41 Torque on a Current Loop: Motors and Meters

Electric motors are among the most common applications of magnetic forces. They power devices ranging from electric toothbrushes and infusion pumps to MRI patient tables, ventilators, laboratory centrifuges, and surgical tools. In every electric motor, a current-carrying loop of wire is placed in a magnetic field. The interaction between the current and the magnetic field produces a torque, causing the loop to rotate and converting electrical energy into mechanical work (Figure 41.1).

To understand how a motor works, we examine the magnetic forces acting on each segment of the rectangular loop shown in Figure 41.1. The magnetic field is assumed to be uniform across the loop, which has width [latex]w[/latex] and height [latex]l[/latex].

The forces acting on the top and bottom segments of the loop are directed vertically, parallel to the axis of rotation. Because these forces act along the axis, they do not produce torque. Furthermore, they are equal in magnitude and opposite in direction, so they cancel and produce no net force on the loop.

The situation is different for the two vertical sides of the loop. As illustrated in Figure 41.2(a), the magnetic forces on these sides are also equal in magnitude and opposite in direction. Although the net force remains zero, each force acts at a distance from the axis of rotation, producing a torque. Since both torques act in the same rotational direction, they add together.

Torque is defined by

where [latex]F[/latex] is the applied force, [latex]r[/latex] is the distance from the axis of rotation to the point where the force acts, and [latex]\theta[/latex] is the angle between the position vector and the force.

For each side of the loop, the distance from the axis is [latex]r=w/2[/latex]. Adding the torques from both sides gives

Each vertical segment has length [latex]l[/latex] and is perpendicular to the magnetic field, so the magnetic force on each side is

Substituting this expression into the torque equation gives

Most practical motors contain many loops of wire rather than just one. If the coil has [latex]N[/latex] turns, the total torque is multiplied by [latex]N[/latex]. Since the area enclosed by each loop is

the torque on a current-carrying loop becomes

This equation applies to loops of any shape placed in a uniform magnetic field. Here, [latex]N[/latex] is the number of turns, [latex]I[/latex] is the current, [latex]A[/latex] is the area of each loop, [latex]B[/latex] is the magnetic field strength, and [latex]\theta[/latex] is the angle between the magnetic field and a line perpendicular to the plane of the loop.

A useful feature of this arrangement is that the magnetic forces on opposite sides of the loop always cancel, producing no net force. Instead of causing the loop to translate, the magnetic field produces only a torque, allowing the loop to rotate smoothly.

The torque calculated in the previous example is the maximum possible torque. As the loop rotates, the angle [latex]\theta[/latex] changes, causing the torque to decrease. When the plane of the loop becomes perpendicular to the magnetic field, [latex]\theta=0^\circ[/latex], the torque becomes zero. If the current remained unchanged, the torque would reverse direction after the loop passed this position, causing the coil to slow down and simply oscillate back and forth instead of continuing to rotate.

To produce continuous rotation, electric motors automatically reverse the direction of the current every half turn using a device called a commutator and stationary electrical contacts called brushes. Reversing the current reverses the magnetic forces on the coil, keeping the torque in the same rotational direction throughout the motion.

The same physical principle is used in many analog measuring instruments, including galvanometers, analog ammeters, voltmeters, and some automobile gauges. As shown in Figure 41.4, these devices resemble small electric motors, but instead of rotating continuously, the coil turns only through a limited angle.

The magnets are specially shaped so that the magnetic field remains nearly perpendicular to the current-carrying coil over its operating range. As a result, the magnetic torque depends primarily on the current rather than on the coil's orientation. A small spring provides an opposing torque, and the pointer comes to rest when the spring torque balances the magnetic torque. Because the magnetic torque is proportional to the current, the pointer deflection is also proportional to the current, producing a linear measurement scale. Increasing the loop area, magnetic field strength, or number of turns improves the instrument's sensitivity, allowing it to detect very small currents.