21.6

The Torque on a Current-Carrying Coil

We have seen that a current-carrying wire can experience a force when placed in a magnetic field. If a loop of wire is suspended properly in a magnetic field, the magnetic force produces a torque that tends to rotate the loop. This torque is responsible for the operation of a widely used type of electric motor.
Figure 21-21a shows a rectangular loop of wire attached to a vertical shaft. The shaft is mounted such that it is free to rotate in a uniform magnetic field. When there is a current in the loop, the loop rotates because magnetic forces act on the two vertical sides, labeled 1 and 2 in the drawing. Part b shows a top view of the loop and the magnetic forces and that act on the two sides. These two forces have the same magnitude, but an application of RHR-1 shows that they point in opposite directions, so the loop experiences no net force. The loop does, however, experience a net torque that tends to rotate it in a clockwise fashion about the vertical shaft. Figure 21-22a shows that the torque is maximum when the normal to the plane of the loop is perpendicular to the field. In contrast, part b shows that the torque is zero when the normal is parallel to the field. When a current-carrying loop is placed in a magnetic field, the loop tends to rotate such that its normal becomes aligned with the magnetic field. In this respect, a current loop behaves like a magnet (e.g., a compass needle) suspended in a magnetic field, since a magnet also rotates to align itself with the magnetic field.

Figure 21-21 (a) A current-carrying loop of wire, which can rotate about a vertical shaft, is situated in a magnetic field. (b) A top view of the loop. The current in side 1 is directed out of the page (), while that in side 2 is directed into the page (). The current in side 1 experiences a force that is opposite to the force exerted on side 2. The two forces produce a clockwise torque about the shaft.

It is possible to determine the magnitude of the torque on the loop. From Equation 21.3 the magnetic force on each vertical side has a magnitude of , where L is the length of side 1 or side 2, and because the current I always remains perpendicular to the magnetic field as the loop rotates. As Section 9.1 discusses, the torque produced by a force is the product of the magnitude of the force and the lever arm. In Figure 21-21b the lever arm is the perpendicular distance from the line of action of the force to the shaft. This distance is given by , where w is the width of the loop, and is the angle between the normal to the plane of the loop and the direction of the magnetic field. The net torque is the sum of the torques on the two sides, so

In this result the product Lw has been replaced by the area A of the loop. If the wire is wrapped so as to form a coil containing N loops, each of area A, the force on each side is N times larger, and the torque becomes proportionally greater:

| (21.4)

Equation 21.4 has been derived for a rectangular coil, but it is valid for any shape of flat coil, such as a circular coil. The torque depends on the geometric properties of the coil and the current in it through the quantity NIA. This quantity is known as the magnetic moment of the coil, and its unit is ampere·meter2. The greater the magnetic moment of a current-carrying coil, the greater the torque that the coil experiences when placed in a magnetic field. Example 6 discusses the torque that a magnetic field applies to such a coil.

| Example 6 | | | The Torque Exerted on a Current-Carrying Coil

A coil of wire has an area of , consists of 100 loops or turns, and contains a current of 0.045 A. The coil is placed in a uniform magnetic field of magnitude 0.15 T. (a) Determine the magnetic moment of the coil. (b) Find the maximum torque that the magnetic field can exert on the coil.
Reasoning and Solution

a. | The magnetic moment of the coil is

b. | According to Equation 21.4, the torque is the product of the magnetic moment NIA and . However, the maximum torque occurs when , so

The physics of a direct-current electric motor. The electric motor is found in many devices, such as CD players, tape decks, automobiles, washing machines, and air conditioners. Figure 21-23 shows that a direct-current (dc) motor consists of a coil of wire placed in a magnetic field and free to rotate about a vertical shaft. The coil of wire contains many turns and is wrapped around an iron cylinder that rotates with the coil, although these features have been omitted to simplify the drawing. The coil and iron cylinder assembly is known as the armature. Each end of the wire coil is attached to a metallic half-ring. Rubbing against each of the half-rings is a graphite contact called a brush. While the half-rings rotate with the coil, the graphite brushes remain stationary. The two half-rings and the associated brushes are referred to as a split-ring commutator (see below).

Figure 21-22 (a) Maximum torque occurs when the normal to the plane of the loop is perpendicular to the magnetic field. (b) The torque is zero when the normal is parallel to the field.

Figure 21-23 The basic components of a dc motor. A CD platter is shown as it might be attached to the motor.

The operation of a motor can be understood by considering Figure 21-24. In part a the current from the battery enters the coil through the left brush and half-ring, goes around the coil, and then leaves through the right half-ring and brush. Consistent with RHR-1, the directions of the magnetic forces and on the two sides of the coil are as shown in the drawing. These forces produce the torque that turns the coil. Eventually the coil reaches the position shown in part b of the drawing. In this position the half-rings momentarily lose electrical contact with the brushes, so that there is no current in the coil and no applied torque. However, like any moving object, the rotating coil does not stop immediately, for its inertia carries it onward. When the half-rings reestablish contact with the brushes, there again is a current in the coil, and a magnetic torque again rotates the coil in the same direction. The split-ring commutator ensures that the current is always in the proper direction to yield a torque that produces a continuous rotation of the coil.

Figure 21-24 (a) When a current exists in the coil, the coil experiences a torque. (b) Because of its inertia, the coil continues to rotate when there is no current.

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