Fictitious Charges Don’t Cause Torque: Mansuripur’s Paradox

There’s been some talk lately about Mansuripur’s Paradox, e.g., see Slashdot.

For those not interested in the fine detail, there’s a very simple explanation as to why there isn’t any real paradox involved.  I’m not sure whether the debate is significant for electrical engineers; it may well be true, as Mansuripur suggests, that the Einstein-Laub equations are more appropriate than the Lorentz law for the purposes of electrical engineering.  (I have no opinion on that question.)  What should be pointed out, though, is that from a fundamental physics point of view there’s really nothing at all to see here.  (I believe that Mansuripur understands this [1], but I’m not at all sure that the journalists do!)

Let’s start with a quote from one of the articles (it looks like the paper is a bit more subtle, but the upshot is might be [2] the same): “Now imagine how things look from a “moving frame of reference” in which the charge and magnet both glide by at a steady speed. Thanks to the weird effects of relativity, the magnet appears to have more positive charge on one side and more negative charge on the other.”

Now, it’s true that there’s an electric field, and for some purposes it may be convenient to imagine that this is due to charges on either side of the magnet. But these charges are fictitious. They aren’t really there, as can be easily shown by observing that charge is a scalar, and hence the charge distribution in the magnet cannot be dependent on the frame of reference. Since they aren’t there, it’s hardly surprising that the external electric field doesn’t apply a force to them.

So, basically, a fiction that happens to be convenient in electric engineering is incompatible with relativity; or, if you prefer, in order to make fictitious charges compatible with relativity you also have to either have fictitious angular momentum, or modify the Lorentz force law.  As far as fundamental physics is concerned, this is not a paradox.


[1] I may be wrong about this; see comments to my question on Stack Exchange.

[2] The comments and linked question also suggest that I might have misunderstood the source of the supposed torque in the original paper.  There’s still nothing indicating any evidence of a real paradox.  I’ll update again if I learn anything new.

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