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On the webpage of a lecture at the University of Virginia, I found this derivation of the magnetic force using Lorentz contraction and electrostatic force. They approximate the gamma factor to the order of ##v^2## and get the correct result. How is it possible to get to the same result using an approximation as one would get without approximating anything?

Of course, if multiple approximations are made to terms that get subtracted or divided, the errors introduced by the approximations might cancel out. But I can't see this happening here, using the same approximation for the increase in positive charge density and decrease in negative charge density and adding them should even increase the error.

What am I missing here?

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# I Derivation of magnetic force using approximated gamma-factor

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