Current sensing coil...

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Hi.
If a wire conducting 1 Ampere of alternating current passes inside this 1000 turns coil; will an AC voltage show up at the coil terminals ?
 

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You get zero voltage. The magnetic field lines are surrounding the wire in circles. To get an induced voltage, the coil axis must be parallel to the field lines, in other words oriented perpendicular to the wire, but not in it's plane, e.g. the coil besides the wire in a crossed position.

An ideal coreless AC current sensor has the form of a Rogowski coil. https://en.wikipedia.org/wiki/Rogowski_coil
 


but the Rogowski coil do induces the voltage..
 

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