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Meir Achuz
#2
Feb6-13, 10:00 AM
Sci Advisor
HW Helper
PF Gold
P: 2,015
The 1/2 in your first equation arises because the integral is over all current distributions. The 1/2 takes care of double counting that takes place in that case.
The B field in the second equation is due to only external currents (not part of mu), so the 1/2 does not arise.

The + sign in the first equation is due to the fact that the current is kept constant by an EMF source that provides energy to keep the current constant. Then, the force is given by +grad U. In the second equation, no energy is supplied, so F= -grad U.
Each case thus gives the same force.

You can look at <http://arxiv.org/pdf/0707.3421.pdf>