They do, but it’s not quite what you think.
The main magnet is, indeed, extremely strong – however, it doesn’t change. Maxwell’s laws of electromagnetism say that a changing magnetic field creates an electric field. From which it follows that a magnetic field which doesn’t change doesn’t create an electric field.
However, if you were to walk past an MRI scanner, then as you move through the magnetic field, from a weak area into a strong area, then your body experiences a changing magnetic field, and a tiny amount of electricity is generated in the body. This isn’t enough to be a problem – you wouldn’t feel it – but as a precaution, there are legal limits on how much electricity can be generated, and so MRI workers are advised to stick to “speed limits” when walking past an MRI scanner (no running close by).
However, although an MRI scanner has a strong unchanging magnet, it actually has a number of weaker magnets which can be changed. The scanner uses these to focus on individual pixels in the image, so in order to get a detailed image reasonably quickly, these magnets have to be changed very quickly (10s or hundreds of times per second) These magnets create rapidly changing magnetic fields during a scan, and this does create electrical currents in the body. These usually aren’t much of a problem, and you usually don’t feel them.
However, sometimes, if the scanner is working very hard and fast – or if you are very tall, then enough electrical current can be produced to cause muscles to twitch. Scanners have legal limits as to how much muscle twitching they are allowed to cause. Actually getting electrical shocks or affecting the heart requires a ton more electrical current, so the legal limit on the design of scanners means that there is absolutely no chance of causing heart disturbances or painful shocks.
Finally, there is a 3rd magnetic field produced by an MRI scanner – this changes 10s or 100s of millions of times per second – this also produces a lot of of electrical current. However, the electrical currents it produces change so quickly, that no biological process can respond that fast – so even though there is quite a lot of current produced, it can’t affect muscles, nerves, brain or the heart. However, electrical currents do produce heat, and this current produces heat in the body. If the scanner is working very fast and hard, then this heat production can be considerable – literally as much heat as going to the gym for some light aerobics.
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