Force on a Closed Circuit in a Magnetic Field B

Section 10.5.2 established that, for a wire of arbitrary shape in a UNIFORM magnetic field, the total force depends only on the vector displacement between the wire's two endpoints. A particularly important special case follows immediately when the wire forms a CLOSED circuit -- that is, when the "starting point" and "ending point" of the wire are one and the same. Extending the general result of Section 10.5.2 to integrate all the way around a closed loop $C$,

and, for a UNIFORM field $B$, pulling $\vec{B}$ outside the integral exactly as before,

Now consider the bracketed term, $\oint_C d\vec{l}$ -- the vector sum of every infinitesimal displacement element $d\vec{l}$, added up all the way around a CLOSED path. Since a closed path, by definition, starts and ends at the exact same point, the total (net) displacement around the whole loop is necessarily zero -- however winding or complicated the loop's shape, you end up back exactly where you started, so the vector sum of all the little steps taken along the way must cancel out perfectly. That is,

$\oint_C d\vec{l} = 0 \quad \Rightarrow \quad \vec{F}_m = 0 \quad (\text{for uniform } \vec{B}).$ …