21.5

The Force on a Current in a Magnetic Field

As we have seen, a charge moving through a magnetic field can experience a magnetic force. Since an electric current is a collection of moving charges, a current in the presence of a magnetic field can also experience a magnetic force. In Figure 21-17, for instance, a current-carrying wire is placed between the poles of a magnet. When the direction of the current I is as shown, the moving charges experience a magnetic force that pushes the wire to the right in the drawing. The direction of the force is determined in the usual manner by using RHR-1, with the minor modification that the direction of the velocity of a positive charge is replaced by the direction of the conventional current I. If the current in the drawing were reversed by switching the leads to the battery, the direction of the force would be reversed, and the wire would be pushed to the left.

Figure 21-17 The wire carries a current I, and the bottom segment of the wire is oriented perpendicular to a magnetic field . A magnetic force deflects the wire to the right.

When a charge moves through a magnetic field, the magnitude of the force that acts on the charge is (Equation 21.1). With the aid of Figure 21-18, this expression can be put into a form that is more suitable for use with an electric current. The drawing shows a wire of length L that carries a current I. The wire is oriented at an angle with respect to a magnetic field . This picture is similar to Figure 21-8c, except that now the charges move in a wire. The magnetic force exerted on this length of wire is the net force acting on the total amount of charge moving in the wire. Suppose that an amount of conventional positive charge travels the length of the wire in a time interval . The magnitude of the magnetic force on this amount of charge is given by Equation 21.1 as . Multiplying and dividing the right side of this equation by , we find that

According to Equation 20.1, the term is the current I in the wire, and the term is the length L of the wire. With these two substitutions, the expression for the magnetic force exerted on a current-carrying wire becomes

| (21.3)

As in the case of a single charge traveling in a magnetic field, the magnetic force on a current-carrying wire is a maximum when the wire is oriented perpendicular to the field and vanishes when the current is parallel or antiparallel to the field . The direction of the magnetic force is given by RHR-1, as Figure 21-18 indicates.

Figure 21-18 The current I in the wire, oriented at an angle with respect to a magnetic field , is acted upon by a magnetic force .

Check Your Understanding 3

The same current-carrying wire is placed in the same magnetic field in four different orientations (see the drawing). Rank the orientations according to the magnitude of the magnetic force exerted on the wire, largest to smallest. (The answer is given at the end of the book.)

Background: The magnetic force exerted on a current-carrying wire depends on the current, the length of the wire, the magnetic field, and the orientation of the wire relative to the field.
For similar questions (including calculational counterparts), consult Self-Assessment Test 21.2, which is described at the end of Section 21.8.

Most loudspeakers operate on the principle that a magnetic field exerts a force on a current-carrying wire. Figure 21-19a shows a speaker design that consists of three basic parts: a cone, a voice coil, and a permanent magnet. The cone is mounted so it can vibrate back and forth. When vibrating, it pushes and pulls on the air in front of it, thereby creating sound waves. Attached to the apex of the cone is the voice coil, which is a hollow cylinder around which coils of wire are wound. The voice coil is slipped over one of the poles of the stationary permanent magnet (the north pole in the drawing) and can move freely. The two ends of the voice-coil wire are connected to the speaker terminals on the back panel of a receiver.

Figure 21-19 (a) An “exploded” view of one type of speaker design, which shows a cone, a voice coil, and a permanent magnet. (b) Because of the current in the voice coil (shown as and ), the magnetic field causes a force to be exerted on the voice coil and cone.

The physics of a loudspeaker.

The receiver acts as an ac generator, sending an alternating current to the voice coil. The alternating current interacts with the magnetic field to generate an alternating force that pushes and pulls on the voice coil and the attached cone. To see how the magnetic force arises, consider Figure 21-19b, which is a cross-sectional view of the voice coil and the magnet. In the cross-sectional view, the current is directed into the page in the upper half of the voice coil () and out of the page in the lower half (). In both cases the magnetic field is perpendicular to the current, so the maximum possible force is exerted on the wire. An application of RHR-1 to both the upper and lower halves of the voice coil shows that the magnetic force in the drawing is directed to the right, causing the cone to accelerate in that direction. One-half of a cycle later when the current is reversed, the direction of the magnetic force is also reversed, and the cone accelerates to the left. If, for example, the alternating current from the receiver has a frequency of 1000 Hz, the alternating magnetic force causes the cone to vibrate back and forth at the same frequency, and a 1000-Hz sound wave is produced. Thus, it is the magnetic force on a current-carrying wire that is responsible for converting an electrical signal into a sound wave. In Example 5 a typical force and acceleration in a loudspeaker are determined.

Problem solving insight Whenever the current in a wire reverses direction, the force exerted on the wire by a given magnetic field also reverses direction.

| Example 5 | | | The Force and Acceleration in a Loudspeaker

The voice coil of a speaker has a diameter of d = 0.025 m, contains 55 turns of wire, and is placed in a 0.10-T magnetic field. The current in the voice coil is 2.0 A. (a) Determine the magnetic force that acts on the coil and cone. (b) The voice coil and cone have a combined mass of 0.020 kg. Find their acceleration.
Reasoning The magnetic force that acts on the current-carrying voice coil is given by Equation 21.3 as . The effective length L of the wire in the voice coil is very nearly the number of turns N times the circumference of one turn: . The acceleration of the voice coil and cone is given by Newton’s second law as the magnetic force divided by the combined mass.
Solution

a. | Since the magnetic field acts perpendicular to all parts of the wire, and the force on the voice coil is

| (21.3)

b. | According to Newton’s second law, the acceleration of the voice coil and cone is

| (4.1)

This acceleration is more than four times the acceleration due to gravity.

Need more practice?

Interactive LearningWare21.2
Two conducting rails are 1.6 m apart and are parallel to the ground at the same height. As the drawing shows, a 0.20-kg aluminum rod is lying on top of the rails, and a 0.050-T magnetic field points upward, perpendicular to the ground. There is a current / in the rod, directed as in the drawing. The coefficient of static friction between the rod and each rail is . How much current is needed to make the rod begin moving, and in which direction will it move?

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Magnetohydrodynamic (MHD) propulsion is a revolutionary type of propulsion system that does not use propellers to power ships and submarines, but, instead, uses a magnetic force on a current. The method eliminates motors, drive shafts, and gears, as well as propellers, so it promises to be a low-noise system with great reliability at relatively low cost. Figure 21-20a shows the first ship to use MHD technology, the Yamato 1. In part b the schematic side view shows one of the two MHD propulsion units that are mounted underneath the vessel. Seawater enters the front of the unit and is expelled from the rear. A jet engine uses air in a similar fashion, taking it in in the front and pushing it out at the back to propel the plane forward.

Figure 21-20 (a) The Yamato 1 is the first ship to use magnetohydrodynamic (MHD) propulsion. (© Dennis Budd Gray) (b) A schematic side view of the Yamato 1. (c) In the MHD unit, the magnetic force exerted on the current pushes water out the back.

The physics of magnetohydrodynamic propulsion.

Figure 21-20c presents an enlarged view of a propulsion unit. An electromagnet (see Section 21.7) within the vessel uses superconducting wire to produce a strong magnetic field. Electrodes (metal plates) mounted on either side of the unit are attached to a dc electric generator. The generator sends an electric current through the seawater, from one electrode to the other, perpendicular to the magnetic field. Consistent with RHR-1, the field applies a force to the current, as the drawing shows. The magnetic force pushes the seawater, similar to the way in which it pushes the wire in Figure 21-17. As a result, a water jet is expelled from the tube. Since the MHD unit exerts a magnetic force on the water, the water exerts a force on the propulsion unit and the vessel attached to it. According to Newton’s third law, this “reaction” force is equal in magnitude, but opposite in direction, to the magnetic force, and it is the reaction force that provides the thrust to drive the vessel.

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