or electromagnets where there is a clearly defined north and south pole.

It's not so much that it either attracts or repels but that it's always in one direction, either continuously to the left or the right depending on the magnet's orientation.

It's not really trying to move away (repel) or towards (attract) anything. It's just trying to move to the left or the right, and whether this constitutes "towards" or "away" depends on where their starting positions are relative to each other.

enter image description here Taken from: https://www.feynmanlectures.caltech.edu/II_01.html#Ch1-S2

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