Transcript Torque On A Current Loop In A Uniform Magnetic Field

Torque On A Current Loop In A
Uniform Magnetic Field
AP Physics C
Montwood High School
R. Casao
• With the knowledge that a
force is exerted on a
current-carrying conductor
when the conductor is
placed in an external
magnetic field, this force
can produce a torque on a
current loop placed in the
magnetic field.
• Consider a rectangular loop
carrying a current I in the
presence of a uniform
magnetic field in the plane
of the loop.
• The forces on the sides
of length b are zero
since these wires are
parallel to the field (ds
X B = 0 for these sides).
• The magnitude of the
forces on the sides of
length a is given by:
F2 = F4 = I·a·B
• The direction of F2 is
out of the page; the
direction of F4 is into the
page.
Tmax
Tmax
Tmax
• If viewing the loop from
the end, and we assume
that the loop is pivoted so
that it can rotate about
point O, the two forces
produce a torque about O
that rotates the loop
clockwise.
b
b
• The magnitude of the
 F2   F4 
torque, which is maximum
2
2
 b
 b in this position, is:
 Ia B  Ia B
Tmax = I·a·b·B:
2
2

 Ia bB
• The area of the loop
A = a·b; Tmax = I·A·B
• If the current direction
were reversed, the
forces would reverse
their directions and the
rotational tendency
would be
counterclockwise.
• Suppose the magnetic
field makes an angle q
with respect to a line
perpendicular to the
plane of the loop (the
dashed line).
• The vector A which is
perpendicular to the plane of
the loop is the area vector A
of the loop.
• Only the forces F2 and F4
contribute to the torque about
the axis of rotation O. The
other two forces on the loop
would not produce a rotation
as these forces would be
equal in magnitude and
opposite in direction and
would also pass through the
axis of rotation O, making the
torque arm 0 m.
• The forces F2 and F4
form a couple and
produce a torque
about any point.
• We will use point O
as the pivot point and
I’m going to resolve
the forces in to a
component that is
parallel to the axis of
the loop and a
component that is
perpendicular to the
axis of the loop.
• The parallel component
of the force Fp will not
produce a rotation
because it passes
through the pivot point.
• The perpendicular
component of the force
F will produce the
torque that will cause
the loop to rotate.
T  F  r
• For each perpendicular
force:
– r = b/2
– F  = F2·sin q
– T = F2·sin q·b/2
• Net torque about
point O:
T = 2·F2·sin q·b/2
T = F2·b·sin q
• Remember:
T = I·a·B·b·sin q
F2 = I·a·B
T = I·A·B·sin q
• The torque has a maximum value I·A·B
when the magnetic field is parallel to the
plane of the loop (angle q between A and B
= 90°).
• The torque is 0 N·m when the magnetic
field is perpendicular to the plane of the
loop (angle q between A and B = 0° =
180°).
• The loop tends to rotate to smaller values
of q (so that the area vector A rotates aligns
with the magnetic field vector B).
• To express the torque as a vector cross product:
 
T  I  A x B  I  A  B  sin θ


• The direction of the area vector A is determined by the right
hand rule: rotate the fingers of the right hand in the direction
of the current in the loop, the thumb points in the direction of
the area vector A.
• The direction of the torque is also given by the right hand
rule: point the fingers of the right hand in the direction of the
area vector A, the palm points in the direction of the
magnetic field B, the thumb points in the direction of the
torque.
• The product I·A is defined as the magnetic moment m of the
loop; m = I·A
• Unit for magnetic moment m = A·m2.
• T = m x B (for a single loop)
• This is the same type of torque that acts on
an electric dipole moment p in an external
electric field to align the dipole moment with
the electric field E; T = p X E
• If a coil in a magnetic field B has N loops of
the same dimensions, the magnetic moment
and the torque on the coil will be N times
greater than a single loop.
T = N·I·A·B = N·(m X B)=N·m·B·sin q
The Plane of the Loop
• Most problems refer
to the plane of the
loop.
• The plane of the loop
is the plane that lies
along a straight line
between the opposing
currents in the
conducting loop.
• The plane of the loop
is perpendicular to the
area vector.
The Plane of the Loop
• Maximum torque occurs when
the plane of the loop is
parallel to the magnetic field
B.
– Angle between plane of loop and
B is 0°.
– Angle between area vector A
and B is 90°.
• Zero torque occurs when the
plane of the loop is
perpendicular to the magnetic
field B.
– Angle between plane of loop and
B is 90°.
– Angle between area vector A
and B is 0°.
Work and Torque
• Work = Torque·angular displacement q
• Generally, the angular displacement q is
measured in radians.
• For a displacement from qi to qf:
 dW   T  dθ  N  I  A  B  sin θ  dθ
θf
θf
θi
θi
W  N  I  A  B  θ sin θ  dθ
θf
i
W  N  I  A  B   cos θ  θ
θf
i
W  N  I  A  B   cos θ f  cos θ i 
W  N  I  A  B   cos θ f  cos θ i 
Potential Energy and Torque
• The potential energy U of a system of a
magnetic dipole in a magnetic field depends on
the orientation of the dipole in the magnetic field.


U  μ  B  μ  B  cos θ
• The system has its lowest U when m points in
the direction of B; the angle between m and
B = 0°.
• The system has its highest U when m points in
the opposite direction of B; the angle between m
and B = 180°.