Consider a superconducting loop of wire carrying a current of I. Let F
denote the magnetic flux linked by this current-carrying loop.

Consider a region of space positioned some distance from the loop. Let
V denote the volume of this region. Assume the volume is small enough
so that the magnetic flux density generated by the loop is
substantially uniform within the volume. Let B denote the value of the
magnetic flux density within the region.

The shape of the volume is approximately ?pillbox,? which means that
the surface of the volume is comprised of three subsurfaces. The first
and second subsurfaces are aligned perpendicular to the magnetic flux
density. Magnetic flux enters the region through the first subsurface,
and exits through the second. The third surface connects the first and
second surface, and is everywhere parallel to the magnetic flux
density. The magnetic flux linked by the first subsurface and the
second are equal and opposite. Let F_1 denote this magnet flux.

Let H denote the magnetic field in the region. Let the medium in which
this experiment takes place be linear, in that the magnetic flux
density and magnetic field are proportional, and each are also
proportional to the current I. Let u denote the proportionality
constant between B and H, i.e. B = uH.

The energy stored in the magnetic field integrated across all space is given by
 
E_tot = FI/2.

The energy stored in the region is given by the product
of the energy density and the volume of the region:

E_region = VB^2/(2u).

The total energy is greater than the energy stored in the region:

E_tot = FI/2 > E_region.

Question: True or false? Is F_1I/2> E_region? If true, prove it. If
false, give a counterexample.