A non-uniform magnetic field is generated by a permanent magnet fixed on the moving support. The resonating metallic bar is fixed at one end (cantilever) and two NdFeB magnets (which work as a one bigger magnet) are placed in contact creating a strong magnetic field in the surrounding.

When the moving support approaches the magnetic tip, the end of the cantilever is repelled because the two north poles are placed facing each other. The resonance frequency is also affected (Time 00:43). The relative change of resonance frequency depends on the first spatial derivative of the magnetic force, which is proportional to the magnetization of the oscillating magnet and so to the second order spatial derivative of the external magnetic field, in this case that generated by the magnet fixed on the moving support.
To monitor the oscillation of the magnetized cantilever, we used a smartphone equipped with a magnetic field sensor. The oscillating magnets produce an oscillating magnetic field on the smartphone that is detected by the sensor. We used Phyphox®, an open-source software created at the 2nd Institute of Physics of the RWTH Aachen University, to analyze the data collected by the magnetic field sensor. Thanks to the “magnetic spectrum analyzer” app, we are able to measure the fundamental peak of the resonating bar in real time!
When we reverse the magnet direction of the moving support (Time 1:03), the resonance frequency decreases as expected by the fact the second order magnetic field derivative changes its sign.
This principle is at the base of the OXiNEMS magnetometers we are developing for biomagnetism [The patent no. US11415642 “A device for sensing a magnetic field” owned by the National Research Council (CNR), the University of Chieti-Pescara and Quantified Air B.V. (NL)] and consumer applications "MEMS Magnetometer Using Magnetic Flux Concentrators and Permanent Magnets" by Federico Maspero, Gabriele Gatani, Simone Cuccurullo, Riccardo Bertacco. 2021 IEEE 34th International Conference on Micro Electro Mechanical Systems (MEMS).

The same principle is also the basis of Magnetic Force Microscopy with a Scanning Probe Microscope, where nano and micro magnetic domains at surfaces are measured by a microscopic AFM tip with a magnetic element on the apex. A reference on MFM from the OXiNEMS project, “Impact of magnetic domains on magnetic flux concentrators" by Federico Maspero, Simone Cuccurullo, Dhavalkumar Mungpara, Alexander Schwarz, Riccardo Bertacco.

This experiment has been created in the framework of the 2022 edition of the Genoa Science Festival for the exhibition “All we are magnetic” organized by the OXiNEMS project aiming to show some of the basic principles of magnetic field detectors in consumer applications and in the field of biomagnetism, in particular the current applications in Magnetoencephalography.

Credits:
Experiment: Luca Pellegrino (National Research Council (CNR), Italy), Emilio Bellingeri (CNR), Filippo Sozzi (CNR), Alexander Schwarz (University of Hamburg, Germany)
Exhibit realization: Manuele Gargano (CNR), Filippo Novara (CNR), Alberto Ravazzolo (CNR)
Screenplay: Luca Pellegrino, Filippo Sozzi (CNR)
Editing: Luca Pellegrino

We acknowledge discussions with all the OXiNEMS members!