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Paleomagnetism can be measured as the magnetic field at the time of rock forming is preserved in some minerals.

Could the paleomagnetism and the present magnetic field somehow influence the movements of tectonic plates?

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The electromagnetic force and related field is a strong force at very small distances (governs the way the proton and electron are held to an atom) but is relatively weak over large distances. I don't see how that small force could act on processes involved in the convection of the mantle. If memory serves, the iron in magmas preserves the force of the electromagnetic field on earth at the time the magma cools to form rock. It leaves its imprint but has no affect on the process or origin of magmatism much less the convection that drives plate tectonics

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The forces of a magnetic field on materials that are not ferromagnetic and not electric conductors are negligible compared to pressure, tension, and buoyancy. It is likely safe to ignore them for the upper mantle.

However, the lower mantle is a semiconductor with a significant thermal gradient, which induces electrical gradient via thermoelectricity. This paper suggests this effect can be strong enough to actually be a source of geomagnetism. If true, it is also strong enough to interact with geomagnetic field to produce mechanical forces, which would manifest as a depth-oriented gradient of East-west drag.

Another paper claims both geomagnetism and plate tectonics have the same cause: Global coupling at 660 km is proposed to explain plate tectonics and the generation of the earth's magnetic field.

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