21.7

Magnetic Fields Produced by Currents

We have seen that a current-carrying wire can experience a magnetic force when placed in a magnetic field that is produced by an external source, such as a permanent magnet. A current-carrying wire also produces a magnetic field of its own, as we will see in this section. Hans Christian Oersted (1777–1851) first discovered this effect in 1820 when he observed that a current-carrying wire influences the orientation of a nearby compass needle. The compass needle aligns itself with the net magnetic field produced by the current and the magnetic field of the earth. Oersted’s discovery, which linked the motion of electric charges with the creation of a magnetic field, marked the beginning of an important discipline called electromagnetism.

| A Long, Straight Wire

Figure 21-25a illustrates Oersted’s discovery with a very long, straight wire. When a current is present, the compass needles point in a circular pattern about the wire. The pattern indicates that the magnetic field lines produced by the current are circles centered on the wire. If the direction of the current is reversed, the needles also reverse their directions, indicating that the direction of the magnetic field has reversed. The direction of the field can be obtained by using Right-Hand Rule No. 2 (RHR-2), as part b of the drawing indicates:

> Right-Hand Rule No. 2. Curl the fingers of the right hand into the shape of a half-circle. Point the thumb in the direction of the conventional current I, and the tips of the fingers will point in the direction of the magnetic field .

Figure 21-25 (a) A very long, straight, current-carrying wire produces magnetic field lines that are circular about the wire, as indicated by the compass needles. (b) With the thumb of the right hand (R.H.) along the current I, the curled fingers point in the direction of the magnetic field, according to RHR-2.

Experimentally, it is found that the magnitude B of the magnetic field produced by an infinitely long, straight wire is directly proportional to the current I and inversely proportional to the radial distance r from the wire: . As usual, this proportionality is converted into an equation by introducing a proportionality constant, which, in this instance, is written as . Thus, the magnitude of the magnetic field is

| (21.5)

The constant is known as the permeability of free space, and its value is . The magnetic field becomes stronger nearer the wire, where r is smaller. Therefore, the field lines near the wire are closer together than those located farther away, where the field is weaker. Figure 21-26 shows the pattern of field lines.

Figure 21-26 The magnetic field becomes stronger as the radial distance r decreases, so the field lines are closer together near the wire.

The magnetic field that surrounds a current-carrying wire can exert a force on a moving charge, as the next example illustrates.

A N A L Y Z I N G M U L T I P L E - C O N C E P T P R O B L E M S

| Example 7 | | | A Current Exerts a Magnetic Force on a Moving Charge

Figure 21-27 shows a very long, straight wire carrying a current of 3.0 A. A particle has a charge of and is moving parallel to the wire at a distance of 0.050 m. The speed of the particle is 280 m/s. Determine the magnitude and direction of the magnetic force exerted on the charged particle by the current in the wire.

Figure 21-27 The moving positive charge q0 experiences a magnetic force because of the magnetic field produced by the current in the wire.

Reasoning The current generates a magnetic field in the space around the wire. The charged particle moves in the presence of this field and, therefore, can experience a magnetic force. The magnitude of this force is given by Equation 21.1, and the direction can be determined by applying RHR-1 (see Section 21.2). Note in Figure 21-27 that the magnetic field produced by the current lies in a plane that is perpendicular to both the wire and the velocity of the particle. Thus, the angle between and is .
Knowns and Unknowns The following list summarizes the data that are given:

Description | Symbol | Value | Comment
Explicit Data |  |  |
Current in wire | I | 3.0 A |
Electric charge of particle | q0 | +6.5 × 10−6 C |
Distance of particle from wire | r | 0.050 m | Particle moves parallel to wire; see Figure 21-27.
Speed of particle | v | 280 m/s |
Implicit Data |  |  |
Directional angle of particle velocity with respect to magnetic field | θ | 90.0° | Particle moves parallel to wire; see Reasoning.
Unknown Variable |  |  |
Magnitude of magnetic force exerted on particle | F | ? |

Modeling the Problem

| Step 1 Magnetic Force on Particle
| The magnitude F of the magnetic force acting on the charged particle is given at the right by Equation 21.1, where is the magnitude of the charge, v is the particle speed, B is the magnitude of the magnetic field produced by the wire, and is the angle between the particle velocity and the magnetic field. Values are given for , v, and . The value of B, however, is unknown, and we determine it in Step 2.

| (21.1)

| Step 2 The Magnetic Field Produced by the Wire
| The magnitude B of the magnetic field produced by a current I in an infinitely long, straight wire is given by Equation 21.5:

| (21.5)

where is the permeability of free space and r is the distance from the wire. This expression can be substituted into Equation 21.1, as shown at the right.

Solution Combining the results of each step algebraically, we find that

The magnitude of the magnetic force on the charged particle is

The direction of the magnetic force is predicted by RHR-1 and, as shown in Figure 21-27, is radially inward toward the wire.
Related Homework: Problem 50

Need more practice?

Interactive LearningWare21.3
A long, straight wire is oriented in the north–south direction, and the current in the wire is directed to the north. The horizontal component of the earth’s magnetic field is and points due north. A small horizontal compass needle is located directly below the wire, 1.9 cm from it. The compass needle points 35° north of west. What is the current in the wire?

We have now seen that an electric current can create a magnetic field of its own. Earlier, we have also seen that an electric current can experience a force created by another magnetic field. Therefore, the magnetic field that one current creates can exert a force on another nearby current. Examples 8 and 9 deal with this magnetic interaction between currents.

A N A L Y Z I N G M U L T I P L E - C O N C E P T P R O B L E M S

| Example 8 | | | Two Current-Carrying Wires Exert Magnetic Forces on One Another

Figure 21-28 shows two parallel, straight wires that are very long. The wires are separated by a distance of 0.065 m and carry currents of I1 = 15 A and I2 = 7.0 A. Find the magnitude and direction of the force that the magnetic field of wire 1 applies to a 1.5-m section of wire 2 when the currents have (a) opposite and (b) the same directions.

Figure 21-28 (a) Two long, parallel wires carrying currents I1 and I2 in opposite directions repel each other. (b) The wires attract each other when the currents are in the same direction.

Reasoning The current I2 is situated in the magnetic field produced by the current I1. Therefore, a length L of wire 2 experiences a magnetic force due to this field. The magnitude of the force is given by Equation 21.3 and the direction by RHR-1. Note in Figure 21-28 that the direction of the magnetic field produced by the current I1 is upward at the location of wire 2, as part a of the figure shows. The direction can be obtained using RHR-2 (thumb of right hand along I1 curled fingers point upward at wire 2 and indicate the direction of ). Thus, is perpendicular to wire 2, and it follows that the angle between and I2 is .
Knowns and Unknowns The following data are available:

Description | Symbol | Value | Comment
Explicit Data |  |  |
Separation between wires | r | 0.065 m | Wires are parallel; see Figure 21-28.
Current in wire 1 | I1 | 15 A |
Current in wire 2 | I2 | 7.0 A |
Length of section of wire 2 | L | 1.5 m |
Implicit Data |  |  |
Directional angle of I2 with respect to magnetic field produced by I1 |  | 90.0° | Wires are parallel; see Reasoning.
Unknown Variable |  |  |
Magnitude of magnetic force exerted on section of wire 2 | F | ? |

Modeling the Problem

| Step 1 Magnetic Force on Section of Wire 2
| The magnitude F of the magnetic force acting on the section of wire 2 is given at the right by Equation 21.3. In this expression I2 is the current in wire 2, L is the length of the section, B is the magnitude of the magnetic field produced by I1 and is the angle between I2 and the magnetic field. We have values for I2, L, and . To determine B, we turn to Step 2.

| (21.3)

| Step 1 Magnetic Field Produced by the Current in Wire 1
| The magnitude B of the magnetic field produced by the current I1 is given by Equation 21.5:

| (21.5)

where is the permeability of free space and r is the distance from the wire. This expression can be substituted into Equation 21.3, as shown at the right.

Solution Combining the results of each step algebraically, we find that

a. | The magnitude of the magnetic force on the 1.5-m section of wire 2 is

The direction of the magnetic force is away from wire 1, as Figure 21-28a indicates. The direction is found by using RHR-1 (fingers of the right hand extend upward along , thumb points along I2, palm pushes in the direction of the force ).
In a like manner, the current in wire 2 also creates a magnetic field that produces a force on wire 1. Reasoning similar to that above shows that a 1.5-m section of wire 1 is repelled from wire 2 with a force that also has a magnitude of . Thus, each wire generates a force on the other, and if the currents are in opposite directions, the wires repel each other. The fact that the two wires exert oppositely directed forces of equal magnitude on each other is consistent with Newton’s third law, the action–reaction law.*
b. | If the current in wire 2 is reversed, as Figure 21-28b shows, wire 2 is attracted to wire 1 because the direction of the magnetic force is reversed. However, the magnitude of the force is the same as that calculated in part a. Likewise, wire 1 is attracted to wire 2. Two parallel wires carrying currents in the same direction attract each other.

Related Homework: Problems 53, 54

| Conceptual Example 9 | | | The Net Force That a Current-Carrying Wire Exerts on a Current-Carrying Coil

Figure 21-29 shows a very long, straight wire carrying a current I1 and a rectangular coil carrying a current I2. The wire and the coil lie in the same plane, with the wire parallel to the long sides of the rectangle. Is the coil attracted to or repelled from the wire?

Figure 21-29 A very long, straight wire carries a current I1, and a rectangular coil carries a current I2. The dashed line is parallel to the wire and locates a small segment on each short side of the coil.

Reasoning and Solution The current in the straight wire exerts a force on each of the four sides of the coil. The net force acting on the coil is the vector sum of these four forces. To decide whether the net force is attractive or repulsive, we need to consider the directions and magnitudes of the individual forces. In the long side of the coil near the wire, the current I2 has the same direction as the current I1, and we have just seen in Example 8 that two such currents attract each other. In the long side of the coil farthest from the wire, I2 has a direction opposite to that of I1 and, according to Example 8, they repel one another. However, the attractive force is stronger than the repulsive force because the magnetic field produced by the current I1 is stronger at shorter distances than at greater distances. Consequently, we reach the preliminary conclusion that the coil is attracted to the wire.
But what about the forces that act on the two short sides? Consider a small segment of each of the short sides, located at the same distance from the straight wire, as indicated by the dashed line in Figure 21-29. Each of these segments experiences the same magnetic field from the current I1. RHR-2 shows that this field is directed downward into the plane of the paper, so that it is perpendicular to the current I2 in each segment. But the directions of I2 in the segments are opposite. As a result, RHR-1 reveals that the magnetic field from the straight wire applies a force to one segment that is opposite to that applied to the other segment. Thus, the forces acting on the two short sides of the coil cancel, and our preliminary conclusion is valid; the coil is attracted to the wire.
Related Homework: Problem 54

| A Loop of Wire

If a current-carrying wire is bent into a circular loop, the magnetic field lines around the loop have the pattern shown in Figure 21-30a. At the center of a loop of radius R, the magnetic field is perpendicular to the plane of the loop and has the value , where I is the current in the loop. Often, the loop consists of N turns of wire that are wound sufficiently close together that they form a flat coil with a single radius. In this case, the magnetic fields of the individual turns add together to give a net field that is N times greater than that of a single loop. For such a coil the magnetic field at the center is

| (21.6)

The direction of the magnetic field at the center of the loop can be determined with the help of RHR-2. If the thumb of the right hand is pointed in the direction of the current and the curled fingers are placed at the center of the loop, as in Figure 21-30b, the fingers indicate that the magnetic field points from right to left.

Figure 21-30 (a) The magnetic field lines in the vicinity of a current-carrying circular loop. (b) The direction of the magnetic field at the center of the loop is given by RHR-2.

Example 10 shows how the magnetic fields produced by the current in a loop of wire and the current in a long, straight wire combine to form a net magnetic field.

| Example 10 | | | Finding the Net Magnetic Field

A long, straight wire carries a current of I1 = 8.0 A. As Figure 21-31a illustrates, a circular loop of wire lies immediately to the right of the straight wire. The loop has a radius of R = 0.030 m and carries a current of I2 = 2.0 A. Assuming that the thickness of the wires is negligible, find the magnitude and direction of the net magnetic field at the center C of the loop.

Figure 21-31 (a) A long, straight wire carrying a current I1 lies next to a circular loop that carries a current I2. (b) The magnetic fields at the center C of the loop produced by the straight wire () and the loop ().

Reasoning The net magnetic field at the point C is the sum of two contributions: (1) the field produced by the long, straight wire, and (2) the field produced by the circular loop. An application of RHR-2 shows that at point C the field is directed upward, perpendicular to the plane containing the straight wire and the loop (see part b of the drawing). Similarly, RHR-2 shows that the magnetic field is directed downward, opposite to the direction of .
Solution If we choose the upward direction in Figure 21-31b as positive, the net magnetic field at the point C is

The net field is positive, so it is directed upward, perpendicular to the plane.

Problem solving insight Do not confuse the formula for the magnetic field produced at the center of a circular loop with that of a very long, straight wire. The formulas are similar, differing only by a factor of in the denominator.

Check Your Understanding 4

Each of the four drawings shows the same three concentric loops of wire. The currents in the loops have the same magnitude I and have the directions shown. Rank the magnitude of the net magnetic field produced at the center of each of the four drawings, largest to smallest. (The answer is given at the end of the book.)

Background: A circular loop produces a magnetic field at its center, the magnitude of which depends on the current and the radius of the loop. The direction of the magnetic field can be found with the aid of Right-Hand Rule No. 2.
For similar questions (including calculational counterparts), consult Self-Assessment Test 21.2, which is described at the end of Section 21.8.

A comparison of the magnetic field lines around the current loop in Figure 21-30a with those in the vicinity of the short bar magnet in Figure 21-32a shows that the two patterns are quite similar. Not only are the patterns similar, but the loop itself behaves as a bar magnet with a “north pole” on one side and a “south pole” on the other side. To emphasize that the loop may be imagined to be a bar magnet, Figure 21-32b includes a “phantom” bar magnet at the center of the loop. The side of the loop that acts like a north pole can be determined with the aid of RHR-2: curl the fingers of the right hand into the shape of a half-circle, point the thumb along the current I, and place the curled fingers at the center of the loop. The fingers not only point in the direction of , but they also point toward the north pole.

Figure 21-32 (a) The field lines around the bar magnet resemble those around the loop in Figure 21-30a. (b) The current loop can be imagined to be a phantom bar magnet with a north pole and a south pole.

Because a current-carrying loop acts like a bar magnet, two adjacent loops can be either attracted to or repelled from each other, depending on the relative directions of the currents. Figure 21-33 includes a “phantom” magnet for each loop and shows that the loops are attracted to each other when the currents are in the same direction and repelled from each other when the currents are in opposite directions. This behavior is analogous to that of the two long, straight wires discussed in Example 8 (see Figure 21-28).

Figure 21-33 (a) The two current loops attract each other if the directions of the currents are the same and (b) repel each other if the directions are opposite. The “phantom” magnets help explain the attraction and repulsion.

| A Solenoid

A solenoid is a long coil of wire in the shape of a helix (see Figure 21-34). If the wire is wound so the turns are packed close to each other and the solenoid is long compared to its diameter, the magnetic field lines have the appearance shown in the drawing. Notice that the field inside the solenoid and away from its ends is nearly constant in magnitude and directed parallel to the axis. The direction of the field inside the solenoid is given by RHR-2, just as it is for a circular current loop. The magnitude of the magnetic field in the interior of a long solenoid is

| (21.7)

where n is the number of turns per unit length of the solenoid and I is the current. If, for example, the solenoid contains 100 turns and has a length of 0.05 m, the number of turns per unit length is n = (100 turns)/(0.05 m) = 2000 turns/m. The magnetic field outside the solenoid is not constant and is much weaker than the interior field. In fact, the magnetic field outside is nearly zero if the length of the solenoid is much greater than its diameter.

Figure 21-34 A solenoid and a cross-sectional view of it, showing the magnetic field lines and the north and south poles.

As with a single loop of wire, a solenoid can also be imagined to be a bar magnet, for the solenoid is just an array of connected current loops. And, as with a circular current loop, the location of the north pole can be determined with RHR-2. Figure 21-34 shows that the left end of the solenoid acts as a north pole, and the right end behaves as a south pole. Solenoids are often referred to as electromagnets, and they have several advantages over permanent magnets. For one thing, the strength of the magnetic field can be altered by changing the current and/or the number of turns per unit length. Furthermore, the north and south poles of an electromagnet can be readily switched by reversing the current.
The physics of magnetic resonance imaging (MRI). Applications of the magnetic field produced by a current-carrying solenoid are widespread. An exciting medical application is in magnetic resonance imaging (MRI). With this technique, detailed pictures of the internal parts of the body can be obtained in a noninvasive way that involves none of the risks inherent in the use of X-rays. Figure 21-35 shows a patient in position in a magnetic resonance imaging machine. The circular opening visible in the photograph provides access to the interior of a solenoid, which is typically made from superconducting wire. The superconducting wire facilitates the use of a large current to produce a strong magnetic field. In the presence of this field, the nuclei of certain atoms can be made to behave as tiny radio transmitters and emit radio waves similar to those used by FM stations. The hydrogen atom, which is so prevalent in the human body, can be made to behave in this fashion. The strength of the magnetic field determines where a given collection of hydrogen atoms will “broadcast” on an imaginary FM dial. With a magnetic field that has a slightly different strength at different places, it is possible to associate the location on this imaginary FM dial with a physical location within the body. Computer processing of these locations produces the magnetic resonance image. When hydrogen atoms are used in this way, the image is essentially a map showing their distribution within the body. Remarkably detailed images can now be obtained, such as that in Figure 21-36. They provide doctors with a powerful diagnostic tool that complements those from X-ray and other techniques. Surgeons can now perform operations more accurately by stepping inside specially designed MRI scanners along with the patient and seeing live images of the area into which they are cutting.

Figure 21-35 A magnetic resonance imaging (MRI) machine. The circular opening in which the patient is positioned provides access to the interior of a solenoid. MRI scans of the brain can be seen on the monitor. (David Mendelsohn/Masterfile)

The physics of television screens and computer display monitors. Television sets and computer display monitors use electromagnets (solenoids) to produce images by exerting magnetic forces on moving electrons. An evacuated glass tube, called a cathode-ray tube (CRT), contains an electron gun that sends a narrow beam of high-speed electrons toward the screen of the tube, as Figure 21-37a illustrates. The inner surface of the screen is covered with a phosphor coating, and when the electrons strike it, they generate a spot of visible light. This spot is called a pixel (a contraction of “picture element”).

Figure 21-36 Magnetic resonance imaging provides one way to diagnose brain disorders. This image shows the brain of a patient with an Arnold-Chiari deformity. In this congenital anomaly, an abnormally elongated cerebellum and medulla oblongata (the red-orange-yellow region just to the right of the top of the spine) protrude down into the spinal canal. (©ISM/Phototake)

Figure 21-37 (a) A cathode-ray tube contains an electron gun, a magnetic field for deflecting the electron beam, and a phosphor-coated screen. A color TV actually uses three guns, although only one is shown here for clarity. (b) The image is formed by scanning the electron beam across the screen. (c) The red, green, and blue phosphors of a color TV.

To create a black-and-white picture, the electron beam is scanned rapidly from left to right across the screen. As the beam makes each horizontal scan, the number of electrons per second striking the screen is changed by electronics controlling the electron gun, making the scan line brighter in some places and darker in others. When the beam reaches the right side of the screen, it is turned off and returned to the left side slightly below where it started (see part b of the figure). The beam is then scanned across the next line, and so on. In current TV sets, a complete picture consists of 525 scan lines (or 625 in Europe) and is formed in of a second. High-definition TV sets have about 1100 scan lines, giving a much sharper, more detailed picture.
The electron beam is deflected by a pair of electromagnets placed around the neck of the tube, between the electron gun and the screen. One electromagnet is responsible for producing the horizontal deflection of the beam and the other for the vertical deflection. For clarity, Figure 21-37a shows the net magnetic field generated by the electromagnets at one instant, and not the electromagnets themselves. The electric current in the electromagnets produces a net magnetic field that exerts a force on the moving electrons, causing their trajectories to bend and reach different points on the screen. Changing the current changes the field, so the electrons can be deflected to any point on the screen.
A color TV operates with three electron guns instead of one. And the single phosphor of a black-and-white TV is replaced by a large number of three-dot clusters of phosphors that glow red, green, and blue when struck by an electron beam, as indicated in Figure 21-37c. Each red, green, and blue color in a cluster is produced when electrons from one of the three guns strike the corresponding phosphor dot. The three dots are so close together that, from a normal viewing distance, they cannot be separately distinguished. Red, green, and blue are primary colors, so virtually all other colors can be created by varying the intensities of the three beams focused on a cluster.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.