Problem 22.12

22.12.
The magnetic force on a segment $d\vec{s}$ of the loop is given by
$d\vec{F} = I d\vec{s} \times \vec{B}$
So, the total force on the loop is given by
$\vec{F} = \oint I d\vec{s} \times \vec{B}$
where we write the $\oint$ because, the loop is closed, so the starting and ending points for our integration are the same point.
Now, since our $\vec{B}$ field and the current I are constant, we can write this as
$\vec{F} = (\oint d\vec{s}) I \times \vec{B}$
But, since the term
$\oint d\vec{s}$
is the net displacement between the starting and ending points of our loop, we find that since these points are identical,
$\oint d\vec{s} = 0$,
so, the net force is zero as well.
$\vec{F_{m}} = 0$