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
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 (for the radius):
Expression for radius, in terms of
`n',`hbar',`Z',`m', and `e'
Query, is the above formula correct?