Use an electromagnet for reading induction as well
You can use a hall sensor from an old computer fan. 

If the magnet is not moving you are not going to be able to sense it with the coil of your electromagnet. 
You can also decouple de input of your arduino with a cap in series with the coil, but again if the magnet is not fast enough you will have to use a bigger cap. 

You can also make a voltage divider between your coil and sense from there. When the 12V are applied just add some code to offset the input signal.