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While playing with voltage multiplication and capacitors, I began to think if my experiments could damage my electrical equipment (eg. mobile phone) by creating strong magnetic fields over a short duration. (for example when experimenting with coil gun technology)

So far, I have restrained myself to human-safe voltages for many reasons.

Yet, I'm still curious how much electrical power would one need to create magnetic field strong enough to damage or disturb surrounding electronics. This means field strong enough to create unexpected electrical potentials and currents in devices.

Can this happen without some heavy technologies? Is there maybe something that prevents this from happening at all? (strong magnets near mobile phones seem to do no harm...)

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The key to understanding EMP is that this is an induced effect, so you need rapid changes in magnetic field, $\frac{dB}{dt}$. In order to generate a rapidly changing field, you have to have a rapidly changing current in an inductor - as you may recall, $$V = -L \frac{dI}{dt}$$ For this rapid change in current you not only need a high voltage - you need a low inductance source of energy. A capacitor could be such a source - but it's not likely you have one lying around that stores the energy needed.

It is in principle possible for a large capacitive discharge to create a spike in magnetic field that induces currents; but most modern electronic equipment is at least somewhat protected from this, so that arcing of an electric motor (brushes) and things like that won't damage every cell phone near by.

This is not to say you couldn't do it - but electrostatic damage (the kind that gives you a shock when you touch a door handle on a dry winter's day) is typically much more likely to damage electronics.

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