# Magnetism

## Connection for AP® Courses

Magnetism plays a major role in your everyday life. All electric motors, with uses as diverse as powering refrigerators, starting cars, and moving elevators, contain magnets. Magnetic resonance imaging (MRI) has become an important diagnostic tool in the field of medicine, and the use of magnetism to explore brain activity is a subject of contemporary research and development. Other applications of magnetism include computer memory, levitation of high-speed trains, the aurora borealis, and, of course, the first important historical use of magnetism: navigation. You will find all of these applications of magnetism linked by a small number of underlying principles.
In this chapter, you will learn that both the internal properties of an object and the movement of charged particles can generate a magnetic field, and you will learn why all magnetic fields have a north and south pole. You will also learn how magnetic fields exert forces on objects, resulting in the magnetic alignment that makes a compass work. You will learn how we use this principle to weigh the smallest of subatomic particles with precision and contain superheated plasma to facilitate nuclear fusion.
Big Idea 1 Objects and systems have properties such as mass and charge. Systems may have internal structure.
Enduring Understanding 1.E Materials have many macroscopic properties that result from the arrangement and interactions of the atoms and molecules that make up the material.
Essential Knowledge 1.E.5 Matter has a property called magnetic permeability.
Essential Knowledge 1.E.6 Matter has a property called magnetic dipole moment.
Big Idea 2 Fields existing in space can be used to explain interactions.
Enduring Understanding 2.D A magnetic field is caused by a magnet or a moving electrically charged object. Magnetic fields observed in nature always seem to be produced either by moving charged objects or by magnetic dipoles or combinations of dipoles and never by single poles.
Essential Knowledge 2.D.1 The magnetic field exerts a force on a moving electrically charged object. That magnetic force is perpendicular to the direction of the velocity of the object and to the magnetic field and is proportional to the magnitude of the charge, the magnitude of the velocity, and the magnitude of the magnetic field. It also depends on the angle between the velocity and the magnetic field vectors. Treatment is quantitative for angles of 0°, 90°, or 180° and qualitative for other angles.
Essential Knowledge 2.D.2 The magnetic field vectors around a straight wire that carries electric current are tangent to concentric circles centered on that wire. The field has no component toward the current-carrying wire.
Essential Knowledge 2.D.3 A magnetic dipole placed in a magnetic field, such as the ones created by a magnet or the Earth, will tend to align with the magnetic field vector.
Essential Knowledge 2.D.4 Ferromagnetic materials contain magnetic domains that are themselves magnets.
Big Idea 3 The interactions of an object with other objects can be described by forces.
Enduring Understanding 3.C At the macroscopic level, forces can be categorized as either long-range (action-at-a-distance) forces or contact forces.
Essential Knowledge 3.C.3 A magnetic force results from the interaction of a moving charged object or a magnet with other moving charged objects or another magnet.
Big Idea 4 Interactions between systems can result in changes in those systems.
Enduring Understanding 4.E The electric and magnetic properties of a system can change in response to the presence of, or changes in, other objects or systems.
Essential Knowledge 4.E.1 The magnetic properties of some materials can be affected by magnetic fields at the system. Students should focus on the underlying concepts and not the use of the vocabulary.


# Magnetism

## Magnets

### Learning Objectives

By the end of this section, you will be able to:
1. Describe the difference between the north and south poles of a magnet.
2. Describe how magnetic poles interact with each other.
All magnets attract iron, such as that in a refrigerator door. However, magnets may attract or repel other magnets. Experimentation shows that all magnets have two poles. If freely suspended, one pole will point toward the north. The two poles are thus named the north magnetic pole and the south magnetic pole (or more properly, north-seeking and south-seeking poles, for the attractions in those directions).
The fact that magnetic poles always occur in pairs of north and south is true from the very large scale—for example, sunspots always occur in pairs that are north and south magnetic poles—all the way down to the very small scale. Magnetic atoms have both a north pole and a south pole, as do many types of subatomic particles, such as electrons, protons, and neutrons.

### Section Summary

1. Magnetism is a subject that includes the properties of magnets, the effect of the magnetic force on moving charges and currents, and the creation of magnetic fields by currents.
2. There are two types of magnetic poles, called the north magnetic pole and south magnetic pole.
3. North magnetic poles are those that are attracted toward the Earth’s geographic north pole.
4. Like poles repel and unlike poles attract.
5. Magnetic poles always occur in pairs of north and south—it is not possible to isolate north and south poles.

### Conceptual Questions


# Magnetism

## Ferromagnets and Electromagnets

### Learning Objectives

By the end of this section, you will be able to:
1. Define ferromagnet.
2. Describe the role of magnetic domains in magnetization.
3. Explain the significance of the Curie temperature.
4. Describe the relationship between electricity and magnetism.

### Ferromagnets

Only certain materials, such as iron, cobalt, nickel, and gadolinium, exhibit strong magnetic effects. Such materials are called ferromagnetic, after the Latin word for iron, ferrum. A group of materials made from the alloys of the rare earth elements are also used as strong and permanent magnets; a popular one is neodymium. 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), they can also be magnetized themselves—that is, they can be induced to be magnetic or made into permanent 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 . (This results in the attraction of the previously unmagnetized material to the magnet.) What happens on a microscopic scale is illustrated in . The regions within the material called domains act like small bar magnets. Within domains, the 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 (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.
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 the 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 1043 K , 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.

### Electromagnets

Early in the 19th century, it was 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 charges had any connection with magnets. Electromagnetism is the use of electric current to make magnets. 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 ).
 shows that the response of iron filings to a current-carrying coil and to a permanent bar magnet. The patterns are similar. In fact, electromagnets and ferromagnets have the same basic characteristics—for example, they have north and south poles that cannot be separated and for which like poles repel and unlike poles attract.
Combining a ferromagnet with an electromagnet can produce particularly strong magnetic effects. (See .) Whenever strong magnetic effects are needed, such as lifting scrap metal, or in particle accelerators, electromagnets are enhanced by ferromagnetic materials. Limits to how strong the magnets can be made are imposed by coil resistance (it will overheat and melt at sufficiently high current), and so superconducting magnets may be employed. These are still limited, because superconducting properties are destroyed by too great a magnetic field.
 shows a few uses of combinations of electromagnets and ferromagnets. Ferromagnetic materials can act as memory devices, because the orientation of the magnetic fields of small domains can be reversed or erased. Magnetic information storage on videotapes and computer hard drives are among the most common applications. This property is vital in our digital world.

### Current: The Source of All Magnetism

An electromagnet creates magnetism with an electric current. In later sections we explore this more quantitatively, finding the strength and direction of magnetic fields created by various currents. But what about ferromagnets?  shows models of how electric currents create magnetism at the submicroscopic level. (Note that we cannot directly observe the paths of individual electrons about atoms, and so a model or visual image, consistent with all direct observations, is made. We can directly observe the electron’s orbital angular momentum, its spin momentum, and subsequent magnetic moments, all of which are explained with electric-current-creating subatomic magnetism.) Currents, including those associated with other submicroscopic particles like protons, allow us to explain ferromagnetism and all other magnetic effects. Ferromagnetism, for example, results from an internal cooperative alignment of electron spins, possible in some materials but not in others.
Crucial to the statement that electric current is the source of all magnetism is the fact that it is impossible to separate north and south magnetic poles. (This is far different from the case of positive and negative charges, which are easily separated.) A current loop always produces a magnetic dipole—that is, a magnetic field that acts like a north pole and south pole pair. Since isolated north and south magnetic poles, called magnetic monopoles, are not observed, currents are used to explain all magnetic effects. If magnetic monopoles did exist, then we would have to modify this underlying connection that all magnetism is due to electrical current. There is no known reason that magnetic monopoles should not exist—they are simply never observed—and so searches at the subnuclear level continue. If they do not exist, we would like to find out why not. If they do exist, we would like to see evidence of them.

### Test Prep for AP Courses

### Section Summary

1. Magnetic poles always occur in pairs of north and south—it is not possible to isolate north and south poles.
2. All magnetism is created by electric current.
3. Ferromagnetic materials, such as iron, are those that exhibit strong magnetic effects.
4. The atoms in ferromagnetic materials act like small magnets (due to currents within the atoms) and can be aligned, usually in millimeter-sized regions called domains.
5. Domains can grow and align on a larger scale, producing permanent magnets. Such a material is magnetized, or induced to be magnetic.
6. Above a material’s Curie temperature, thermal agitation destroys the alignment of atoms, and ferromagnetism disappears.
7. Electromagnets employ electric currents to make magnetic fields, often aided by induced fields in ferromagnetic materials.


# Magnetism

## Magnetic Fields and Magnetic Field Lines

### Learning Objectives

By the end of this section, you will be able to:
1. Define magnetic field and describe the magnetic field lines of various magnetic fields.
Einstein is said to have been fascinated by a compass as a child, perhaps musing on how the needle felt a force without direct physical contact. His ability to think deeply and clearly about action at a distance, particularly for gravitational, electric, and magnetic forces, later enabled him to create his revolutionary theory of relativity. Since magnetic forces act at a distance, we define a magnetic field to represent magnetic forces. The pictorial representation of magnetic field lines is very useful in visualizing the strength and direction of the magnetic field. As shown in , the direction of magnetic field lines  is defined to be the direction in which the north end of a compass needle points. The magnetic field is traditionally called the .
Small compasses used to test a magnetic field will not disturb it. (This is analogous to the way we tested electric fields with a small test charge. In both cases, the fields represent only the object creating them and not the probe testing them.)  shows how the magnetic field appears for a current loop and a long straight wire, as could be explored with small compasses. A small compass placed in these fields will align itself parallel to the field line at its location, with its north pole pointing in the direction of B. Note the symbols used for field into and out of the paper.
Extensive exploration of magnetic fields has revealed a number of hard-and-fast rules. We use magnetic field lines to represent the field (the lines are a pictorial tool, not a physical entity in and of themselves). The properties of magnetic field lines can be summarized by these rules:
1. The direction of the magnetic field is tangent to the field line at any point in space. A small compass will point in the direction of the field line.
2. The strength of the field is proportional to the closeness of the lines. It is exactly proportional to the number of lines per unit area perpendicular to the lines (called the areal density).
3. Magnetic field lines can never cross, meaning that the field is unique at any point in space.
4. Magnetic field lines are continuous, forming closed loops without beginning or end. They go from the north pole to the south pole.
The last property is related to the fact that the north and south poles cannot be separated. It is a distinct difference from electric field lines, which begin and end on the positive and negative charges. If magnetic monopoles existed, then magnetic field lines would begin and end on them.

### Section Summary

1. Magnetic fields can be pictorially represented by magnetic field lines, the properties of which are as follows:
1. The field is tangent to the magnetic field line.
2. Field strength is proportional to the line density.
3. Field lines cannot cross.
4. Field lines are continuous loops.

### Conceptual Questions


# Magnetism

## Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field

### Learning Objectives

By the end of this section, you will be able to:
1. Describe the effects of magnetic fields on moving charges.
2. Use the right hand rule 1 to determine the velocity of a charge, the direction of the magnetic field, and the direction of the magnetic force on a moving charge.
3. Calculate the magnetic force on a moving charge.
What is the mechanism by which one magnet exerts a force on another? The answer is related to the fact that all magnetism is caused by current, the flow of charge. Magnetic fields exert forces on moving charges, and so they exert forces on other magnets, all of which have moving charges.

### Right Hand Rule 1

The magnetic force on a moving charge is one of the most fundamental known. Magnetic force is as important as the electrostatic or Coulomb force. Yet the magnetic force is more complex, in both the number of factors that affects it and in its direction, than the relatively simple Coulomb force. The magnitude of the magnetic force  on a charge  moving at a speed  in a magnetic field of strength  is given by
where  is the angle between the directions of  and  This force is often called the Lorentz force. In fact, this is how we define the magnetic field strength —in terms of the force on a charged particle moving in a magnetic field. The SI unit for magnetic field strength  is called the tesla (T) after the eccentric but brilliant inventor Nikola Tesla (1856–1943). To determine how the tesla relates to other SI units, we solve  for .
Because

     is unitless, the tesla is
(note that C/s = A).
Another smaller unit, called the gauss (G), where , is sometimes used. The strongest permanent magnets have fields near 2 T; superconducting electromagnets may attain 10 T or more. The Earth’s magnetic field on its surface is only about , or 0.5 G.
The direction of the magnetic force  is perpendicular to the plane formed by  and , as determined by the right hand rule 1 (or RHR-1), which is illustrated in . RHR-1 states that, to determine the direction of the magnetic force on a positive moving charge, you point the thumb of the right hand in the direction of , the fingers in the direction of , and a perpendicular to the palm points in the direction of . One way to remember this is that there is one velocity, and so the thumb represents it. There are many field lines, and so the fingers represent them. The force is in the direction you would push with your palm. The force on a negative charge is in exactly the opposite direction to that on a positive charge.

### Test Prep for AP Courses

### Section Summary

1. Magnetic fields exert a force on a moving charge q, the magnitude of which is

where  is the angle between the directions of  and .

2. The SI unit for magnetic field strength  is the tesla (T), which is related to other units by

3. The direction of the force on a moving charge is given by right hand rule 1 (RHR-1): Point the thumb of the right hand in the direction of , the fingers in the direction of , and a perpendicular to the palm points in the direction of .
4. The force is perpendicular to the plane formed by  and . Since the force is zero if  is parallel to , charged particles often follow magnetic field lines rather than cross them.

### Conceptual Questions

### Problems & Exercises


# Magnetism

## Force on a Moving Charge in a Magnetic Field: Examples and Applications

### Learning Objectives

By the end of this section, you will be able to:
1. Describe the effects of a magnetic field on a moving charge.
2. Calculate the radius of curvature of the path of a charge that is moving in a magnetic field.
Magnetic force can cause a charged particle to move in a circular or spiral path. Cosmic rays are energetic charged particles in outer space, some of which approach the Earth. They can be forced into spiral paths by the Earth’s magnetic field. Protons in giant accelerators are kept in a circular path by magnetic force. The bubble chamber photograph in  shows charged particles moving in such curved paths. The curved paths of charged particles in magnetic fields are the basis of a number of phenomena and can even be used analytically, such as in a mass spectrometer.
So does the magnetic force cause circular motion? Magnetic force is always perpendicular to velocity, so that it does no work on the charged particle. The particle’s kinetic energy and speed thus remain constant. The direction of motion is affected, but not the speed. This is typical of uniform circular motion. The simplest case occurs when a charged particle moves perpendicular to a uniform -field, such as shown in . (If this takes place in a vacuum, the magnetic field is the dominant factor determining the motion.) Here, the magnetic force supplies the centripetal force . Noting that , we see that .
Because the magnetic force  supplies the centripetal force , we have
Solving for  yields
Here,  is the radius of curvature of the path of a charged particle with mass  and charge , moving at a speed  perpendicular to a magnetic field of strength . If the velocity is not perpendicular to the magnetic field, then  is the component of the velocity perpendicular to the field. The component of the velocity parallel to the field is unaffected, since the magnetic force is zero for motion parallel to the field. This produces a spiral motion rather than a circular one.
 shows how electrons not moving perpendicular to magnetic field lines follow the field lines. The component of velocity parallel to the lines is unaffected, and so the charges spiral along the field lines. If field strength increases in the direction of motion, the field will exert a force to slow the charges, forming a kind of magnetic mirror, as shown below.
The properties of charged particles in magnetic fields are related to such different things as the Aurora Australis or Aurora Borealis and particle accelerators. Charged particles approaching magnetic field lines may get trapped in spiral orbits about the lines rather than crossing them, as seen above. Some cosmic rays, for example, follow the Earth’s magnetic field lines, entering the atmosphere near the magnetic poles and causing the southern or northern lights through their ionization of molecules in the atmosphere. This glow of energized atoms and molecules is seen in Introduction to Magnetism. Those particles that approach middle latitudes must cross magnetic field lines, and many are prevented from penetrating the atmosphere. Cosmic rays are a component of background radiation; consequently, they give a higher radiation dose at the poles than at the equator.
Some incoming charged particles become trapped in the Earth’s magnetic field, forming two belts above the atmosphere known as the Van Allen radiation belts after the discoverer James A. Van Allen, an American astrophysicist. (See .) Particles trapped in these belts form radiation fields (similar to nuclear radiation) so intense that piloted space flights avoid them and satellites with sensitive electronics are kept out of them. In the few minutes it took lunar missions to cross the Van Allen radiation belts, astronauts received radiation doses more than twice the allowed annual exposure for radiation workers. Other planets have similar belts, especially those having strong magnetic fields like Jupiter.
Back on Earth, we have devices that employ magnetic fields to contain charged particles. Among them are the giant particle accelerators that have been used to explore the substructure of matter. (See .) Magnetic fields not only control the direction of the charged particles, they also are used to focus particles into beams and overcome the repulsion of like charges in these beams.
Thermonuclear fusion (like that occurring in the Sun) is a hope for a future clean energy source. One of the most promising devices is the tokamak, which uses magnetic fields to contain (or trap) and direct the reactive charged particles. (See .) Less exotic, but more immediately practical, amplifiers in microwave ovens use a magnetic field to contain oscillating electrons. These oscillating electrons generate the microwaves sent into the oven.

Mass spectrometers have a variety of designs, and many use magnetic fields to measure mass. The curvature of a charged particle’s path in the field is related to its mass and is measured to obtain mass information. (See More Applications of Magnetism.) Historically, such techniques were employed in the first direct observations of electron charge and mass. Today, mass spectrometers (sometimes coupled with gas chromatographs) are used to determine the make-up and sequencing of large biological molecules.

### Test Prep for AP Courses

### Section Summary

1. Magnetic force can supply centripetal force and cause a charged particle to move in a circular path of radius

where  is the component of the velocity perpendicular to  for a charged particle with mass  and charge .

### Conceptual Questions

### Problems & Exercises

If you need additional support for these problems, see More Applications of Magnetism.


# Magnetism

## The Hall Effect

### Learning Objectives

By the end of this section, you will be able to:
1. Describe the Hall effect.
2. Calculate the Hall emf across a current-carrying conductor.
We have seen effects of a magnetic field on free-moving charges. The magnetic field also affects charges moving in a conductor. One result is the Hall effect, which has important implications and applications.
 shows what happens to charges moving through a conductor in a magnetic field. The field is perpendicular to the electron drift velocity and to the width of the conductor. Note that conventional current is to the right in both parts of the figure. In part (a), electrons carry the current and move to the left. In part (b), positive charges carry the current and move to the right. Moving electrons feel a magnetic force toward one side of the conductor, leaving a net positive charge on the other side. This separation of charge creates a voltage , known as the Hall emf, across the conductor. The creation of a voltage across a current-carrying conductor by a magnetic field is known as the Hall effect, after Edwin Hall, the American physicist who discovered it in 1879.
One very important use of the Hall effect is to determine whether positive or negative charges carries the current. Note that in (b), where positive charges carry the current, the Hall emf has the sign opposite to when negative charges carry the current. Historically, the Hall effect was used to show that electrons carry current in metals and it also shows that positive charges carry current in some semiconductors. The Hall effect is used today as a research tool to probe the movement of charges, their drift velocities and densities, and so on, in materials. In 1980, it was discovered that the Hall effect is quantized, an example of quantum behavior in a macroscopic object.
The Hall effect has other uses that range from the determination of blood flow rate to precision measurement of magnetic field strength. To examine these quantitatively, we need an expression for the Hall emf, , across a conductor. Consider the balance of forces on a moving charge in a situation where , , and  are mutually perpendicular, such as shown in . Although the magnetic force moves negative charges to one side, they cannot build up without limit. The electric field caused by their separation opposes the magnetic force, , and the electric force, , eventually grows to equal it. That is,
or
Note that the electric field  is uniform across the conductor because the magnetic field  is uniform, as is the conductor. For a uniform electric field, the relationship between electric field and voltage is , where  is the width of the conductor and  is the Hall emf. Entering this into the last expression gives
Solving this for the Hall emf yields
where  is the Hall effect voltage across a conductor of width  through which charges move at a speed .
One of the most common uses of the Hall effect is in the measurement of magnetic field strength . Such devices, called Hall probes, can be made very small, allowing fine position mapping. Hall probes can also be made very accurate, usually accomplished by careful calibration. Another application of the Hall effect is to measure fluid flow in any fluid that has free charges (most do). (See .) A magnetic field applied perpendicular to the flow direction produces a Hall emf  as shown. Note that the sign of  depends not on the sign of the charges, but only on the directions of  and . The magnitude of the Hall emf is , where  is the pipe diameter, so that the average velocity  can be determined from  providing the other factors are known.

### Test Prep for AP Courses

### Section Summary

1. The Hall effect is the creation of voltage , known as the Hall emf, across a current-carrying conductor by a magnetic field.
2. The Hall emf is given by

for a conductor of width  through which charges move at a speed .

### Conceptual Questions

### Problems & Exercises


# Magnetism

## Magnetic Force on a Current-Carrying Conductor

### Learning Objectives

By the end of this section, you will be able to:
1. Describe the effects of a magnetic force on a current-carrying conductor.
2. Calculate the magnetic force on a current-carrying conductor.
Because charges ordinarily cannot escape a conductor, the magnetic force on charges moving in a conductor is transmitted to the conductor itself.

We can derive an expression for the magnetic force on a current by taking a sum of the magnetic forces on individual charges. (The forces add because they are in the same direction.) The force on an individual charge moving at the drift velocity  is given by . Taking  to be uniform over a length of wire  and zero elsewhere, the total magnetic force on the wire is then , where  is the number of charge carriers in the section of wire of length . Now, , where  is the number of charge carriers per unit volume and  is the volume of wire in the field. Noting that , where  is the cross-sectional area of the wire, then the force on the wire is . Gathering terms,
Because  (see Current),
is the equation for magnetic force on a length , as shown in . If we divide both sides of this expression by , we find that the magnetic force per unit length of wire in a uniform field is . The direction of this force is given by RHR-1, with the thumb in the direction of the current . Then, with the fingers in the direction of , a perpendicular to the palm points in the direction of , as in .
Magnetic force on current-carrying conductors is used to convert electric energy to work. (Motors are a prime example—they employ loops of wire and are considered in the next section.) Magnetohydrodynamics (MHD) is the technical name given to a clever application where magnetic force pumps fluids without moving mechanical parts. (See .)
A strong magnetic field is applied across a tube and a current is passed through the fluid at right angles to the field, resulting in a force on the fluid parallel to the tube axis as shown. The absence of moving parts makes this attractive for moving a hot, chemically active substance, such as the liquid sodium employed in some nuclear reactors. Experimental artificial hearts are testing with this technique for pumping blood, perhaps circumventing the adverse effects of mechanical pumps. (Cell membranes, however, are affected by the large fields needed in MHD, delaying its practical application in humans.) MHD propulsion for nuclear submarines has been proposed, because it could be considerably quieter than conventional propeller drives. The deterrent value of nuclear submarines is based on their ability to hide and survive a first or second nuclear strike. As we slowly disassemble our nuclear weapons arsenals, the submarine branch will be the last to be decommissioned because of this ability (See .) Existing MHD drives are heavy and inefficient—much development work is needed.

### Section Summary

1. The magnetic force on current-carrying conductors is given by

where  is the current,  is the length of a straight conductor in a uniform magnetic field , and  is the angle between  and . The force follows RHR-1 with the thumb in the direction of .

### Conceptual Questions

### Problems & Exercises


# Magnetism

## Torque on a Current Loop: Motors and Meters

### Learning Objectives

By the end of this section, you will be able to:
1. Describe how motors and meters work in terms of torque on a current loop.
2. Calculate the torque on a current-carrying loop in a magnetic field.
Motors are the most common application of magnetic force on current-carrying wires. Motors have loops of wire in a magnetic field. When current is passed through the loops, the magnetic field exerts torque on the loops, which rotates a shaft. Electrical energy is converted to mechanical work in the process. (See .)
Let us examine the force on each segment of the loop in  to find the torques produced about the axis of the vertical shaft. (This will lead to a useful equation for the torque on the loop.) We take the magnetic field to be uniform over the rectangular loop, which has width  and height . First, we note that the forces on the top and bottom segments are vertical and, therefore, parallel to the shaft, producing no torque. Those vertical forces are equal in magnitude and opposite in direction, so that they also produce no net force on the loop.  shows views of the loop from above. Torque is defined as , where  is the force,  is the distance from the pivot that the force is applied, and  is the angle between  and . As seen in (a), right hand rule 1 gives the forces on the sides to be equal in magnitude and opposite in direction, so that the net force is again zero. However, each force produces a clockwise torque. Since , the torque on each vertical segment is , and the two add to give a total torque.
Now, each vertical segment has a length  that is perpendicular to , so that the force on each is . Entering  into the expression for torque yields
If we have a multiple loop of  turns, we get  times the torque of one loop. Finally, note that the area of the loop is ; the expression for the torque becomes
This is the torque on a current-carrying loop in a uniform magnetic field. This equation can be shown to be valid for a loop of any shape. The loop carries a current , has  turns, each of area , and the perpendicular to the loop makes an angle  with the field . The net force on the loop is zero.
The torque found in the preceding example is the maximum. As the coil rotates, the torque decreases to zero at . The torque then reverses its direction once the coil rotates past . (See (d).) This means that, unless we do something, the coil will oscillate back and forth about equilibrium at . To get the coil to continue rotating in the same direction, we can reverse the current as it passes through  with automatic switches called brushes. (See .)
Meters, such as those in analog fuel gauges on a car, are another common application of magnetic torque on a current-carrying loop.  shows that a meter is very similar in construction to a motor. The meter in the figure has its magnets shaped to limit the effect of  by making  perpendicular to the loop over a large angular range. Thus the torque is proportional to  and not . A linear spring exerts a counter-torque that balances the current-produced torque. This makes the needle deflection proportional to . If an exact proportionality cannot be achieved, the gauge reading can be calibrated. To produce a galvanometer for use in analog voltmeters and ammeters that have a low resistance and respond to small currents, we use a large loop area , high magnetic field , and low-resistance coils.

### Section Summary

1. The torque  on a current-carrying loop of any shape in a uniform magnetic field. is

where  is the number of turns,  is the current,  is the area of the loop,  is the magnetic field strength, and  is the angle between the perpendicular to the loop and the magnetic field.

### Conceptual Questions

### Problems & Exercises


# Magnetism

## Magnetic Fields Produced by Currents: Ampere’s Law

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate current that produces a magnetic field.
2. Use the right hand rule 2 to determine the direction of current or the direction of magnetic field loops.
How much current is needed to produce a significant magnetic field, perhaps as strong as the Earth’s field? Surveyors will tell you that overhead electric power lines create magnetic fields that interfere with their compass readings. Indeed, when Oersted discovered in 1820 that a current in a wire affected a compass needle, he was not dealing with extremely large currents. How does the shape of wires carrying current affect the shape of the magnetic field created? We noted earlier that a current loop created a magnetic field similar to that of a bar magnet, but what about a straight wire or a toroid (doughnut)? How is the direction of a current-created field related to the direction of the current? Answers to these questions are explored in this section, together with a brief discussion of the law governing the fields created by currents.

### Magnetic Field Created by a Long Straight Current-Carrying Wire: Right Hand Rule 2

Magnetic fields have both direction and magnitude. As noted before, one way to explore the direction of a magnetic field is with compasses, as shown for a long straight current-carrying wire in . Hall probes can determine the magnitude of the field. The field around a long straight wire is found to be in circular loops. The right hand rule 2 (RHR-2) emerges from this exploration and is valid for any current segment—point the thumb in the direction of the current, and the fingers curl in the direction of the magnetic field loops created by it.
The magnetic field strength (magnitude) produced by a long straight current-carrying wire is found by experiment to be
where  is the current,  is the shortest distance to the wire, and the constant  is the permeability of free space.
 is one of the basic constants in nature. We will see later that  is related to the speed of light.) Since the wire is very long, the magnitude of the field depends only on distance from the wire , not on position along the wire.

### Ampere’s Law and Others

The magnetic field of a long straight wire has more implications than you might at first suspect. Each segment of current produces a magnetic field like that of a long straight wire, and the total field of any shape current is the vector sum of the fields due to each segment. The formal statement of the direction and magnitude of the field due to each segment is called the Biot-Savart law. Integral calculus is needed to sum the field for an arbitrary shape current. This results in a more complete law, called Ampere’s law, which relates magnetic field and current in a general way. Ampere’s law in turn is a part of Maxwell’s equations, which give a complete theory of all electromagnetic phenomena. Considerations of how Maxwell’s equations appear to different observers led to the modern theory of relativity, and the realization that electric and magnetic fields are different manifestations of the same thing. Most of this is beyond the scope of this text in both mathematical level, requiring calculus, and in the amount of space that can be devoted to it. But for the interested student, and particularly for those who continue in physics, engineering, or similar pursuits, delving into these matters further will reveal descriptions of nature that are elegant as well as profound. In this text, we shall keep the general features in mind, such as RHR-2 and the rules for magnetic field lines listed in Magnetic Fields and Magnetic Field Lines, while concentrating on the fields created in certain important situations.

### Magnetic Field Produced by a Current-Carrying Circular Loop

The magnetic field near a current-carrying loop of wire is shown in . Both the direction and the magnitude of the magnetic field produced by a current-carrying loop are complex. RHR-2 can be used to give the direction of the field near the loop, but mapping with compasses and the rules about field lines given in Magnetic Fields and Magnetic Field Lines are needed for more detail. There is a simple formula for the magnetic field strength at the center of a circular loop. It is
where  is the radius of the loop. This equation is very similar to that for a straight wire, but it is valid only at the center of a circular loop of wire. The similarity of the equations does indicate that similar field strength can be obtained at the center of a loop. One way to get a larger field is to have  loops; then, the field is . Note that the larger the loop, the smaller the field at its center, because the current is farther away.

### Magnetic Field Produced by a Current-Carrying Solenoid

A solenoid is a long coil of wire (with many turns or loops, as opposed to a flat loop). Because of its shape, the field inside a solenoid can be very uniform, and also very strong. The field just outside the coils is nearly zero.  shows how the field looks and how its direction is given by RHR-2.
The magnetic field inside of a current-carrying solenoid is very uniform in direction and magnitude. Only near the ends does it begin to weaken and change direction. The field outside has similar complexities to flat loops and bar magnets, but the magnetic field strength inside a solenoid is simply
where  is the number of loops per unit length of the solenoid , with  being the number of loops and  the length). Note that  is the field strength anywhere in the uniform region of the interior and not just at the center. Large uniform fields spread over a large volume are possible with solenoids, as  implies.
There are interesting variations of the flat coil and solenoid. For example, the toroidal coil used to confine the reactive particles in tokamaks is much like a solenoid bent into a circle. The field inside a toroid is very strong but circular. Charged particles travel in circles, following the field lines, and collide with one another, perhaps inducing fusion. But the charged particles do not cross field lines and escape the toroid. A whole range of coil shapes are used to produce all sorts of magnetic field shapes. Adding ferromagnetic materials produces greater field strengths and can have a significant effect on the shape of the field. Ferromagnetic materials tend to trap magnetic fields (the field lines bend into the ferromagnetic material, leaving weaker fields outside it) and are used as shields for devices that are adversely affected by magnetic fields, including the Earth’s magnetic field.

### Test Prep for AP Courses

### Section Summary

1. The strength of the magnetic field created by current in a long straight wire is given by

where  is the current,  is the shortest distance to the wire, and the constant  is the permeability of free space.
2. The direction of the magnetic field created by a long straight wire is given by right hand rule 2 (RHR-2): Point the thumb of the right hand in the direction of current, and the fingers curl in the direction of the magnetic field loops created by it.
3. The magnetic field created by current following any path is the sum (or integral) of the fields due to segments along the path (magnitude and direction as for a straight wire), resulting in a general relationship between current and field known as Ampere’s law.
4. The magnetic field strength at the center of a circular loop is given by

where  is the radius of the loop. This equation becomes  for a flat coil of  loops. RHR-2 gives the direction of the field about the loop. A long coil is called a solenoid.
5. The magnetic field strength inside a solenoid is

where  is the number of loops per unit length of the solenoid. The field inside is very uniform in magnitude and direction.

### Conceptual Questions


# Magnetism

## Magnetic Force between Two Parallel Conductors

### Learning Objectives

By the end of this section, you will be able to:
1. Describe the effects of the magnetic force between two conductors.
2. Calculate the force between two parallel conductors.
You might expect that there are significant forces between current-carrying wires, since ordinary currents produce significant magnetic fields and these fields exert significant forces on ordinary currents. But you might not expect that the force between wires is used to define the ampere. It might also surprise you to learn that this force has something to do with why large circuit breakers burn up when they attempt to interrupt large currents.
The force between two long straight and parallel conductors separated by a distance  can be found by applying what we have developed in preceding sections.  shows the wires, their currents, the fields they create, and the subsequent forces they exert on one another. Let us consider the field produced by wire 1 and the force it exerts on wire 2 (call the force ). The field due to  at a distance  is given to be
This field is uniform along wire 2 and perpendicular to it, and so the force  it exerts on wire 2 is given by  with :
By Newton’s third law, the forces on the wires are equal in magnitude, and so we just write  for the magnitude of . (Note that .) Since the wires are very long, it is convenient to think in terms of , the force per unit length. Substituting the expression for  into the last equation and rearranging terms gives
 is the force per unit length between two parallel currents  and  separated by a distance . The force is attractive if the currents are in the same direction and repulsive if they are in opposite directions.
This force is responsible for the pinch effect in electric arcs and plasmas. The force exists whether the currents are in wires or not. In an electric arc, where currents are moving parallel to one another, there is an attraction that squeezes currents into a smaller tube. In large circuit breakers, like those used in neighborhood power distribution systems, the pinch effect can concentrate an arc between plates of a switch trying to break a large current, burn holes, and even ignite the equipment. Another example of the pinch effect is found in the solar plasma, where jets of ionized material, such as solar flares, are shaped by magnetic forces.
The operational definition of the ampere is based on the force between current-carrying wires. Note that for parallel wires separated by 1 meter with each carrying 1 ampere, the force per meter is
Since  is exactly  by definition, and because
, the force per meter is exactly . This is the basis of the operational definition of the ampere.
Infinite-length straight wires are impractical and so, in practice, a current balance is constructed with coils of wire separated by a few centimeters. Force is measured to determine current. This also provides us with a method for measuring the coulomb. We measure the charge that flows for a current of one ampere in one second. That is, . For both the ampere and the coulomb, the method of measuring force between conductors is the most accurate in practice.

### Test Prep for AP Courses

### Section Summary

1. The force between two parallel currents  and , separated by a distance , has a magnitude per unit length given by

2. The force is attractive if the currents are in the same direction, repulsive if they are in opposite directions.

### Conceptual Questions

### Problems & Exercises


# Magnetism

## More Applications of Magnetism

### Learning Objectives

By the end of this section, you will be able to:
1. Describe some applications of magnetism.

### Mass Spectrometry

The curved paths followed by charged particles in magnetic fields can be put to use. A charged particle moving perpendicular to a magnetic field travels in a circular path having a radius .
It was noted that this relationship could be used to measure the mass of charged particles such as ions. A mass spectrometer is a device that measures such masses. Most mass spectrometers use magnetic fields for this purpose, although some of them have extremely sophisticated designs. Since there are five variables in the relationship, there are many possibilities. However, if , , and  can be fixed, then the radius of the path  is simply proportional to the mass  of the charged particle. Let us examine one such mass spectrometer that has a relatively simple design. (See .) The process begins with an ion source, a device like an electron gun. The ion source gives ions their charge, accelerates them to some velocity , and directs a beam of them into the next stage of the spectrometer. This next region is a velocity selector that only allows particles with a particular value of  to get through.
The velocity selector has both an electric field and a magnetic field, perpendicular to one another, producing forces in opposite directions on the ions. Only those ions for which the forces balance travel in a straight line into the next region. If the forces balance, then the electric force  equals the magnetic force , so that . Noting that  cancels, we see that
is the velocity particles must have to make it through the velocity selector, and further, that  can be selected by varying  and . In the final region, there is only a uniform magnetic field, and so the charged particles move in circular arcs with radii proportional to particle mass. The paths also depend on charge , but since  is in multiples of electron charges, it is easy to determine and to discriminate between ions in different charge states.
Mass spectrometry today is used extensively in chemistry and biology laboratories to identify chemical and biological substances according to their mass-to-charge ratios. In medicine, mass spectrometers are used to measure the concentration of isotopes used as tracers. Usually, biological molecules such as proteins are very large, so they are broken down into smaller fragments before analyzing. Recently, large virus particles have been analyzed as a whole on mass spectrometers. Sometimes a gas chromatograph or high-performance liquid chromatograph provides an initial separation of the large molecules, which are then input into the mass spectrometer.

### Cathode Ray Tubes—CRTs—and the Like

What do non-flat-screen TVs, old computer monitors, x-ray machines, and the 2-mile-long Stanford Linear Accelerator have in common? All of them accelerate electrons, making them different versions of the electron gun. Many of these devices use magnetic fields to steer the accelerated electrons.  shows the construction of the type of cathode ray tube (CRT) found in some TVs, oscilloscopes, and old computer monitors. Two pairs of coils are used to steer the electrons, one vertically and the other horizontally, to their desired destination.

### Magnetic Resonance Imaging

Magnetic resonance imaging (MRI) is one of the most useful and rapidly growing medical imaging tools. It non-invasively produces two-dimensional and three-dimensional images of the body that provide important medical information with none of the hazards of x-rays. MRI is based on an effect called nuclear magnetic resonance (NMR) in which an externally applied magnetic field interacts with the nuclei of certain atoms, particularly those of hydrogen (protons). These nuclei possess their own small magnetic fields, similar to those of electrons and the current loops discussed earlier in this chapter.
When placed in an external magnetic field, such nuclei experience a torque that pushes or aligns the nuclei into one of two new energy states—depending on the orientation of its spin (analogous to the N pole and S pole in a bar magnet). Transitions from the lower to higher energy state can be achieved by using an external radio frequency signal to “flip” the orientation of the small magnets. (This is actually a quantum mechanical process. The direction of the nuclear magnetic field is quantized as is energy in the radio waves. We will return to these topics in later chapters.) The specific frequency of the radio waves that are absorbed and reemitted depends sensitively on the type of nucleus, the chemical environment, and the external magnetic field strength. Therefore, this is a resonance phenomenon in which nuclei in a magnetic field act like resonators (analogous to those discussed in the treatment of sound in Oscillatory Motion and Waves) that absorb and reemit only certain frequencies. Hence, the phenomenon is named nuclear magnetic resonance (NMR).
NMR has been used for more than 50 years as an analytical tool. It was formulated in 1946 by F. Bloch and E. Purcell, with the 1952 Nobel Prize in Physics going to them for their work. Over the past two decades, NMR has been developed to produce detailed images in a process now called magnetic resonance imaging (MRI), a name coined to avoid the use of the word “nuclear” and the concomitant implication that nuclear radiation is involved. (It is not.) The 2003 Nobel Prize in Medicine went to P. Lauterbur and P. Mansfield for their work with MRI applications.
The largest part of the MRI unit is a superconducting magnet that creates a magnetic field, typically between 1 and 2 T in strength, over a relatively large volume. MRI images can be both highly detailed and informative about structures and organ functions. It is helpful that normal and non-normal tissues respond differently for slight changes in the magnetic field. In most medical images, the protons that are hydrogen nuclei are imaged. (About 2/3 of the atoms in the body are hydrogen.) Their location and density give a variety of medically useful information, such as organ function, the condition of tissue (as in the brain), and the shape of structures, such as vertebral disks and knee-joint surfaces. MRI can also be used to follow the movement of certain ions across membranes, yielding information on active transport, osmosis, dialysis, and other phenomena. With excellent spatial resolution, MRI can provide information about tumors, strokes, shoulder injuries, infections, etc.
An image requires position information as well as the density of a nuclear type (usually protons). By varying the magnetic field slightly over the volume to be imaged, the resonant frequency of the protons is made to vary with position. Broadcast radio frequencies are swept over an appropriate range and nuclei absorb and reemit them only if the nuclei are in a magnetic field with the correct strength. The imaging receiver gathers information through the body almost point by point, building up a tissue map. The reception of reemitted radio waves as a function of frequency thus gives position information. These “slices” or cross sections through the body are only several mm thick. The intensity of the reemitted radio waves is proportional to the concentration of the nuclear type being flipped, as well as information on the chemical environment in that area of the body. Various techniques are available for enhancing contrast in images and for obtaining more information. Scans called T1, T2, or proton density scans rely on different relaxation mechanisms of nuclei. Relaxation refers to the time it takes for the protons to return to equilibrium after the external field is turned off. This time depends upon tissue type and status (such as inflammation).
While MRI images are superior to x rays for certain types of tissue and have none of the hazards of x rays, they do not completely supplant x-ray images. MRI is less effective than x rays for detecting breaks in bone, for example, and in imaging breast tissue, so the two diagnostic tools complement each other. MRI images are also expensive compared to simple x-ray images and tend to be used most often where they supply information not readily obtained from x rays. Another disadvantage of MRI is that the patient is totally enclosed with detectors close to the body for about 30 minutes or more, leading to claustrophobia. It is also difficult for the obese patient to be in the magnet tunnel. New “open-MRI” machines are now available in which the magnet does not completely surround the patient.
Over the last decade, the development of much faster scans, called “functional MRI” (fMRI), has allowed us to map the functioning of various regions in the brain responsible for thought and motor control. This technique measures the change in blood flow for activities (thought, experiences, action) in the brain. The nerve cells increase their consumption of oxygen when active. Blood hemoglobin releases oxygen to active nerve cells and has somewhat different magnetic properties when oxygenated than when deoxygenated. With MRI, we can measure this and detect a blood oxygen-dependent signal. Most of the brain scans today use fMRI.

### Other Medical Uses of Magnetic Fields

Currents in nerve cells and the heart create magnetic fields like any other currents. These can be measured but with some difficulty since their strengths are about  to  less than the Earth’s magnetic field. Recording of the heart’s magnetic field as it beats is called a magnetocardiogram (MCG), while measurements of the brain’s magnetic field is called a magnetoencephalogram (MEG). Both give information that differs from that obtained by measuring the electric fields of these organs (ECGs and EEGs), but they are not yet of sufficient importance to make these difficult measurements common.
In both of these techniques, the sensors do not touch the body. MCG can be used in fetal studies, and is probably more sensitive than echocardiography. MCG also looks at the heart’s electrical activity whose voltage output is too small to be recorded by surface electrodes as in EKG. It has the potential of being a rapid scan for early diagnosis of cardiac ischemia (obstruction of blood flow to the heart) or problems with the fetus.
MEG can be used to identify abnormal electrical discharges in the brain that produce weak magnetic signals. Therefore, it looks at brain activity, not just brain structure. It has been used for studies of Alzheimer’s disease and epilepsy. Advances in instrumentation to measure very small magnetic fields have allowed these two techniques to be used more in recent years. What is used is a sensor called a SQUID, for superconducting quantum interference device. This operates at liquid helium temperatures and can measure magnetic fields thousands of times smaller than the Earth’s.
Finally, there is a burgeoning market for magnetic cures in which magnets are applied in a variety of ways to the body, from magnetic bracelets to magnetic mattresses. The best that can be said for such practices is that they are apparently harmless, unless the magnets get close to the patient’s computer or magnetic storage disks. Claims are made for a broad spectrum of benefits from cleansing the blood to giving the patient more energy, but clinical studies have not verified these claims, nor is there an identifiable mechanism by which such benefits might occur.

### Section Summary

1. Crossed (perpendicular) electric and magnetic fields act as a velocity filter, giving equal and opposite forces on any charge with velocity perpendicular to the fields and of magnitude

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Connection for AP® Courses

Nature’s displays of symmetry are beautiful and alluring. As shown in Figure 23.2, a butterfly’s wings exhibit an appealing symmetry in a complex system. The laws of physics display symmetries at the most basic level – these symmetries are a source of wonder and imply deeper meaning. Since we place a high value on symmetry, we look for it when we explore nature. The remarkable thing is that we find it.
This chapter supports Big Idea 4, illustrating how electric and magnetic changes can take place in a system due to interactions with other systems. The hint of symmetry between electricity and magnetism found in the preceding chapter will be elaborated upon in this chapter. Specifically, we know that a current creates a magnetic field. If nature is symmetric in this case, then perhaps a magnetic field can create a current. Historically, it was very shortly after Oersted discovered that currents cause magnetic fields that other scientists asked the following question: Can magnetic fields cause currents? The answer was soon found by experiment to be yes. In 1831, some 12 years after Oersted’s discovery, the English scientist Michael Faraday (1791–1862) and the American scientist Joseph Henry (1797–1878) independently demonstrated that magnetic fields can produce currents. The basic process of generating emfs (electromotive forces), and hence currents, with magnetic fields is known as induction; this process is also called “magnetic induction” to distinguish it from charging by induction, which utilizes the Coulomb force.
Today, currents induced by magnetic fields are essential to our technological society. The ubiquitous generator – found in automobiles, on bicycles, in nuclear power plants, and so on – uses magnetism to generate current. Other devices that use magnetism to induce currents include pickup coils in electric guitars, transformers of every size, certain microphones, airport security gates, and damping mechanisms on sensitive chemical balances. Explanations and examples in this chapter will help you understand current induction via magnetic interactions in mechanical systems (Enduring Understanding 4.E, Essential Knowledge 4.E.2). You will also learn how the behavior of AC circuits depends strongly on the effect of magnetic fields on currents.
The content of this chapter supports:
Big Idea 4 Interactions between systems can result in changes in those systems.
Enduring Understanding 4.E The electric and magnetic properties of a system can change in response to the presence of, or changes in, other objects or systems.
Essential Knowledge 4.E.2 Changing magnetic flux induces an electric field that can establish an induced emf in a system.


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Induced Emf and Magnetic Flux

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate the flux of a uniform magnetic field through a loop of arbitrary orientation.
2. Describe methods to produce an electromotive force (emf) with a magnetic field or magnet and a loop of wire.
The apparatus used by Faraday to demonstrate that magnetic fields can create currents is illustrated in . When the switch is closed, a magnetic field is produced in the coil on the top part of the iron ring and transmitted to the coil on the bottom part of the ring. The galvanometer is used to detect any current induced in the coil on the bottom. It was found that each time the switch is closed, the galvanometer detects a current in one direction in the coil on the bottom. (You can also observe this in a physics lab.) Each time the switch is opened, the galvanometer detects a current in the opposite direction. Interestingly, if the switch remains closed or open for any length of time, there is no current through the galvanometer. Closing and opening the switch induces the current. It is the change in magnetic field that creates the current. More basic than the current that flows is the emf that causes it. The current is a result of an emf induced by a changing magnetic field, whether or not there is a path for current to flow.
An experiment easily performed and often done in physics labs is illustrated in . An emf is induced in the coil when a bar magnet is pushed in and out of it. Emfs of opposite signs are produced by motion in opposite directions, and the emfs are also reversed by reversing poles. The same results are produced if the coil is moved rather than the magnet—it is the relative motion that is important. The faster the motion, the greater the emf, and there is no emf when the magnet is stationary relative to the coil.
The method of inducing an emf used in most electric generators is shown in . A coil is rotated in a magnetic field, producing an alternating current emf, which depends on rotation rate and other factors that will be explored in later sections. Note that the generator is remarkably similar in construction to a motor (another symmetry).
So we see that changing the magnitude or direction of a magnetic field produces an emf. Experiments revealed that there is a crucial quantity called the magnetic flux,  , given by
where  is the magnetic field strength over an area , at an angle

    with the perpendicular to the area as shown in . Any change in magnetic flux  This process is defined to be electromagnetic induction. Units of magnetic flux  are . As seen in , ⊥, which is the component of  perpendicular to the area . Thus magnetic flux is , the product of the area and the component of the magnetic field perpendicular to it.
All induction, including the examples given so far, arises from some change in magnetic flux . For example, Faraday changed  and hence  when opening and closing the switch in his apparatus (shown in ). This is also true for the bar magnet and coil shown in . When rotating the coil of a generator, the angle  and, hence,  is changed. Just how great an emf and what direction it takes depend on the change in  and how rapidly the change is made, as examined in the next section.

### Test Prep for AP Courses

### Section Summary

1. The crucial quantity in induction is magnetic flux , defined to be , where  is the magnetic field strength over an area  at an angle  with the perpendicular to the area.
2. Units of magnetic flux  are .
3. Any change in magnetic flux  induces an emf—the process is defined to be electromagnetic induction.

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Faraday’s Law of Induction: Lenz’s Law

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate emf, current, and magnetic fields using Faraday’s Law.
2. Explain the physical results of Lenz’s Law

### Faraday’s and Lenz’s Law

Faraday’s experiments showed that the emf induced by a change in magnetic flux depends on only a few factors. First, emf is directly proportional to the change in flux . Second, emf  is greatest when the change in time  is smallest—that is, emf  is inversely proportional to . Finally, if a coil has

    turns, an emf will be produced that is  times greater than for a single coil, so that emf  is directly proportional to . The equation for the emf induced by a change in magnetic flux is
This relationship is known as Faraday’s law of induction. The units for emf are volts, as is usual.
The minus sign in Faraday’s law of induction is very important. The minus means that the emf creates a current I and magnetic field B that oppose the change in flux . The direction (given by the minus sign) of the emfis so important that it is called Lenz’s law after the Russian Heinrich Lenz (1804–1865), who, like Faraday and Henry,independently investigated aspects of induction. Faraday was aware of the direction, but Lenz stated it so clearly that he is credited for its discovery. (See .)
For practice, apply these steps to the situations shown in  and to others that are part of the following text material.

### Applications of Electromagnetic Induction

There are many applications of Faraday’s Law of induction, as we will explore in this chapter and others. At this juncture, let us mention several that have to do with data storage and magnetic fields. A very important application has to do with audio and video recording tapes. A plastic tape, coated with iron oxide, moves past a recording head. This recording head is basically a round iron ring about which is wrapped a coil of wire—an electromagnet (). A signal in the form of a varying input current from a microphone or camera goes to the recording head. These signals (which are a function of the signal amplitude and frequency) produce varying magnetic fields at the recording head. As the tape moves past the recording head, the magnetic field orientations of the iron oxide molecules on the tape are changed thus recording the signal. In the playback mode, the magnetized tape is run past another head, similar in structure to the recording head. The different magnetic field orientations of the iron oxide molecules on the tape induces an emf in the coil of wire in the playback head. This signal then is sent to a loudspeaker or video player.
Similar principles apply to computer hard drives, except at a much faster rate. Here recordings are on a coated, spinning disk. Read heads historically were made to work on the principle of induction. However, the input information is carried in digital rather than analog form – a series of 0’s or 1’s are written upon the spinning hard drive. Today, most hard drive readout devices do not work on the principle of induction, but use a technique known as giant magnetoresistance. (The discovery that weak changes in a magnetic field in a thin film of iron and chromium could bring about much larger changes in electrical resistance was one of the first large successes of nanotechnology.) Another application of induction is found on the magnetic stripe on the back of your personal credit card as used at the grocery store or the ATM machine. This works on the same principle as the audio or video tape mentioned in the last paragraph in which a head reads personal information from your card.
Another application of electromagnetic induction is when electrical signals need to be transmitted across a barrier. Consider the cochlear implant shown below. Sound is picked up by a microphone on the outside of the skull and is used to set up a varying magnetic field. A current is induced in a receiver secured in the bone beneath the skin and transmitted to electrodes in the inner ear. Electromagnetic induction can be used in other instances where electric signals need to be conveyed across various media.
Another contemporary area of research in which electromagnetic induction is being successfully implemented (and with substantial potential) is transcranial magnetic simulation. A host of disorders, including depression and hallucinations can be traced to irregular localized electrical activity in the brain. In transcranial magnetic stimulation, a rapidly varying and very localized magnetic field is placed close to certain sites identified in the brain. Weak electric currents are induced in the identified sites and can result in recovery of electrical functioning in the brain tissue.
Sleep apnea (“the cessation of breath”) affects both adults and infants (especially premature babies and it may be a cause of sudden infant deaths [SID]). In such individuals, breath can stop repeatedly during their sleep. A cessation of more than 20 seconds can be very dangerous. Stroke, heart failure, and tiredness are just some of the possible consequences for a person having sleep apnea. The concern in infants is the stopping of breath for these longer times. One type of monitor to alert parents when a child is not breathing uses electromagnetic induction. A wire wrapped around the infant’s chest has an alternating current running through it. The expansion and contraction of the infant’s chest as the infant breathes changes the area through the coil. A pickup coil located nearby has an alternating current induced in it due to the changing magnetic field of the initial wire. If the child stops breathing, there will be a change in the induced current, and so a parent can be alerted.

### Section Summary

1. Faraday’s law of induction states that the emfinduced by a change in magnetic flux is
    when flux changes by

2. If emf is induced in a coil,

   is its number of turns.
3. The minus sign means that the emf creates a current  and magnetic field  that oppose the change in flux  —this opposition is known as Lenz’s law.

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Motional Emf

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate emf, force, magnetic field, and work due to the motion of an object in a magnetic field.
As we have seen, any change in magnetic flux induces an emf opposing that change—a process known as induction. Motion is one of the major causes of induction. For example, a magnet moved toward a coil induces an emf, and a coil moved toward a magnet produces a similar emf. In this section, we concentrate on motion in a magnetic field that is stationary relative to the Earth, producing what is loosely called motional emf.
One situation where motional emf occurs is known as the Hall effect and has already been examined. Charges moving in a magnetic field experience the magnetic force , which moves opposite charges in opposite directions and produces an . We saw that the Hall effect has applications, including measurements of  and . We will now see that the Hall effect is one aspect of the broader phenomenon of induction, and we will find that motional emf can be used as a power source.
Consider the situation shown in . A rod is moved at a speed

along a pair of conducting rails separated by a distance

in a uniform magnetic field . The rails are stationary relative to  and are connected to a stationary resistor . The resistor could be anything from a light bulb to a voltmeter. Consider the area enclosed by the moving rod, rails, and resistor.  is perpendicular to this area, and the area is increasing as the rod moves. Thus the magnetic flux enclosed by the rails, rod, and resistor is increasing. When flux changes, an emf is induced according to Faraday’s law of induction.
To find the magnitude of emf induced along the moving rod, we use Faraday’s law of induction without the sign:
Here and below, “emf” implies the magnitude of the emf. In this equation,  and the flux . We have

     and , since  is perpendicular to

  .
Now , since  is uniform. Note that the area swept out by the rod is . Entering these quantities into the expression for  emf yields
Finally, note that , the velocity of the rod. Entering this into the last expression shows that
is the motional emf. This is the same expression given for the Hall effect previously.
To find the direction of the induced field, the direction of the current, and the polarity of the induced emf, we apply Lenz’s law as explained in Faraday's Law of Induction: Lenz's Law. (See (b).) Flux is increasing, since the area enclosed is increasing. Thus the induced field must oppose the existing one and be out of the page. And so the RHR-2 requires that I be counterclockwise, which in turn means the top of the rod is positive as shown.
Motional emf also occurs if the magnetic field moves and the rod (or other object) is stationary relative to the Earth (or some observer). We have seen an example of this in the situation where a moving magnet induces an emf in a stationary coil. It is the relative motion that is important. What is emerging in these observations is a connection between magnetic and electric fields. A moving magnetic field produces an electric field through its induced emf. We already have seen that a moving electric field produces a magnetic field—moving charge implies moving electric field and moving charge produces a magnetic field.
Motional emfs in the Earth’s weak magnetic field are not ordinarily very large, or we would notice voltage along metal rods, such as a screwdriver, during ordinary motions. For example, a simple calculation of the motional emf of a 1 m rod moving at 3.0 m/s perpendicular to the Earth’s field gives . This small value is consistent with experience. There is a spectacular exception, however. In 1992 and 1996, attempts were made with the space shuttle to create large motional emfs. The Tethered Satellite was to be let out on a 20 km length of wire as shown in , to create a 5 kV emf by moving at orbital speed through the Earth’s field. This emf could be used to convert some of the shuttle’s kinetic and potential energy into electrical energy if a complete circuit could be made. To complete the circuit, the stationary ionosphere was to supply a return path for the current to flow. (The ionosphere is the rarefied and partially ionized atmosphere at orbital altitudes. It conducts because of the ionization. The ionosphere serves the same function as the stationary rails and connecting resistor in , without which there would not be a complete circuit.) Drag on the current in the cable due to the magnetic force  does the work that reduces the shuttle’s kinetic and potential energy and allows it to be converted to electrical energy. The tests were both unsuccessful. In the first, the cable hung up and could only be extended a couple of hundred meters; in the second, the cable broke when almost fully extended.  indicates feasibility in principle.

### Section Summary

1. An emf induced by motion relative to a magnetic field

 is called a motional emf and is given by

where  is the length of the object moving at speed  relative to the field.

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Eddy Currents and Magnetic Damping

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the magnitude and direction of an induced eddy current, and the effect this will have on the object it is induced in.
2. Describe several applications of magnetic damping.

### Eddy Currents and Magnetic Damping

As discussed in Motional Emf, motional emf is induced when a conductor moves in a magnetic field or when a magnetic field moves relative to a conductor. If motional emf can cause a current loop in the conductor, we refer to that current as an eddy current. Eddy currents can produce significant drag, called magnetic damping, on the motion involved. Consider the apparatus shown in , which swings a pendulum bob between the poles of a strong magnet. (This is another favorite physics lab activity.) If the bob is metal, there is significant drag on the bob as it enters and leaves the field, quickly damping the motion. If, however, the bob is a slotted metal plate, as shown in (b), there is a much smaller effect due to the magnet. There is no discernible effect on a bob made of an insulator. Why is there drag in both directions, and are there any uses for magnetic drag?
 shows what happens to the metal plate as it enters and leaves the magnetic field. In both cases, it experiences a force opposing its motion. As it enters from the left, flux increases, and so an eddy current is set up (Faraday’s law) in the counterclockwise direction (Lenz’s law), as shown. Only the right-hand side of the current loop is in the field, so that there is an unopposed force on it to the left (RHR-1). When the metal plate is completely inside the field, there is no eddy current if the field is uniform, since the flux remains constant in this region. But when the plate leaves the field on the right, flux decreases, causing an eddy current in the clockwise direction that, again, experiences a force to the left, further slowing the motion. A similar analysis of what happens when the plate swings from the right toward the left shows that its motion is also damped when entering and leaving the field.
When a slotted metal plate enters the field, as shown in , an emf is induced by the change in flux, but it is less effective because the slots limit the size of the current loops. Moreover, adjacent loops have currents in opposite directions, and their effects cancel. When an insulating material is used, the eddy current is extremely small, and so magnetic damping on insulators is negligible. If eddy currents are to be avoided in conductors, then they can be slotted or constructed of thin layers of conducting material separated by insulating sheets.

### Applications of Magnetic Damping

One use of magnetic damping is found in sensitive laboratory balances. To have maximum sensitivity and accuracy, the balance must be as friction-free as possible. But if it is friction-free, then it will oscillate for a very long time. Magnetic damping is a simple and ideal solution. With magnetic damping, drag is proportional to speed and becomes zero at zero velocity. Thus the oscillations are quickly damped, after which the damping force disappears, allowing the balance to be very sensitive. (See .) In most balances, magnetic damping is accomplished with a conducting disc that rotates in a fixed field.
Since eddy currents and magnetic damping occur only in conductors, recycling centers can use magnets to separate metals from other materials. Trash is dumped in batches down a ramp, beneath which lies a powerful magnet. Conductors in the trash are slowed by magnetic damping while nonmetals in the trash move on, separating from the metals. (See .) This works for all metals, not just ferromagnetic ones. A magnet can separate out the ferromagnetic materials alone by acting on stationary trash.
Other major applications of eddy currents are in metal detectors and braking systems in trains and roller coasters. Portable metal detectors () consist of a primary coil carrying an alternating current and a secondary coil in which a current is induced. An eddy current will be induced in a piece of metal close to the detector which will cause a change in the induced current within the secondary coil, leading to some sort of signal like a shrill noise. Braking using eddy currents is safer because factors such as rain do not affect the braking and the braking is smoother. However, eddy currents cannot bring the motion to a complete stop, since the force produced decreases with speed. Thus, speed can be reduced from say 20 m/s to 5 m/s, but another form of braking is needed to completely stop the vehicle. Generally, powerful rare earth magnets such as neodymium magnets are used in roller coasters.  shows rows of magnets in such an application. The vehicle has metal fins (normally containing copper) which pass through the magnetic field slowing the vehicle down in much the same way as with the pendulum bob shown in .
Induction cooktops have electromagnets under their surface. The magnetic field is varied rapidly producing eddy currents in the base of the pot, causing the pot and its contents to increase in temperature. Induction cooktops have high efficiencies and good response times but the base of the pot needs to be ferromagnetic, iron or steel for induction to work.

### Section Summary

1. Current loops induced in moving conductors are called eddy currents.
2. They can create significant drag, called magnetic damping.

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Electric Generators

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate the emf induced in a generator.
2. Calculate the peak emf which can be induced in a particular generator system.
Electric generators induce an emf by rotating a coil in a magnetic field, as briefly discussed in Induced Emf and Magnetic Flux. We will now explore generators in more detail. Consider the following example.
The emf calculated in  is the average over one-fourth of a revolution. What is the emf at any given instant? It varies with the angle between the magnetic field and a perpendicular to the coil. We can get an expression for emf as a function of time by considering the motional emf on a rotating rectangular coil of width  and height  in a uniform magnetic field, as illustrated in .
Charges in the wires of the loop experience the magnetic force, because they are moving in a magnetic field. Charges in the vertical wires experience forces parallel to the wire, causing currents. But those in the top and bottom segments feel a force perpendicular to the wire, which does not cause a current. We can thus find the induced emf by considering only the side wires. Motional emf is given to be , where the velocity v is perpendicular to the magnetic field . Here the velocity is at an angle  with , so that its component perpendicular to  is  (see ). Thus in this case the emf induced on each side is , and they are in the same direction. The total emf around the loop is then
This expression is valid, but it does not give emf as a function of time. To find the time dependence of emf, we assume the coil rotates at a constant angular velocity . The angle  is related to angular velocity by , so that
Now, linear velocity

 is related to angular velocity

   by . Here , so that , and
Noting that the area of the loop is , and allowing for  loops, we find that
is the emf induced in a generator coil of  turns and area  rotating at a constant angular velocity

 in a uniform magnetic field . This can also be expressed as
where
is the maximum (peak) emf. Note that the frequency of the oscillation is , and the period is .  shows a graph of emf as a function of time, and it now seems reasonable that AC voltage is sinusoidal.
The fact that the peak emf, , makes good sense. The greater the number of coils, the larger their area, and the stronger the field, the greater the output voltage. It is interesting that the faster the generator is spun (greater ), the greater the emf. This is noticeable on bicycle generators—at least the cheaper varieties. One of the authors as a juvenile found it amusing to ride his bicycle fast enough to burn out his lights, until he had to ride home lightless one dark night.
 shows a scheme by which a generator can be made to produce pulsed DC. More elaborate arrangements of multiple coils and split rings can produce smoother DC, although electronic rather than mechanical means are usually used to make ripple-free DC.
In real life, electric generators look a lot different than the figures in this section, but the principles are the same. The source of mechanical energy that turns the coil can be falling water (hydropower), steam produced by the burning of fossil fuels, or the kinetic energy of wind.  shows a cutaway view of a steam turbine; steam moves over the blades connected to the shaft, which rotates the coil within the generator.
Generators illustrated in this section look very much like the motors illustrated previously. This is not coincidental. In fact, a motor becomes a generator when its shaft rotates. Certain early automobiles used their starter motor as a generator. In Back Emf, we shall further explore the action of a motor as a generator.

### Test Prep for AP Courses

### Section Summary

1. An electric generator rotates a coil in a magnetic field, inducing an emfgiven as a function of time by

where  is the area of an -turn coil rotated at a constant angular velocity  in a uniform magnetic field .
2. The peak emf  of a generator is

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Back Emf

### Learning Objectives

By the end of this section, you will be able to:
1. Explain what back emf is and how it is induced.
It has been noted that motors and generators are very similar. Generators convert mechanical energy into electrical energy, whereas motors convert electrical energy into mechanical energy. Furthermore, motors and generators have the same construction. When the coil of a motor is turned, magnetic flux changes, and an emf (consistent with Faraday’s law of induction) is induced. The motor thus acts as a generator whenever its coil rotates. This will happen whether the shaft is turned by an external input, like a belt drive, or by the action of the motor itself. That is, when a motor is doing work and its shaft is turning, an emf is generated. Lenz’s law tells us the emf opposes any change, so that the input emf that powers the motor will be opposed by the motor’s self-generated emf, called the back emf of the motor. (See .)
Back emf is the generator output of a motor, and so it is proportional to the motor’s angular velocity . It is zero when the motor is first turned on, meaning that the coil receives the full driving voltage and the motor draws maximum current when it is on but not turning. As the motor turns faster and faster, the back emf grows, always opposing the driving emf, and reduces the voltage across the coil and the amount of current it draws. This effect is noticeable in a number of situations. When a vacuum cleaner, refrigerator, or washing machine is first turned on, lights in the same circuit dim briefly due to the

   drop produced in feeder lines by the large current drawn by the motor. When a motor first comes on, it draws more current than when it runs at its normal operating speed. When a mechanical load is placed on the motor, like an electric wheelchair going up a hill, the motor slows, the back emf drops, more current flows, and more work can be done. If the motor runs at too low a speed, the larger current can overheat it (via resistive power in the coil, ), perhaps even burning it out. On the other hand, if there is no mechanical load on the motor, it will increase its angular velocity  until the back emf is nearly equal to the driving emf. Then the motor uses only enough energy to overcome friction.
Consider, for example, the motor coils represented in . The coils have a  equivalent resistance and are driven by a 48.0 V emf. Shortly after being turned on, they draw a current  and, thus, dissipate  of energy as heat transfer. Under normal operating conditions for this motor, suppose the back emf is 40.0 V. Then at operating speed, the total voltage across the coils is 8.0 V (48.0 V minus the 40.0 V back emf), and the current drawn is . Under normal load, then, the power dissipated is . The latter will not cause a problem for this motor, whereas the former 5.76 kW would burn out the coils if sustained.

### Section Summary

1. Any rotating coil will have an induced emf—in motors, this is called back emf, since it opposes the emf input to the motor.

### Conceptual Questions

### Problems & Exercises


# Electromagnetic Induction, AC Circuits, and Electrical Technologies

## Transformers

### Learning Objectives

By the end of this section, you will be able to:
1. Explain how a transformer works.
2. Calculate voltage, current, and/or number of turns given the other quantities.
Transformers do what their name implies—they transform voltages from one value to another (The term voltage is used rather than emf, because transformers have internal resistance). For example, many cell phones, laptops, video games, and power tools and small appliances have a transformer built into their plug-in unit (like that in ) that changes 120 V or 240 V AC into whatever voltage the device uses. Transformers are also used at several points in the power distribution systems, such as illustrated in . Power is sent long distances at high voltages, because less current is required for a given amount of power, and this means less line loss, as was discussed previously. But high voltages pose greater hazards, so that transformers are employed to produce lower voltage at the user’s location.
The type of transformer considered in this text—see —is based on Faraday’s law of induction and is very similar in construction to the apparatus Faraday used to demonstrate magnetic fields could cause currents. The two coils are called the primary and secondary coils. In normal use, the input voltage is placed on the primary, and the secondary produces the transformed output voltage. Not only does the iron core trap the magnetic field created by the primary coil, its magnetization increases the field strength. Since the input voltage is AC, a time-varying magnetic flux is sent to the secondary, inducing its AC output voltage.
For the simple transformer shown in , the output voltage  depends almost entirely on the input voltage  and the ratio of the number of loops in the primary and secondary coils. Faraday’s law of induction for the secondary coil gives its induced output voltage  to be
where  is the number of loops in the secondary coil and /  is the rate of change of magnetic flux. Note that the output voltage equals the induced emf (), provided coil resistance is small (a reasonable assumption for transformers). The cross-sectional area of the coils is the same on either side, as is the magnetic field strength, and so

is the same on either side. The input primary voltage  is also related to changing flux by
The reason for this is a little more subtle. Lenz’s law tells us that the primary coil opposes the change in flux caused by the input voltage , hence the minus sign (This is an example of self-inductance, a topic to be explored in some detail in later sections). Assuming negligible coil resistance, Kirchhoff’s loop rule tells us that the induced emf exactly equals the input voltage. Taking the ratio of these last two equations yields a useful relationship:
This is known as the transformer equation, and it simply states that the ratio of the secondary to primary voltages in a transformer equals the ratio of the number of loops in their coils.
The output voltage of a transformer can be less than, greater than, or equal to the input voltage, depending on the ratio of the number of loops in their coils. Some transformers even provide a variable output by allowing connection to be made at different points on the secondary coil. A step-up transformer is one that increases voltage, whereas a step-down transformer decreases voltage. Assuming, as we have, that resistance is negligible, the electrical power output of a transformer equals its input. This is nearly true in practice—transformer efficiency often exceeds 99%. Equating the power input and output,
Rearranging terms gives
Combining this with , we find that
is the relationship between the output and input currents of a transformer. So if voltage increases, current decreases. Conversely, if voltage decreases, current increases.
The fact that transformers are based on Faraday’s law of induction makes it clear why we cannot use transformers to change DC voltages. If there is no change in primary voltage, there is no voltage induced in the secondary. One possibility is to connect DC to the primary coil through a switch. As the switch is opened and closed, the secondary produces a voltage like that in . This is not really a practical alternative, and AC is in common use wherever it is necessary to increase or decrease voltages.
Transformers have many applications in electrical safety systems, which are discussed in Electrical Safety: Systems and Devices.

### Test Prep for AP Courses

### Section Summary

1. Transformers use induction to transform voltages from one value to another.
2. For a transformer, the voltages across the primary and secondary coils are related by

where  and  are the voltages across primary and secondary coils having  and  turns.
3. The currents  and  in the primary and secondary coils are related by .
4. A step-up transformer increases voltage and decreases current, whereas a step-down transformer decreases voltage and increases current.

### Conceptual Questions

### Problems & Exercises