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A time-dependent current (either DC or AC) that creates a changing magnetic field around the wire, in absence of other materials, only lose current in heat proportional to ohmic resistance of wire itself.

In our paradigm there are several nearby permanent magnets that constantly keep a magnetic field perpendicularly and parallel to copper wire which their strength is greater than magnetic field generated by changing current.

Hence changing magnetic field of time-dependent current might lose some potential by inducing eddy currents on nearby permanent magnets (which is trivial in nonconductive ferrite magnets) or hysteresis on magnet body's crystal structures. But does the magnetic field of permanent magnets also enforce moving current inside the wire creep and lose more potential to overcome applied field? or current loses because of nearby permanent magnets in general is much lower than ohmic resistance of wire and we can ignore it?

Are there equations to accurately calculate total current lose due to presence of permanent magnets in such circumstances when we know applied flux density by magnets and current density inside wire?

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The static magnetic field itself will not induce any current or voltage in a wire. (Else we would have been able to use simple wires as magnetometers for our smartphones, in stead of the complicated chips they have now).

The only effect you will notice is, as you say the EM wave(s) created by the cable coupling into anything they can couple into with the right directionality and causing some loss. This, however, is even negligible if you run a wire above a copper plate for the more likely frequencies. If you go into radio frequency and such those effects might be noticeable on their own, but then you're already also in the domain of skinning and parasitic capacitance and all such, which I would expect to be more pronounced anyway.

Now, if you were to take away the skinning effect and make a few loops, let's say 1.5 or more, you can couple into some metals, but then we'd just call that an inductor to begin with.

I have not, to date, run into a situation where I had to think about static magnetics near a normal straight wire. But that's possibly because any magnetics placed anywhere in my designs usually interact with a coil quite near them on purpose.

But as I am too tired to do any maths to prove it, I'm even more open to standing corrected than usual.

Until further notice, I'd say, just go with resistivity, unless you are working with HF, in which case you are going to have much more serious problems with other things that a permanent magnet here or there.

Note: At 10kHz something in the AWG15 or 16 range will already show a noticeable skin effect, so even outside of radio frequency, if you go beyond 2 or 3kHz at higher power levels, it can be very wise to look up the term "skin effect" on Google.

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I don't know if it would be relevant in your setup, but (also static) magnetic fields can generate voltages in directions orthogonal to the current flow.

It is called Hall effect. Probably it could affect your results if you are doing some precision measurements.

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