detect small ac voltages
Aggiemundo,
First, some symantics nit-picking. It is current, not voltage that flows through the wire. Now, on to your question: I believe that the 4V that you see is due to capacitive coupling between the lamp wire and the coil. This would cause a common mode voltage that would appear on both ends of the coil. The voltage would be seen on the scope due to the difference between the scop common and the AC mains common.
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If you visualize the magnetic field that is induced in the lamp cord due to current flow, the lines of flux appear as circles with the axis of the circles parallel to the lamp wire. Now, if you wind a coil around the lamp wire, the lines of flux are parallel with the windings of the coil. In order to induce a voltage in the coil due to magnetic coupling, the coil needs to be oriented perpendicular to the lamp wire.
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If you want to measure the current with an inductive device, you need to pass the wire through a core. The core should have another winding into which a voltage proportional to current would appear. The voltage will be proportional to the number of turns on the core and the resistance of the load connected to the winding. Remember, you must pass only 1 wire through the core, otherwise the magnetic fields from the two wires will cancel. This is a current transformer.
Google "current transformer". You will get lots of hits.
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Another way to detect current is to use a Hall effect sensor. The output voltage will be small, so you would need to amplify it.
Regards,
Kral