Physics For Scientists And Engineers 6E - part 227
SECTION 29.3 • Torque on a Current Loop in a Uniform Magnetic Field
905
Now suppose that the uniform magnetic field makes an angle ! * 90° with a line
perpendicular to the plane of the loop, as in Figure 29.14. For convenience, we assume
that
B is perpendicular to sides # and $. In this case, the magnetic forces F
1
and
F
3
exerted on sides ! and " cancel each other and produce no torque because they pass
through a common origin. However, the magnetic forces
F
2
and
F
4
acting on sides #
and $ produce a torque about any point. Referring to the end view shown in Figure
29.14, we note that the moment arm of
F
2
about the point O is equal to (b/2) sin !.
Likewise, the moment arm of
F
4
about O is also (b/2) sin !. Because F
2
"
F
4
"
IaB, the
magnitude of the net torque about O is
where A " ab is the area of the loop. This result shows that the torque has its
maximum value IAB when the field is perpendicular to the normal to the plane of the
loop (! " 90°), as we saw when discussing Figure 29.13, and is zero when the field is
parallel to the normal to the plane of the loop (! " 0).
A convenient expression for the torque exerted on a loop placed in a uniform
magnetic field
B is
(29.9)
where
A, the vector shown in Figure 29.14, is perpendicular to the plane of the loop and
has a magnitude equal to the area of the loop. We determine the direction of
A using the
right-hand rule described in Figure 29.15. When you curl the fingers of your right hand
in the direction of the current in the loop, your thumb points in the direction of
A. As
we see in Figure 29.14, the loop tends to rotate in the direction of decreasing values of !
(that is, such that the area vector
A rotates toward the direction of the magnetic field).
The product I
A is defined to be the magnetic dipole moment " (often simply
called the “magnetic moment”) of the loop:
(29.10)
The SI unit of magnetic dipole moment is ampere-meter
2
(A # m
2
). Using this definition,
we can express the torque exerted on a current-carrying loop in a magnetic field
B as
(29.11)
Note that this result is analogous to Equation 26.18, # "
p ! E, for the torque exerted
on an electric dipole in the presence of an electric field
E, where p is the electric
dipole moment.
# $ " !
B
" "
I
A
# "
I
A ! B
" IAB sin !
" IaB
#
b
2
sin !
$
'
IaB
#
b
2
sin !
$
"
IabB sin !
) "
F
2
b
2
sin ! ' F
4
b
2
sin !
F
2
F
4
O
B
A
b
2
– sin
θ
b
2
–
θ
θ
θ
#
$
×
Active Figure 29.14 An end view of the loop in
Figure 29.13b rotated through an angle with
respect to the magnetic field. If B is at an angle
!
with respect to vector A, which is perpendicular to
the plane of the loop, the torque is IAB sin
!
where
the magnitude of A is A, the area of the loop.
At the Active Figures link
at http://www.pse6.com, you
can choose the current in the
loop, the magnetic field, and
the initial orientation of the loop
and observe the subsequent
motion.
Torque on a current loop in a
magnetic field
Magnetic dipole moment of a
current loop
Torque on a magnetic moment
in a magnetic field
A
I
µ
Figure 29.15 Right-hand rule
for determining the direction of
the vector A. The direction of the
magnetic moment
"
is the same as
the direction of A.