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Physics

20.1 Magnetic Fields, Field Lines, and Force

Physics20.1 Magnetic Fields, Field Lines, and Force
  1. Preface
  2. 1 What is Physics?
    1. Introduction
    2. 1.1 Physics: Definitions and Applications
    3. 1.2 The Scientific Methods
    4. 1.3 The Language of Physics: Physical Quantities and Units
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  3. 2 Motion in One Dimension
    1. Introduction
    2. 2.1 Relative Motion, Distance, and Displacement
    3. 2.2 Speed and Velocity
    4. 2.3 Position vs. Time Graphs
    5. 2.4 Velocity vs. Time Graphs
    6. Key Terms
    7. Section Summary
    8. Key Equations
    9. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    10. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  4. 3 Acceleration
    1. Introduction
    2. 3.1 Acceleration
    3. 3.2 Representing Acceleration with Equations and Graphs
    4. Key Terms
    5. Section Summary
    6. Key Equations
    7. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    8. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  5. 4 Forces and Newton’s Laws of Motion
    1. Introduction
    2. 4.1 Force
    3. 4.2 Newton's First Law of Motion: Inertia
    4. 4.3 Newton's Second Law of Motion
    5. 4.4 Newton's Third Law of Motion
    6. Key Terms
    7. Section Summary
    8. Key Equations
    9. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    10. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  6. 5 Motion in Two Dimensions
    1. Introduction
    2. 5.1 Vector Addition and Subtraction: Graphical Methods
    3. 5.2 Vector Addition and Subtraction: Analytical Methods
    4. 5.3 Projectile Motion
    5. 5.4 Inclined Planes
    6. 5.5 Simple Harmonic Motion
    7. Key Terms
    8. Section Summary
    9. Key Equations
    10. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    11. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  7. 6 Circular and Rotational Motion
    1. Introduction
    2. 6.1 Angle of Rotation and Angular Velocity
    3. 6.2 Uniform Circular Motion
    4. 6.3 Rotational Motion
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  8. 7 Newton's Law of Gravitation
    1. Introduction
    2. 7.1 Kepler's Laws of Planetary Motion
    3. 7.2 Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity
    4. Key Terms
    5. Section Summary
    6. Key Equations
    7. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    8. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  9. 8 Momentum
    1. Introduction
    2. 8.1 Linear Momentum, Force, and Impulse
    3. 8.2 Conservation of Momentum
    4. 8.3 Elastic and Inelastic Collisions
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  10. 9 Work, Energy, and Simple Machines
    1. Introduction
    2. 9.1 Work, Power, and the Work–Energy Theorem
    3. 9.2 Mechanical Energy and Conservation of Energy
    4. 9.3 Simple Machines
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  11. 10 Special Relativity
    1. Introduction
    2. 10.1 Postulates of Special Relativity
    3. 10.2 Consequences of Special Relativity
    4. Key Terms
    5. Section Summary
    6. Key Equations
    7. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    8. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  12. 11 Thermal Energy, Heat, and Work
    1. Introduction
    2. 11.1 Temperature and Thermal Energy
    3. 11.2 Heat, Specific Heat, and Heat Transfer
    4. 11.3 Phase Change and Latent Heat
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  13. 12 Thermodynamics
    1. Introduction
    2. 12.1 Zeroth Law of Thermodynamics: Thermal Equilibrium
    3. 12.2 First law of Thermodynamics: Thermal Energy and Work
    4. 12.3 Second Law of Thermodynamics: Entropy
    5. 12.4 Applications of Thermodynamics: Heat Engines, Heat Pumps, and Refrigerators
    6. Key Terms
    7. Section Summary
    8. Key Equations
    9. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    10. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  14. 13 Waves and Their Properties
    1. Introduction
    2. 13.1 Types of Waves
    3. 13.2 Wave Properties: Speed, Amplitude, Frequency, and Period
    4. 13.3 Wave Interaction: Superposition and Interference
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  15. 14 Sound
    1. Introduction
    2. 14.1 Speed of Sound, Frequency, and Wavelength
    3. 14.2 Sound Intensity and Sound Level
    4. 14.3 Doppler Effect and Sonic Booms
    5. 14.4 Sound Interference and Resonance
    6. Key Terms
    7. Section Summary
    8. Key Equations
    9. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    10. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  16. 15 Light
    1. Introduction
    2. 15.1 The Electromagnetic Spectrum
    3. 15.2 The Behavior of Electromagnetic Radiation
    4. Key Terms
    5. Section Summary
    6. Key Equations
    7. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    8. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  17. 16 Mirrors and Lenses
    1. Introduction
    2. 16.1 Reflection
    3. 16.2 Refraction
    4. 16.3 Lenses
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  18. 17 Diffraction and Interference
    1. Introduction
    2. 17.1 Understanding Diffraction and Interference
    3. 17.2 Applications of Diffraction, Interference, and Coherence
    4. Key Terms
    5. Section Summary
    6. Key Equations
    7. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    8. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  19. 18 Static Electricity
    1. Introduction
    2. 18.1 Electrical Charges, Conservation of Charge, and Transfer of Charge
    3. 18.2 Coulomb's law
    4. 18.3 Electric Field
    5. 18.4 Electric Potential
    6. 18.5 Capacitors and Dielectrics
    7. Key Terms
    8. Section Summary
    9. Key Equations
    10. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    11. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  20. 19 Electrical Circuits
    1. Introduction
    2. 19.1 Ohm's law
    3. 19.2 Series Circuits
    4. 19.3 Parallel Circuits
    5. 19.4 Electric Power
    6. Key Terms
    7. Section Summary
    8. Key Equations
    9. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    10. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  21. 20 Magnetism
    1. Introduction
    2. 20.1 Magnetic Fields, Field Lines, and Force
    3. 20.2 Motors, Generators, and Transformers
    4. 20.3 Electromagnetic Induction
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  22. 21 The Quantum Nature of Light
    1. Introduction
    2. 21.1 Planck and Quantum Nature of Light
    3. 21.2 Einstein and the Photoelectric Effect
    4. 21.3 The Dual Nature of Light
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  23. 22 The Atom
    1. Introduction
    2. 22.1 The Structure of the Atom
    3. 22.2 Nuclear Forces and Radioactivity
    4. 22.3 Half Life and Radiometric Dating
    5. 22.4 Nuclear Fission and Fusion
    6. 22.5 Medical Applications of Radioactivity: Diagnostic Imaging and Radiation
    7. Key Terms
    8. Section Summary
    9. Key Equations
    10. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Performance Task
    11. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  24. 23 Particle Physics
    1. Introduction
    2. 23.1 The Four Fundamental Forces
    3. 23.2 Quarks
    4. 23.3 The Unification of Forces
    5. Key Terms
    6. Section Summary
    7. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Performance Task
    8. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  25. A | Reference Tables
  26. Index

Section Learning Objectives

By the end of this section, you will be able to do the following:

  • Summarize properties of magnets and describe how some nonmagnetic materials can become magnetized
  • Describe and interpret drawings of magnetic fields around permanent magnets and current-carrying wires
  • Calculate the magnitude and direction of magnetic force in a magnetic field and the force on a current-carrying wire in a magnetic field

Teacher Support

Teacher Support

The learning objectives in this section will help your students master the following standards:

  • (5) The student knows the nature of forces in the physical world. The student is expected to:
    • (G) investigate and describe the relationship between electric and magnetic fields in applications such as generators, motors, and transformers.

In addition, the High School Physics Laboratory Manual addresses content in this section in the lab titled: Magnetism, as well as the following standards:

  • (5) Science concepts. The student knows the nature of forces in the physical world. The student is expected to:
    • (G) investigate and describe the relationship between electric and magnetic fields in applications such as generators, motors, and transformers.

Section Key Terms

Curie temperature domain electromagnet electromagnetism ferromagnetic
magnetic dipole magnetic field magnetic pole magnetized north pole
permanent magnet right-hand rule solenoid south pole

Magnets and Magnetization

People have been aware of magnets and magnetism for thousands of years. The earliest records date back to ancient times, particularly in the region of Asia Minor called Magnesia—the name of this region is the source of words like magnet. Magnetic rocks found in Magnesia, which is now part of western Turkey, stimulated interest during ancient times. When humans first discovered magnetic rocks, they likely found that certain parts of these rocks attracted bits of iron or other magnetic rocks more strongly than other parts. These areas are called the poles of a magnet. A magnetic pole is the part of a magnet that exerts the strongest force on other magnets or magnetic material, such as iron. For example, the poles of the bar magnet shown in Figure 20.2 are where the paper clips are concentrated.

A bar magnet with paper clips attached to it.
Figure 20.2 A bar magnet with paper clips attracted to the two poles.

If a bar magnet is suspended so that it rotates freely, one pole of the magnet will always turn toward the north, with the opposite pole facing south. This discovery led to the compass, which is simply a small, elongated magnet mounted so that it can rotate freely. An example of a compass is shown Figure 20.3. The pole of the magnet that orients northward is called the north pole, and the opposite pole of the magnet is called the south pole.

A compass on a lanyard.
Figure 20.3 A compass is an elongated magnet mounted in a device that allows the magnet to rotate freely.

The discovery that one particular pole of a magnet orients northward, whereas the other pole orients southward allowed people to identify the north and south poles of any magnet. It was then noticed that the north poles of two different magnets repel each other, and likewise for the south poles. Conversely, the north pole of one magnet attracts the south pole of other magnets. This situation is analogous to that of electric charge, where like charges repel and unlike charges attract. In magnets, we simply replace charge with pole: Like poles repel and unlike poles attract. This is summarized in Figure 20.4, which shows how the force between magnets depends on their relative orientation.

Bar magnets oriented in the same direction repel, whereas if they are oriented in the opposite direction of each other, they attract.
Figure 20.4 Depending on their relative orientation, magnet poles will either attract each other or repel each other.

Consider again the fact that the pole of a magnet that orients northward is called the north pole of the magnet. If unlike poles attract, then the magnetic pole of Earth that is close to the geographic North Pole must be a magnetic south pole! Likewise, the magnetic pole of Earth that is close to the geographic South Pole must be a magnetic north pole. This situation is depicted in Figure 20.5, in which Earth is represented as containing a giant internal bar magnet with its magnetic south pole at the geographic North Pole and vice versa. If we were to somehow suspend a giant bar magnet in space near Earth, then the north pole of the space magnet would be attracted to the south pole of Earth’s internal magnet. This is in essence what happens with a compass needle: Its magnetic north pole is attracted to the magnet south pole of Earth’s internal magnet.

Earth depicted as a giant bar magnet whose magnetic north pole is at its geographic South Pole and vice versa.
Figure 20.5 Earth can be thought of as containing a giant magnet running through its core. The magnetic south pole of Earth’s magnet is at the geographic North Pole, so the north pole of magnets is attracted to the North Pole, which is how the north pole of magnets got their name. Likewise, the south pole of magnets is attracted to the geographic South Pole of Earth.

What happens if you cut a bar magnet in half? Do you obtain one magnet with two south poles and one magnet with two north poles? The answer is no: Each half of the bar magnet has a north pole and a south pole. You can even continue cutting each piece of the bar magnet in half, and you will always obtain a new, smaller magnet with two opposite poles. As shown in Figure 20.6, you can continue this process down to the atomic scale, and you will find that even the smallest particles that behave as magnets have two opposite poles. In fact, no experiment has ever found any object with a single magnetic pole, from the smallest subatomic particle such as electrons to the largest objects in the universe such as stars. Because magnets always have two poles, they are referred to as magnetic dipolesdi means two. Below, we will see that magnetic dipoles have properties that are analogous to electric dipoles.

A magnet cut into smaller and smaller sizes, all with a north and a south pole.
Figure 20.6 All magnets have two opposite poles, from the smallest, such as subatomic particles, to the largest, such as stars.

Watch Physics

Introduction to Magnetism

This video provides an interesting introduction to magnetism and discusses, in particular, how electrons around their atoms contribute to the magnetic effects that we observe.

Grasp Check

Toward which magnetic pole of Earth is the north pole of a compass needle attracted?

  1. The north pole of a compass needle is attracted to the north magnetic pole of Earth, which is located near the geographic North Pole of Earth.
  2. The north pole of a compass needle is attracted to the south magnetic pole of Earth, which is located near the geographic North Pole of Earth.
  3. The north pole of a compass needle is attracted to the north magnetic pole of Earth, which is located near the geographic South Pole of Earth.
  4. The north pole of a compass needle is attracted to the south magnetic pole of Earth, which is located near the geographic South Pole of Earth.

Only certain materials, such as iron, cobalt, nickel, and gadolinium, exhibit strong magnetic effects. Such materials are called ferromagnetic, after the Latin word ferrum for iron. Other materials exhibit weak magnetic effects, which are detectable only with sensitive instruments. Not only do ferromagnetic materials respond strongly to magnets—the way iron is attracted to magnets—but they can also be magnetized themselves—that is, they can be induced to be magnetic or made into permanent magnets (Figure 20.7). A permanent magnet is simply a material that retains its magnetic behavior for a long time, even when exposed to demagnetizing influences.

An iron bar is magnetized when heated and cooled (or tapped when cold) when placed between a magnet’s north and south poles.
Figure 20.7 An unmagnetized piece of iron is placed between two magnets, heated, and then cooled, or simply tapped when cold. The iron becomes a permanent magnet with the poles aligned as shown: Its south pole is adjacent to the north pole of the original magnet, and its north pole is adjacent to the south pole of the original magnet. Note that attractive forces are created between the central magnet and the outer magnets.

When a magnet is brought near a previously unmagnetized ferromagnetic material, it causes local magnetization of the material with unlike poles closest, as in the right side of Figure 20.7. This causes an attractive force, which is why unmagnetized iron is attracted to a magnet.

What happens on a microscopic scale is illustrated in Figure 7(a). Regions within the material called domains act like small bar magnets. Within domains, the magnetic poles of individual atoms are aligned. Each atom acts like a tiny bar magnet. Domains are small and randomly oriented in an unmagnetized ferromagnetic object. In response to an external magnetic field, the domains may grow to millimeter size, aligning themselves, as shown in Figure 7(b). This induced magnetization can be made permanent if the material is heated and then cooled, or simply tapped in the presence of other magnets.

Part (a) shows domains in an unmagnetized ferromagnetic material. Part (b) shows alignment of the domains when the material is magnetized.
Figure 20.8 (a) An unmagnetized piece of iron—or other ferromagnetic material—has randomly oriented domains. (b) When magnetized by an external magnet, the domains show greater alignment, and some grow at the expense of others. Individual atoms are aligned within domains; each atom acts like a tiny bar magnet.

Conversely, a permanent magnet can be demagnetized by hard blows or by heating it in the absence of another magnet. Increased thermal motion at higher temperature can disrupt and randomize the orientation and size of the domains. There is a well-defined temperature for ferromagnetic materials, which is called the Curie temperature, above which they cannot be magnetized. The Curie temperature for iron is 1,043 K (770 °C °C ), which is well above room temperature. There are several elements and alloys that have Curie temperatures much lower than room temperature and are ferromagnetic only below those temperatures.

Snap Lab

Refrigerator Magnets

We know that like magnetic poles repel and unlike poles attract. See if you can show this for two refrigerator magnets. Will the magnets stick if you turn them over? Why do they stick to the refrigerator door anyway? What can you say about the magnetic properties of the refrigerator door near the magnet? Do refrigerator magnets stick to metal or plastic spoons? Do they stick to all types of metal?

Teacher Support

Teacher Support

Holding a magnetic close to an unmagnetized ferromagnetic material will magnetically polarize the ferromagnetic material, causing the atomic magnetic dipoles to orient towards the external magnet. This is similar to electric polarization. Thus, the ferromagnetic material becomes magnetized in the presence of the external magnet, and the two magnets attract each other. For a magnet to stick to the refrigerator door, the door must contain some ferromagnetic material. Magnets will stick to ferrous spoons, for example spoons with iron in them, but not to nonferrous spoons, such as spoons made from Al or Ag, and will not stick to a magnet. Magnets will also not stick to plastic spoons.

Grasp Check

You have one magnet with the north and south poles labeled. How can you use this magnet to identify the north and south poles of other magnets?

  1. If the north pole of a known magnet is repelled by a pole of an unknown magnet on bringing them closer, that pole of unknown magnet is its north pole; otherwise, it is its south pole.
  2. If the north pole of known magnet is attracted to a pole of an unknown magnet on bringing them closer, that pole of unknown magnet is its north pole; otherwise, it is its south pole.

Magnetic Fields

We have thus seen that forces can be applied between magnets and between magnets and ferromagnetic materials without any contact between the objects. This is reminiscent of electric forces, which also act over distances. Electric forces are described using the concept of the electric field, which is a force field around electric charges that describes the force on any other charge placed in the field. Likewise, a magnet creates a magnetic field around it that describes the force exerted on other magnets placed in the field. As with electric fields, the pictorial representation of magnetic field lines is very useful for visualizing the strength and direction of the magnetic field.

As shown in Figure 20.9, the direction of magnetic field lines is defined to be the direction in which the north pole of a compass needle points. If you place a compass near the north pole of a magnet, the north pole of the compass needle will be repelled and point away from the magnet. Thus, the magnetic field lines point away from the north pole of a magnet and toward its south pole.

Two bar magnets with magnetic field lines emanating from them in complete loops.
Figure 20.9 The black lines represent the magnetic field lines of a bar magnet. The field lines point in the direction that the north pole of a small compass would point, as shown at left. Magnetic field lines never stop, so the field lines actually penetrate the magnet to form complete loops, as shown at right.

Magnetic field lines can be mapped out using a small compass. The compass is moved from point to point around a magnet, and at each point, a short line is drawn in the direction of the needle, as shown in Figure 20.10. Joining the lines together then reveals the path of the magnetic field line. Another way to visualize magnetic field lines is to sprinkle iron filings around a magnet. The filings will orient themselves along the magnetic field lines, forming a pattern such as that shown on the right in Figure 20.10.

Virtual Physics

Using a Compass to Map Out the Magnetic Field

This simulation presents you with a bar magnet and a small compass. Begin by dragging the compass around the bar magnet to see in which direction the magnetic field points. Note that the strength of the magnetic field is represented by the brightness of the magnetic field icons in the grid pattern around the magnet. Use the magnetic field meter to check the field strength at several points around the bar magnet. You can also flip the polarity of the magnet, or place Earth on the image to see how the compass orients itself.

Grasp Check

With the slider at the top right of the simulation window, set the magnetic field strength to 100 percent . Now use the magnetic field meter to answer the following question: Near the magnet, where is the magnetic field strongest and where is it weakest? Don’t forget to check inside the bar magnet.

  1. The magnetic field is strongest at the center and weakest between the two poles just outside the bar magnet. The magnetic field lines are densest at the center and least dense between the two poles just outside the bar magnet.
  2. The magnetic field is strongest at the center and weakest between the two poles just outside the bar magnet. The magnetic field lines are least dense at the center and densest between the two poles just outside the bar magnet.
  3. The magnetic field is weakest at the center and strongest between the two poles just outside the bar magnet. The magnetic field lines are densest at the center and least dense between the two poles just outside the bar magnet.
  4. The magnetic field is weakest at the center and strongest between the two poles just outside the bar magnet and the magnetic field lines are least dense at the center and densest between the two poles just outside the bar magnet.
Parts (a) through (d) show the steps in drawing magnetic field lines using a small compass that is moved from point to point around a magnet; the image on the right shows iron filings sprinkled around a magnet.
Figure 20.10 Magnetic field lines can be drawn by moving a small compass from point to point around a magnet. At each point, draw a short line in the direction of the compass needle. Joining the points together reveals the path of the magnetic field lines. Another way to visualize magnetic field lines is to sprinkle iron filings around a magnet, as shown at right.

When two magnets are brought close together, the magnetic field lines are perturbed, just as happens for electric field lines when two electric charges are brought together. Bringing two north poles together—or two south poles—will cause a repulsion, and the magnetic field lines will bend away from each other. This is shown in Figure 20.11, which shows the magnetic field lines created by the two closely separated north poles of a bar magnet. When opposite poles of two magnets are brought together, the magnetic field lines join together and become denser between the poles. This situation is shown in Figure 20.11.

Part (a) shows the field lines when two north poles are brought close together. Part (b) shows the field lines when opposite poles are brought together; they become denser.
Figure 20.11 (a) When two north poles are approached together, the magnetic field lines repel each other and the two magnets experience a repulsive force. The same occurs if two south poles are approached together. (b) If opposite poles are approached together, the magnetic field lines become denser between the poles and the magnets experience an attractive force.

Like the electric field, the magnetic field is stronger where the lines are denser. Thus, between the two north poles in Figure 20.11, the magnetic field is very weak because the density of the magnetic field is almost zero. A compass placed at that point would essentially spin freely if we ignore Earth’s magnetic field. Conversely, the magnetic field lines between the north and south poles in Figure 20.11 are very dense, indicating that the magnetic field is very strong in this region. A compass placed here would quickly align with the magnetic field and point toward the south pole on the right.

Teacher Support

Teacher Support

Misconception Alert

The density of the magnetic field lines in Figure 20.11 indicates the magnitude of the force that would be applied to a small test magnet placed in this field. The density does not indicate the force between the two magnets that create the field. The magnitude of the force between the two magnets is the same in both cases in Figure 20.11. This can be understood by imagining that you place one of the magnets in the field of the other magnet. This situation is symmetrical: The magnetic fields look the same—other than direction—for both situations shown in Figure 20.11. Because the magnets are of equal strength, they perturb the magnetic field of the opposite magnet, which is why the magnetic field must be probed by a small magnetic such as, a compass.

Note that magnets are not the only things that make magnetic fields. Early in the nineteenth century, people discovered that electrical currents cause magnetic effects. The first significant observation was by the Danish scientist Hans Christian Oersted (1777–1851), who found that a compass needle was deflected by a current-carrying wire. This was the first significant evidence that the movement of electric charges had any connection with magnets. An electromagnet is a device that uses electric current to make a magnetic field. These temporarily induced magnets are called electromagnets. Electromagnets are employed for everything from a wrecking yard crane that lifts scrapped cars to controlling the beam of a 90-km-circumference particle accelerator to the magnets in medical-imaging machines (see Figure 20.12).

MRI machine.
Figure 20.12 Instrument for magnetic resonance imaging (MRI). The device uses a cylindrical-coil electromagnet to produce for the main magnetic field. The patient goes into the tunnel on the gurney. (credit: Bill McChesney, Flickr)

The magnetic field created by an electric current in a long straight wire is shown in Figure 20.13. The magnetic field lines form concentric circles around the wire. The direction of the magnetic field can be determined using the right-hand rule. This rule shows up in several places in the study of electricity and magnetism. Applied to a straight current-carrying wire, the right-hand rule says that, with your right thumb pointed in the direction of the current, the magnetic field will be in the direction in which your right fingers curl, as shown in Figure 20.13. If the wire is very long compared to the distance r from the wire, the strength B of the magnetic field is given by

B straightwire = μ 0 I 2πr B straightwire = μ 0 I 2πr
20.1

where I is the current in the wire in amperes. The SI unit for magnetic field is the tesla (T). The symbol μ 0 μ 0 —read “mu-zero”—is a constant called the “permeability of free space” and is given by

μ 0 =4π× 10 −7 Tm/A. μ 0 =4π× 10 −7 Tm/A.
20.2
Demonstration of the right-hand rule, where the thumb points in the direction of the current and the fingers curl in the direction of the magnetic field.
Figure 20.13 This image shows how to use the right-hand rule to determine the direction of the magnetic field created by current flowing through a straight wire. Point your right thumb in the direction of the current, and the magnetic field will be in the direction in which your fingers curl.

Watch Physics

Magnetic Field Due to an Electric Current

This video describes the magnetic field created by a straight current-carrying wire. It goes over the right-hand rule to determine the direction of the magnetic field, and presents and discusses the formula for the strength of the magnetic field due to a straight current-carrying wire.

Grasp Check

A long straight wire is placed on a table top and electric current flows through the wire from right to left. If you look at the wire end-on from the left end, does the magnetic field go clockwise or counterclockwise?

  1. By pointing your right-hand thumb in the direction opposite of current, the right-hand fingers will curl counterclockwise, so the magnetic field will be in the counterclockwise direction.
  2. By pointing your right-hand thumb in the direction opposite of current, the right-hand fingers will curl clockwise, so the magnetic field will be in the clockwise direction.
  3. By pointing your right-hand thumb in the direction of current, the right-hand fingers will curl counterclockwise, so the magnetic field will be in the counterclockwise direction.
  4. By pointing your right-hand thumb in the direction of current, the right-hand fingers will curl clockwise, so the magnetic field will be in the clockwise direction.

Now imagine winding a wire around a cylinder with the cylinder then removed. The result is a wire coil, as shown in Figure 20.14. This is called a solenoid. To find the direction of the magnetic field produced by a solenoid, apply the right-hand rule to several points on the coil. You should be able to convince yourself that, inside the coil, the magnetic field points from left to right. In fact, another application of the right-hand rule is to curl your right-hand fingers around the coil in the direction in which the current flows. Your right thumb then points in the direction of the magnetic field inside the coil: left to right in this case.

A wire coil with current running through it. The magnetic field generated is shown by a red arrow.
Figure 20.14 A wire coil with current running through as shown produces a magnetic field in the direction of the red arrow.

Each loop of wire contributes to the magnetic field inside the solenoid. Because the magnetic field lines must form closed loops, the field lines close the loop outside the solenoid. The magnetic field lines are much denser inside the solenoid than outside the solenoid. The resulting magnetic field looks very much like that of a bar magnet, as shown in Figure 20.15. The magnetic field strength deep inside a solenoid is

B solenoid = μ 0 NI , B solenoid = μ 0 NI ,
20.3

where N is the number of wire loops in the solenoid and is the length of the solenoid.

Part (a) shows iron filings around a solenoid; part (b) shows iron filings around a bar magnet.
Figure 20.15 Iron filings show the magnetic field pattern around (a) a solenoid and (b) a bar magnet. The fields patterns are very similar, especially near the ends of the solenoid and bar magnet.

Virtual Physics

Electromagnets

Use this simulation to visualize the magnetic field made from a solenoid. Be sure to click on the tab that says Electromagnet. You can drive AC or DC current through the solenoid by choosing the appropriate current source. Use the field meter to measure the strength of the magnetic field and then change the number of loops in the solenoid to see how this affects the magnetic field strength.

Grasp Check

Choose the battery as current source and set the number of wire loops to four. With a nonzero current going through the solenoid, measure the magnetic field strength at a point. Now decrease the number of wire loops to two. How does the magnetic field strength change at the point you chose?
  1. There will be no change in magnetic field strength when number of loops reduces from four to two.
  2. The magnetic field strength decreases to half of its initial value when number of loops reduces from four to two.
  3. The magnetic field strength increases to twice of its initial value when number of loops reduces from four to two.
  4. The magnetic field strength increases to four times of its initial value when number of loops reduces from four to two.

Magnetic Force

If a moving electric charge, that is electric current, produces a magnetic field that can exert a force on another magnet, then the reverse should be true by Newton’s third law. In other words, a charge moving through the magnetic field produced by another object should experience a force—and this is exactly what we find. As a concrete example, consider Figure 20.16, which shows a charge q moving with velocity v v through a magnetic field B B between the poles of a permanent magnet. The magnitude F of the force experienced by this charge is

F=qvBsinθ, F=qvBsinθ,
20.4

where θ θ is the angle between the velocity of the charge and the magnetic field.

The direction of the force may be found by using another version of the right-hand rule: First, we join the tails of the velocity vector and a magnetic field vector, as shown in step 1 of Figure 20.16. We then curl our right fingers from v v to B B , as indicated in step (2) of Figure 20.16. The direction in which the right thumb points is the direction of the force. For the charge in Figure 20.16, we find that the force is directed into the page.

Note that the factor sinθ sinθ in the equation F=qvBsinθ F=qvBsinθ means that zero force is applied on a charge that moves parallel to a magnetic field because θ=0 θ=0 and sin0=0 sin0=0 . The maximum force a charge can experience is when it moves perpendicular to the magnetic field, because θ=90° θ=90° and sin90°=1. sin90°=1.

Part (a) shows an electron moving in a uniform magnetic field. Part (b) outlines the steps of the right-hand rule.
Figure 20.16 (a) An electron moves through a uniform magnetic field. (b) Using the right-hand rule, the force on the electron is found to be directed into the page.

Instead of a single charge moving through a magnetic field, consider now a steady current I moving through a straight wire. If we place this wire in a uniform magnetic field, as shown in Figure 20.19, what is the force on the wire or, more precisely, on the electrons in the wire? An electric current involves charges that move. If the charges q move a distance in a time t, then their speed is v=/t. v=/t. Inserting this into the equation F=qvBsinθ F=qvBsinθ gives

F = q( t )Bsinθ = ( q t )Bsinθ. F = q( t )Bsinθ = ( q t )Bsinθ.
20.5

The factor q/t in this equation is nothing more than the current in the wire. Thus, using I=q/t I=q/t , we obtain

F=IBsinθ(1.4). F=IBsinθ(1.4).
20.6

This equation gives the force on a straight current-carrying wire of length in a magnetic field of strength B. The angle θ θ is the angle between the current vector and the magnetic field vector. Note that is the length of wire that is in the magnetic field and for which θ0, θ0, as shown in Figure 20.19.

The direction of the force is determined in the same way as for a single charge. Curl your right fingers from the vector for I to the vector for B, and your right thumb will point in the direction of the force on the wire. For the wire shown in Figure 20.19, the force is directed into the page.

A current-carrying wire in a magnetic field, with force acting on the wire going into the page.
Figure 20.19 A straight wire carrying current I in a magnetic field B. The force exerted on the wire is directed into the page. The length is the length of the wire that is in the magnetic field.

Throughout this section, you may have noticed the symmetries between magnetic effects and electric effects. These effects all fall under the umbrella of electromagnetism, which is the study of electric and magnetic phenomena. We have seen that electric charges produce electric fields, and moving electric charges produce magnetic fields. A magnetic dipole produces a magnetic field, and, as we will see in the next section, moving magnetic dipoles produce an electric field. Thus, electricity and magnetism are two intimately related and symmetric phenomena.

Worked Example

Trajectory of Electron in Magnetic Field

A proton enters a region of constant magnetic field, as shown in Figure 20.20. The magnetic field is coming out of the page. If the electron is moving at 3.0× 10 6 m/s 3.0× 10 6 m/s and the magnetic field strength is 2.0 T, what is the magnitude and direction of the force on the proton?

A proton dot enters (green arrow) a magnetic field (depicted as circles with dots).
Figure 20.20 A proton enters a region of uniform magnetic field. The magnetic field is coming out of the page—the circles with dots represent vector arrow heads coming out of the page.

Strategy

Use the equation F=qvBsinθ F=qvBsinθ to find the magnitude of the force on the proton. The angle between the magnetic field vectors and the velocity vector of the proton is 90°. 90°. The direction of the force may be found by using the right-hand rule.

Discussion

This seems like a very small force. However, the proton has a mass of 1.67× 10 −27 kg 1.67× 10 −27 kg, so its acceleration is a= F m = 9.6× 10 −13 N 1.67× 10 −27 kg =5.7× 10 14  m/s 2 a= F m = 9.6× 10 −13 N 1.67× 10 −27 kg =5.7× 10 14  m/s 2 , or about ten thousand billion times the acceleration due to gravity!

We found that the proton’s initial acceleration as it enters the magnetic field is downward in the plane of the page. Notice that, as the proton accelerates, its velocity remains perpendicular to the magnetic field, so the magnitude of the force does not change. In addition, because of the right-hand rule, the direction of the force remains perpendicular to the velocity. This force is nothing more than a centripetal force: It has a constant magnitude and is always perpendicular to the velocity. Thus, the magnitude of the velocity does not change, and the proton executes circular motion. The radius of this circle may be found by using the kinematics relationship.

F = ma=m v 2 r a = v 2 r r = v 2 a = ( 3.0× 10 6 m/s ) 2 5.7× 10 14  m/s 2 =1.6cm F = ma=m v 2 r a = v 2 r r = v 2 a = ( 3.0× 10 6 m/s ) 2 5.7× 10 14  m/s 2 =1.6cm
20.8

The path of the proton in the magnetic field is shown in Figure 20.22.

A proton dot enters a magnetic field perpendicularly, making a circular motion.
Figure 20.22 When traveling perpendicular to a constant magnetic field, a charged particle will execute circular motion, as shown here for a proton.

Worked Example

Wire with Current in Magnetic Field

Now suppose we run a wire through the uniform magnetic field from the previous example, as shown. If the wire carries a current of 1.0 A in the y ^ y ^ -direction, and the region with magnetic field is 4.0 cm long, what is the force on the wire?

Insertion of a wire into a magnetic field from the previous example.

Strategy

Use equation F=IBsinθ F=IBsinθ to find the magnitude of the force on the wire. The length of the wire inside the magnetic field is 4.0 cm, and the angle between the current direction and the magnetic field direction is 90°. To find the direction of the force, use the right-hand rule as explained just after the equation F=IBsinθ. F=IBsinθ.

Discussion

The direction of the force is the same as the initial direction of the force was in the previous example for a proton. However, because the current in a wire is confined to a wire, the direction in which the charges move does not change. Instead, the entire wire accelerates in the x ^ x ^ -direction. The force on a current-carrying wire in a magnetic field is the basis of all electrical motors, as we will see in the upcoming sections.

Practice Problems

1.

What is the magnitude of the force on an electron moving at 1.0 × 106 m/s perpendicular to a 1.0-T magnetic field?

  1. 0.8 × 10–13 N
  2. 1.6 × 10–14 N
  3. 0.8 × 10–14 N
  4. 1.6 × 10–13 N
2.

A straight 10 cm wire carries 0.40 A and is oriented perpendicular to a magnetic field. If the force on the wire is 0.022 N, what is the magnitude of the magnetic field?

  1. 1.10 × 10–2 T
  2. 0.55 × 10–2 T
  3. 1.10 T
  4. 0.55 T

Check Your Understanding

3.
If two magnets repel each other, what can you conclude about their relative orientation?
  1. Either the south pole of magnet 1 is closer to the north pole of magnet 2 or the north pole of magnet 1 is closer to the south pole of magnet 2.
  2. Either the south poles of both the magnet 1 and magnet 2 are closer to each other or the north poles of both the magnet 1 and magnet 2 are closer to each other.
4.

Describe methods to demagnetize a ferromagnet.

  1. by cooling, heating, or submerging in water
  2. by heating, hammering, and spinning it in external magnetic field
  3. by hammering, heating, and rubbing with cloth
  4. by cooling, submerging in water, or rubbing with cloth
5.
What is a magnetic field?
  1. The directional lines present inside and outside the magnetic material that indicate the magnitude and direction of the magnetic force.
  2. The directional lines present inside and outside the magnetic material that indicate the magnitude of the magnetic force.
  3. The directional lines present inside the magnetic material that indicate the magnitude and the direction of the magnetic force.
  4. The directional lines present outside the magnetic material that indicate the magnitude and the direction of the magnetic force.
6.
Which of the following drawings is correct?
  1. Horseshoe-shaped magnet with field lines pointing north to south and back to north.
  2. Horseshoe-shaped magnet with field lines pointing north to south only.
  3. Horseshoe-shaped magnet with field lines pointing south to north and back to south.
  4. Horseshoe-shaped magnet with field lines pointing south to north only.
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