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Is it possible that a metallic object will not be under the influence of a magnetic field at a certain closeness to a conductor, but will then experience the effect on moving to a particular distance (and onwards)? In other words, is there a threshold radius around a conductor within which the magnetic effect cannot be felt?

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Any object, whether it be magnetized or not (and in particular whether it be ferromagnetic or para/diamagnetic), will only experience magnetic forces when it is placed in an external magnetic field. If you are thinking of a long, straight conductor, then the magnetic field it produces will increase as you get near it (though it does decrease inside the conductor).

That said, there are some configurations of currents that produce a magnetic field "outside" and but none "outside". Two examples are a toroidal solenoid, in which the field is confined to the solenoid interior and is zero in the donut hole, and two long cylindrical, coaxial solenoids carrying the same current in opposite directions, for which the field is confined to the middle region and cancels out in the inner region. A coaxial cable is an example of the latter.

If you are thinking about an extended distribution of currents such as a wire of nonzero thickness, then yes, the field is zero at the centre and it is small nearby. However, you can hardly place a metallic object inside a wire.

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