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I'm trying to learn about hidden momentum. After reading what I could find with a google search, I understand that it is equal to the momentum carried by radiation, calculated with the Poynting vector. I can't really understand how the momentum conservation laws apply to the situations in which there is a magnet(current through the loop) and in which this magnet is turned off. How should i write these laws?
I think I somehow understand the origin of hidden momentum, but I don't see how everything sums up in both situations. Could you please help me understand?

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Well if we neglect the hidden momentum the conservation law of momentum in electromagnetism is simple:

The momentum can be stored in static fields ($D\times B$); the mechanical momentum ($mv$) + electromagnetic momentum ($D\times B$) $= constant$.

The similar formula is valid for angular momentum (where it is not hidden momentum) See Feynman's Lectures on Physics.

In my opinion the relativist hidden momentum is not connected with electromagnetism because it can be removed for that magnetic loop if one place it in a gravitational field for example who vanishes the electrostatic potential energy.

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