Figure 1: A Current Loop in a Magnetic Field
For a magnetic field (vector) 
acting on an arm of a current loop, a square current loop and
a Bohr orbit are similar.
The force on each single charge (q) travelling
in the arm comes from the Lorentz force:
Since 
is perpendicular to (in our case), the cross product
simplifies.
The current 
is given by:
where 
is the cross sectional area of the wire-loop,
and the force on each charge is:
Since the number of charges is 
,
the force on one arm of the loop (of length `a') is
which is reversed on the other (opposite arm) leg of the loop.
The loop is 
in area (and 2 a + 2 b in circumference),
the moment arm about the pivot point is 
if 
is the angle between the loop and the field.
In the Figure 1, the length of the horizontal arms are `a'
while the moment arm is of length `b/2', i.e., from the axis to a
horizontal arm is b/2 (cm).
The torque ( 
) is
but, since a 
b is the area (A) of the loop, we have
Commonly, this torque is related to a magnetic moment equivalent, i.e.,
in analogy with an electric dipole in an electric field. A current loop is equivalent to a magnetic moment, a tiny bar magnet.
We assume that the above would hold for a Bohr orbit.
From Bohr Theory we had
and
so, solving for 
we have
and then the current (= stat-coul/sec) = charge/transit-time
, where (we use twice here, once for the
torque, and once for the period- both usages are common) is the period.
so
but, since the area is 
, we have
which defines the Bohr magnetic moment
known as the Bohr magneton.
We then have
which is, finally,
The energy associated with rotating the current loop (Bohr orbit) about the axis perpendicular to the field (orienting the loop relative to the field), is
i.e.,
which gives
or
