Matter & Interactions 2nd ed. Practice Problems
Aaron Titus | High Point University
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2k80002     Torque on a coil     2k80002
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Question

 

A coil of 20 turns of wire has a radius 0.5 cm and is connected to a power supply so that 0.4 A flows through the coil. At , the coil is at rest, current flows into the page at the top of the coil, and the plane of the coil makes an angle of with respect to a magnetic field of 0.5 T.


Figure: A coil in a uniform magnetic field.

  1. At the instant shown (t=0), what is the torque on the coil?
  2. In what direction will the loop rotate if it is released from rest?
  3. What is the magnetic potential energy of the loop at t=0?
  4. If the system is frictionless, what will be the kinetic energy of the coil when it reaches its equilibrium orientation?
  5. Describe the motion of the coil, even after a long time, if the system is frictionless.

 



Solution

 

(a) The coil acts as a magnetic dipole with dipole moment in a direction perpendicular to the plane of the coil and in the direction given by the right-hand rule, as shown below.


Figure: Magnetic dipole moment of the coil.

The torque on a magnetic dipole is . In this case, and . There are two ways to calculate the torque. First, one can calculate the cross product:

The second way to determine torque is to calculate the magnitude of the torque and then use the right-hand rule to get the direction. First, sketch the magnetic dipole moment and magnetic field vector tail to tail and find the angle between them.


Figure: Angle between the magnetic dipole moment and magnetic field vectors.

The magnitude of the torque on the coil is

Using your right hand, point your fingers in the direction of and rotate toward . Your thumb will point in the direction. Therefore,

(b) According to the angular momentum principle, the change in the angular momentum of the coil will be in the same direction as the torque on the coil. Since it is released from rest, then the final angular momentum after a small time interval will be in the same direction as the torque, in the direction. Thus, the coil will rotate clockwise, toward equilibrium.

(c) The magnetic potential energy of a dipole in a magnetic field is

(d) The equilibrium orientation of the coil is shown below.


Figure: Equilibrium orientation of the coil.

The magnetic potential energy at equilibrium is

Thus, in rotating from its initial position at to the equilibrium position, the change in potential energy of the coil is

Since it is a closed system, Conservation of Energy states that:

In other words, the potential energy lost by the coil in rotating toward equilibrium goes into and increase in kinetic energy. As a result, its angular speed of the coil increases as it rotates toward equilibrium.

(e) The coil will rotate past equilibrium and will then slow down until coming to rest with the orientation shown below.


Figure: Coil has rotated past equilibrium.

It will then rotate back toward equilibrium, speeding up as it rotates. In this way, it will oscillate back and forth around equilibrium.