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Suppose that magnetic wire with resistivity similar to copper but with non-negligible permeability existed. What effect would magnetic wire have on the coil inductance versus a copper coil of the same size/construction?

This is of course a purely academic exercise, since such a material does not exist, though it could perhaps exist for plated wire at high frequencies, due to the skin effect.

EDIT:

I guess it's not entirely academic. isdi pointed out that "high permeability wire" actually exists.

According to this:

Mu Metal Wire is used in the production of inductors and electrical transformers. It has one of the highest magnetic permeability ratings of any metal or alloy, making it an industry standard in applications where metals with high magnetic permeability are needed.

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    \$\begingroup\$ I suppose it would increase the inductance slightly, based on the idea that some of the magnetic field is passing through more permeable material. \$\endgroup\$ Dec 5 '18 at 21:17
  • \$\begingroup\$ This question borders on being off-topic, as it implies the need for physics to define the properties of this "magnetic wire". Common magnet wire is pure copper with an enamel coating, so its properties are known. At best we can offer opinions based on our point of view and expertise on "what if" scenarios, and that is not what we are here for. \$\endgroup\$
    – user105652
    Dec 5 '18 at 21:31
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    \$\begingroup\$ Not academic at all, the stuff is out there, including both homogeneous and plated wire. try googling "high permeability wire" for info. \$\endgroup\$
    – isdi
    Dec 5 '18 at 21:33
  • \$\begingroup\$ How about IRON wire? \$\endgroup\$ Dec 6 '18 at 3:26
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If the structure is identical except for the wire you would get a slightly higher inductance reading as the "core" would now include not only your inductor core but the permeability of the wire.

The Mu Metal Wire you mention is not used to wind the core. It's usually used for magnetic "shielding". You need a material that offers a low reluctance path for magnetic fields to shape them (since you can't shield a magnetic field, only redirect it).

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Two common wire types are ferromagnetic, one being 'copperweld', an iron wire with copper cladding, and the other being maganese-bronze (cuprous alloy, potentially useful as music wire).

Because a solenoid winding has the B field roughly on-axis, the material of the wire will make relatively little difference to the overall B-field created, just some minor ripple next to the wires, but the internal field (to the conducting wire) will show an extraordinary skin depth (very shallow) due to magnetic induction in the wire. This will raise the effective electrical resistance of the coil with increase in frequency. That effect will be much less in copperweld (because the skin is the copper part, not magnetic).

Skin depth is given by

$$ \epsilon = {1 \over \sqrt{(\pi \mu \sigma f )} }$$

where mu is the magnetic susceptibility, and sigma is the electrical conductivity; a susceptibility of 100 would reduce skin depth by 90%.

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