Anti-theft tags contain a tiny mechanical oscillator that is activated by a magnetic field. The detectors at a shop's doorway can detect the oscillation magnetically. The tag can be activated and deactivated by magnetizing or demagnetizing it, respectively.

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Erik Saathoff
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Sources:

http://www.metglas.com/products/magne...

https://www.google.com/patents/WO1999...

https://patents.google.com/patent/WO2...

https://www.google.com/patents/US4510...

https://www.google.com/patents/US4298862

https://en.wikipedia.org/wiki/Magneto...