# Circuits and DC Instruments

## Electromotive Force: Terminal Voltage

### Learning Objectives

By the end of this section, you will be able to:
1. Compare and contrast the voltage and the electromagnetic force of an electric power source.
2. Describe what happens to the terminal voltage, current, and power delivered to a load as internal resistance of the voltage source increases (due to aging of batteries, for example).
3. Explain why it is beneficial to use more than one voltage source connected in parallel.
When you forget to turn off your car lights, they slowly dim as the battery runs down. Why don’t they simply blink off when the battery’s energy is gone? Their gradual dimming implies that battery output voltage decreases as the battery is depleted.
Furthermore, if you connect an excessive number of 12-V lights in parallel to a car battery, they will be dim even when the battery is fresh and even if the wires to the lights have very low resistance. This implies that the battery’s output voltage is reduced by the overload.
The reason for the decrease in output voltage for depleted or overloaded batteries is that all voltage sources have two fundamental parts—a source of electrical energy and an internal resistance. Let us examine both.

### Electromotive Force

You can think of many different types of voltage sources. Batteries themselves come in many varieties. There are many types of mechanical/electrical generators, driven by many different energy sources, ranging from nuclear to wind. Solar cells create voltages directly from light, while thermoelectric devices create voltage from temperature differences.
A few voltage sources are shown in . All such devices create a potential difference and can supply current if connected to a resistance. On the small scale, the potential difference creates an electric field that exerts force on charges, causing current. We thus use the name electromotive force, abbreviated emf.
Emf is not a force at all; it is a special type of potential difference. To be precise, the electromotive force (emf) is the potential difference of a source when no current is flowing. Units of emf are volts.
Electromotive force is directly related to the source of potential difference, such as the particular combination of chemicals in a battery. However, emf differs from the voltage output of the device when current flows. The voltage across the terminals of a battery, for example, is less than the emf when the battery supplies current, and it declines further as the battery is depleted or loaded down. However, if the device’s output voltage can be measured without drawing current, then output voltage will equal emf (even for a very depleted battery).

### Internal Resistance

As noted before, a 12-V truck battery is physically larger, contains more charge and energy, and can deliver a larger current than a 12-V motorcycle battery. Both are lead-acid batteries with identical emf, but, because of its size, the truck battery has a smaller internal resistance . Internal resistance is the inherent resistance to the flow of current within the source itself.
 is a schematic representation of the two fundamental parts of any voltage source. The emf (represented by a script E in the figure) and internal resistance  are in series. The smaller the internal resistance for a given emf, the more current and the more power the source can supply.
The internal resistance  can behave in complex ways. As noted,  increases as a battery is depleted. But internal resistance may also depend on the magnitude and direction of the current through a voltage source, its temperature, and even its history. The internal resistance of rechargeable nickel-cadmium cells, for example, depends on how many times and how deeply they have been depleted.
Why are the chemicals able to produce a unique potential difference? Quantum mechanical descriptions of molecules, which take into account the types of atoms and numbers of electrons in them, are able to predict the energy states they can have and the energies of reactions between them.
In the case of a lead-acid battery, an energy of 2 eV is given to each electron sent to the anode. Voltage is defined as the electrical potential energy divided by charge: . An electron volt is the energy given to a single electron by a voltage of 1 V. So the voltage here is 2 V, since 2 eV is given to each electron. It is the energy produced in each molecular reaction that produces the voltage. A different reaction produces a different energy and, hence, a different voltage.

### Terminal Voltage

The voltage output of a device is measured across its terminals and, thus, is called its terminal voltage. Terminal voltage is given by
where  is the internal resistance and  is the current flowing at the time of the measurement.
 is positive if current flows away from the positive terminal, as shown in . You can see that the larger the current, the smaller the terminal voltage. And it is likewise true that the larger the internal resistance, the smaller the terminal voltage.
Suppose a load resistance  is connected to a voltage source, as in . Since the resistances are in series, the total resistance in the circuit is . Thus the current is given by Ohm’s law to be
We see from this expression that the smaller the internal resistance , the greater the current the voltage source supplies to its load . As batteries are depleted,  increases. If  becomes a significant fraction of the load resistance, then the current is significantly reduced, as the following example illustrates.
Battery testers, such as those in , use small load resistors to intentionally draw current to determine whether the terminal voltage drops below an acceptable level. They really test the internal resistance of the battery. If internal resistance is high, the battery is weak, as evidenced by its low terminal voltage.
Some batteries can be recharged by passing a current through them in the direction opposite to the current they supply to a resistance. This is done routinely in cars and batteries for small electrical appliances and electronic devices, and is represented pictorially in . The voltage output of the battery charger must be greater than the emf of the battery to reverse current through it. This will cause the terminal voltage of the battery to be greater than the emf, since , and  is now negative.

### Multiple Voltage Sources

There are two voltage sources when a battery charger is used. Voltage sources connected in series are relatively simple. When voltage sources are in series, their internal resistances add and their emfs add algebraically. (See .) Series connections of voltage sources are common—for example, in flashlights, toys, and other appliances. Usually, the cells are in series in order to produce a larger total emf.
But if the cells oppose one another, such as when one is put into an appliance backward, the total emf is less, since it is the algebraic sum of the individual emfs.
A battery is a multiple connection of voltaic cells, as shown in . The disadvantage of series connections of cells is that their internal resistances add. One of the authors once owned a 1957 MGA that had two 6-V batteries in series, rather than a single 12-V battery. This arrangement produced a large internal resistance that caused him many problems in starting the engine.
If the series connection of two voltage sources is made into a complete circuit with the emfs in opposition, then a current of magnitude  flows. See , for example, which shows a circuit exactly analogous to the battery charger discussed above. If two voltage sources in series with emfs in the same sense are connected to a load , as in , then

 flows.
 shows two voltage sources with identical emfs in parallel and connected to a load resistance. In this simple case, the total emf is the same as the individual emfs. But the total internal resistance is reduced, since the internal resistances are in parallel. The parallel connection thus can produce a larger current.
Here,  flows through the load, and  is less than those of the individual batteries. For example, some diesel-powered cars use two 12-V batteries in parallel; they produce a total emf of 12 V but can deliver the larger current needed to start a diesel engine.

### Animals as Electrical Detectors

A number of animals both produce and detect electrical signals. Fish, sharks, platypuses, and echidnas (spiny anteaters) all detect electric fields generated by nerve activity in prey. Electric eels produce their own emf through biological cells (electric organs) called electroplaques, which are arranged in both series and parallel as a set of batteries.
Electroplaques are flat, disk-like cells; those of the electric eel have a voltage of 0.15 V across each one. These cells are usually located toward the head or tail of the animal, although in the case of the electric eel, they are found along the entire body. The electroplaques in the South American eel are arranged in 140 rows, with each row stretching horizontally along the body and containing 5,000 electroplaques. This can yield an emf of approximately 600 V, and a current of 1 A—deadly.
The mechanism for detection of external electric fields is similar to that for producing nerve signals in the cell through depolarization and repolarization—the movement of ions across the cell membrane. Within the fish, weak electric fields in the water produce a current in a gel-filled canal that runs from the skin to sensing cells, producing a nerve signal. The Australian platypus, one of the very few mammals that lay eggs, can detect fields of 30 , while sharks have been found to be able to sense a field in their snouts as small as 100  (). Electric eels use their own electric fields produced by the electroplaques to stun their prey or enemies.

### Solar Cell Arrays

Another example dealing with multiple voltage sources is that of combinations of solar cells—wired in both series and parallel combinations to yield a desired voltage and current. Photovoltaic generation (PV), the conversion of sunlight directly into electricity, is based upon the photoelectric effect, in which photons hitting the surface of a solar cell create an electric current in the cell.
Most solar cells are made from pure silicon—either as single-crystal silicon, or as a thin film of silicon deposited upon a glass or metal backing. Most single cells have a voltage output of about 0.5 V, while the current output is a function of the amount of sunlight upon the cell (the incident solar radiation—the insolation). Under bright noon sunlight, a current of about  of cell surface area is produced by typical single-crystal cells.
Individual solar cells are connected electrically in modules to meet electrical-energy needs. They can be wired together in series or in parallel—connected like the batteries discussed earlier. A solar-cell array or module usually consists of between 36 and 72 cells, with a power output of 50 W to 140 W.
The output of the solar cells is direct current. For most uses in a home, AC is required, so a device called an inverter must be used to convert the DC to AC. Any extra output can then be passed on to the outside electrical grid for sale to the utility.

### Test Prep for AP Courses

### Section Summary

1. All voltage sources have two fundamental parts—a source of electrical energy that has a characteristic electromotive force (emf), and an internal resistance .
2. The emf is the potential difference of a source when no current is flowing.
3. The numerical value of the emf depends on the source of potential difference.
4. The internal resistance  of a voltage source affects the output voltage when a current flows.
5. The voltage output of a device is called its terminal voltage  and is given by , where  is the electric current and is positive when flowing away from the positive terminal of the voltage source.
6. When multiple voltage sources are in series, their internal resistances add and their emfs add algebraically.
7. Solar cells can be wired in series or parallel to provide increased voltage or current, respectively.

### Conceptual Questions

### Problem Exercises


# Circuits and DC Instruments

## Kirchhoff’s Rules

### Learning Objectives

By the end of this section, you will be able to:
1. Analyze a complex circuit using Kirchhoff’s rules, using the conventions for determining the correct signs of various terms.
Many complex circuits, such as the one in , cannot be analyzed with the series-parallel techniques developed in Resistors in Series and Parallel and Electromotive Force: Terminal Voltage. There are, however, two circuit analysis rules that can be used to analyze any circuit, simple or complex. These rules are special cases of the laws of conservation of charge and conservation of energy. The rules are known as Kirchhoff’s rules, after their inventor Gustav Kirchhoff (1824–1887).
Explanations of the two rules will now be given, followed by problem-solving hints for applying Kirchhoff’s rules, and a worked example that uses them.

### Kirchhoff’s First Rule

Kirchhoff’s first rule (the junction rule) is an application of the conservation of charge to a junction; it is illustrated in . Current is the flow of charge, and charge is conserved; thus, whatever charge flows into the junction must flow out. Kirchhoff’s first rule requires that  (see figure). Equations like this can and will be used to analyze circuits and to solve circuit problems.

### Kirchhoff’s Second Rule

Kirchhoff’s second rule (the loop rule) is an application of conservation of energy. The loop rule is stated in terms of potential, , rather than potential energy, but the two are related since . Recall that emf is the potential difference of a source when no current is flowing.  In a closed loop, whatever energy is supplied by emf must be transferred into other forms by devices in the loop, since there are no other ways in which energy can be transferred into or out of the circuit.  illustrates the changes in potential in a simple series circuit loop.
Kirchhoff’s second rule requires . Rearranged, this is , which means the emf equals the sum of the  (voltage) drops in the loop.

### Applying Kirchhoff’s Rules

By applying Kirchhoff’s rules, we generate equations that allow us to find the unknowns in circuits. The unknowns may be currents, emfs, or resistances. Each time a rule is applied, an equation is produced. If there are as many independent equations as unknowns, then the problem can be solved. There are two decisions you must make when applying Kirchhoff’s rules. These decisions determine the signs of various quantities in the equations you obtain from applying the rules.
1. When applying Kirchhoff’s first rule, the junction rule, you must label the current in each branch and decide in what direction it is going. For example, in , , and , currents are labeled , , , and , and arrows indicate their directions. There is no risk here, for if you choose the wrong direction, the current will be of the correct magnitude but negative.
2. When applying Kirchhoff’s second rule, the loop rule, you must identify a closed loop and decide in which direction to go around it, clockwise or counterclockwise. For example, in  the loop was traversed in the same direction as the current (clockwise). Again, there is no risk; going around the circuit in the opposite direction reverses the sign of every term in the equation, which is like multiplying both sides of the equation by
 and the following points will help you get the plus or minus signs right when applying the loop rule. Note that the resistors and emfs are traversed by going from a to b. In many circuits, it will be necessary to construct more than one loop. In traversing each loop, one needs to be consistent for the sign of the change in potential. (See .)
1. When a resistor is traversed in the same direction as the current, the change in potential is . (See .)
2. When a resistor is traversed in the direction opposite to the current, the change in potential is . (See .)
3. When an emf is traversed from  to + (the same direction it moves positive charge), the change in potential is +emf. (See .)
4. When an emf is traversed from + to  (opposite to the direction it moves positive charge), the change in potential is

          emf. (See .)
The material in this section is correct in theory. We should be able to verify it by making measurements of current and voltage. In fact, some of the devices used to make such measurements are straightforward applications of the principles covered so far and are explored in the next modules. As we shall see, a very basic, even profound, fact results—making a measurement alters the quantity being measured.

### Test Prep for AP Courses

### Section Summary

1. Kirchhoff’s rules can be used to analyze any circuit, simple or complex.
2. Kirchhoff’s first rule—the junction rule: The sum of all currents entering a junction must equal the sum of all currents leaving the junction.
3. Kirchhoff’s second rule—the loop rule: The algebraic sum of changes in potential around any closed circuit path (loop) must be zero.
4. The two rules are based, respectively, on the laws of conservation of charge and energy.
5. When calculating potential and current using Kirchhoff’s rules, a set of conventions must be followed for determining the correct signs of various terms.
6. The simpler series and parallel rules are special cases of Kirchhoff’s rules.

### Conceptual Questions

### Problem Exercises


# Circuits and DC Instruments

## DC Voltmeters and Ammeters

### Learning Objectives

By the end of this section, you will be able to:
1. Explain why a voltmeter must be connected in parallel with the circuit.
2. Draw a diagram showing an ammeter correctly connected in a circuit.
3. Describe how a galvanometer can be used as either a voltmeter or an ammeter.
4. Find the resistance that must be placed in series with a galvanometer to allow it to be used as a voltmeter with a given reading.
5. Explain why measuring the voltage or current in a circuit can never be exact.
Voltmeters measure voltage, whereas ammeters measure current. Some of the meters in automobile dashboards, digital cameras, cell phones, and tuner-amplifiers are voltmeters or ammeters. (See .) The internal construction of the simplest of these meters and how they are connected to the system they monitor give further insight into applications of series and parallel connections.
Voltmeters are connected in parallel with whatever device’s voltage is to be measured. A parallel connection is used because objects in parallel experience the same potential difference. (See , where the voltmeter is represented by the symbol V.)
Ammeters are connected in series with whatever device’s current is to be measured. A series connection is used because objects in series have the same current passing through them. (See , where the ammeter is represented by the symbol A.)

### Analog Meters: Galvanometers

Analog meters have a needle that swivels to point at numbers on a scale, as opposed to digital meters, which have numerical readouts similar to a hand-held calculator. The heart of most analog meters is a device called a galvanometer, denoted by G. Current flow through a galvanometer, , produces a proportional needle deflection. (This deflection is due to the force of a magnetic field upon a current-carrying wire.)
The two crucial characteristics of a given galvanometer are its resistance and current sensitivity. Current sensitivity is the current that gives a full-scale deflection of the galvanometer’s needle, the maximum current that the instrument can measure. For example, a galvanometer with a current sensitivity of  has a maximum deflection of its needle when  flows through it, reads half-scale when  flows through it, and so on.
If such a galvanometer has a  resistance, then a voltage of only  produces a full-scale reading. By connecting resistors to this galvanometer in different ways, you can use it as either a voltmeter or ammeter that can measure a broad range of voltages or currents.

### Galvanometer as Voltmeter

 shows how a galvanometer can be used as a voltmeter by connecting it in series with a large resistance, . The value of the resistance  is determined by the maximum voltage to be measured. Suppose you want 10 V to produce a full-scale deflection of a voltmeter containing a  galvanometer with a  sensitivity. Then 10 V applied to the meter must produce a current of . The total resistance must be
( is so large that the galvanometer resistance, , is nearly negligible.) Note that 5 V applied to this voltmeter produces a half-scale deflection by producing a  current through the meter, and so the voltmeter’s reading is proportional to voltage as desired.
This voltmeter would not be useful for voltages less than about half a volt, because the meter deflection would be small and difficult to read accurately. For other voltage ranges, other resistances are placed in series with the galvanometer. Many meters have a choice of scales. That choice involves switching an appropriate resistance into series with the galvanometer.

### Galvanometer as Ammeter

The same galvanometer can also be made into an ammeter by placing it in parallel with a small resistance , often called the shunt resistance, as shown in . Since the shunt resistance is small, most of the current passes through it, allowing an ammeter to measure currents much greater than those producing a full-scale deflection of the galvanometer.
Suppose, for example, an ammeter is needed that gives a full-scale deflection for 1.0 A, and contains the same  galvanometer with its  sensitivity. Since  and  are in parallel, the voltage across them is the same.
These  drops are  so that . Solving for , and noting that  is  and  is 0.999950 A, we have

### Taking Measurements Alters the Circuit

When you use a voltmeter or ammeter, you are connecting another resistor to an existing circuit and, thus, altering the circuit. Ideally, voltmeters and ammeters do not appreciably affect the circuit, but it is instructive to examine the circumstances under which they do or do not interfere.
First, consider the voltmeter, which is always placed in parallel with the device being measured. Very little current flows through the voltmeter if its resistance is a few orders of magnitude greater than the device, and so the circuit is not appreciably affected. (See (a).) (A large resistance in parallel with a small one has a combined resistance essentially equal to the small one.) If, however, the voltmeter’s resistance is comparable to that of the device being measured, then the two in parallel have a smaller resistance, appreciably affecting the circuit. (See (b).) The voltage across the device is not the same as when the voltmeter is out of the circuit.
An ammeter is placed in series in the branch of the circuit being measured, so that its resistance adds to that branch. Normally, the ammeter’s resistance is very small compared with the resistances of the devices in the circuit, and so the extra resistance is negligible. (See (a).) However, if very small load resistances are involved, or if the ammeter is not as low in resistance as it should be, then the total series resistance is significantly greater, and the current in the branch being measured is reduced. (See (b).)
A practical problem can occur if the ammeter is connected incorrectly. If it was put in parallel with the resistor to measure the current in it, you could possibly damage the meter; the low resistance of the ammeter would allow most of the current in the circuit to go through the galvanometer, and this current would be larger since the effective resistance is smaller.
One solution to the problem of voltmeters and ammeters interfering with the circuits being measured is to use galvanometers with greater sensitivity. This allows construction of voltmeters with greater resistance and ammeters with smaller resistance than when less sensitive galvanometers are used.
There are practical limits to galvanometer sensitivity, but it is possible to get analog meters that make measurements accurate to a few percent. Note that the inaccuracy comes from altering the circuit, not from a fault in the meter.

### Section Summary

1. Voltmeters measure voltage, and ammeters measure current.
2. A voltmeter is placed in parallel with the voltage source to receive full voltage and must have a large resistance to limit its effect on the circuit.
3. An ammeter is placed in series to get the full current flowing through a branch and must have a small resistance to limit its effect on the circuit.
4. Both can be based on the combination of a resistor and a galvanometer, a device that gives an analog reading of current.
5. Standard voltmeters and ammeters alter the circuit being measured and are thus limited in accuracy.

### Conceptual Questions

### Problem Exercises


# Circuits and DC Instruments

## Null Measurements

### Learning Objectives

By the end of this section, you will be able to:
1. Explain why a null measurement device is more accurate than a standard voltmeter or ammeter.
2. Demonstrate how a Wheatstone bridge can be used to accurately calculate the resistance in a circuit.
Standard measurements of voltage and current alter the circuit being measured, introducing uncertainties in the measurements. Voltmeters draw some extra current, whereas ammeters reduce current flow. Null measurements balance voltages so that there is no current flowing through the measuring device and, therefore, no alteration of the circuit being measured.
Null measurements are generally more accurate but are also more complex than the use of standard voltmeters and ammeters, and they still have limits to their precision. In this module, we shall consider a few specific types of null measurements, because they are common and interesting, and they further illuminate principles of electric circuits.

### The Potentiometer

Suppose you wish to measure the emf of a battery. Consider what happens if you connect the battery directly to a standard voltmeter as shown in . (Once we note the problems with this measurement, we will examine a null measurement that improves accuracy.) As discussed before, the actual quantity measured is the terminal voltage , which is related to the emf of the battery by , where  is the current that flows and  is the internal resistance of the battery.
The emf could be accurately calculated if  were very accurately known, but it is usually not. If the current  could be made zero, then , and so emf could be directly measured. However, standard voltmeters need a current to operate; thus, another technique is needed.
A potentiometer is a null measurement device for measuring potentials (voltages). (See .) A voltage source is connected to a resistor  say, a long wire, and passes a constant current through it. There is a steady drop in potential (an  drop) along the wire, so that a variable potential can be obtained by making contact at varying locations along the wire.
(b) shows an unknown  (represented by script   in the figure) connected in series with a galvanometer. Note that  opposes the other voltage source. The location of the contact point (see the arrow on the drawing) is adjusted until the galvanometer reads zero. When the galvanometer reads zero, , where  is the resistance of the section of wire up to the contact point. Since no current flows through the galvanometer, none flows through the unknown emf, and so  is directly sensed.
Now, a very precisely known standard  is substituted for , and the contact point is adjusted until the galvanometer again reads zero, so that . In both cases, no current passes through the galvanometer, and so the current  through the long wire is the same. Upon taking the ratio ,  cancels, giving
Solving for  gives
Because a long uniform wire is used for , the ratio of resistances  is the same as the ratio of the lengths of wire that zero the galvanometer for each emf. The three quantities on the right-hand side of the equation are now known or measured, and  can be calculated. The uncertainty in this calculation can be considerably smaller than when using a voltmeter directly, but it is not zero. There is always some uncertainty in the ratio of resistances  and in the standard . Furthermore, it is not possible to tell when the galvanometer reads exactly zero, which introduces error into both  and , and may also affect the current .

### Resistance Measurements and the Wheatstone Bridge

There is a variety of so-called ohmmeters that purport to measure resistance. What the most common ohmmeters actually do is to apply a voltage to a resistance, measure the current, and calculate the resistance using Ohm’s law. Their readout is this calculated resistance. Two configurations for ohmmeters using standard voltmeters and ammeters are shown in . Such configurations are limited in accuracy, because the meters alter both the voltage applied to the resistor and the current that flows through it.
The Wheatstone bridge is a null measurement device for calculating resistance by balancing potential drops in a circuit. (See .) The device is called a bridge because the galvanometer forms a bridge between two branches. A variety of bridge devices are used to make null measurements in circuits.
Resistors  and  are precisely known, while the arrow through  indicates that it is a variable resistance. The value of  can be precisely read. With the unknown resistance  in the circuit,  is adjusted until the galvanometer reads zero. The potential difference between points b and d is then zero, meaning that b and d are at the same potential. With no current running through the galvanometer, it has no effect on the rest of the circuit. So the branches abc and adc are in parallel, and each branch has the full voltage of the source. That is, the  drops along abc and adc are the same. Since b and d are at the same potential, the  drop along ad must equal the  drop along ab. Thus,
Again, since b and d are at the same potential, the  drop along dc must equal the  drop along bc. Thus,
Taking the ratio of these last two expressions gives
Canceling the currents and solving for Rx yields
This equation is used to calculate the unknown resistance when current through the galvanometer is zero. This method can be very accurate (often to four significant digits), but it is limited by two factors. First, it is not possible to get the current through the galvanometer to be exactly zero. Second, there are always uncertainties in , , and , which contribute to the uncertainty in .

### Section Summary

1. Null measurement techniques achieve greater accuracy by balancing a circuit so that no current flows through the measuring device.
2. One such device, for determining voltage, is a potentiometer.
3. Another null measurement device, for determining resistance, is the Wheatstone bridge.
4. Other physical quantities can also be measured with null measurement techniques.

### Conceptual questions

### Problem Exercises


# Circuits and DC Instruments

## DC Circuits Containing Resistors and Capacitors

When you use a flash camera, it takes a few seconds to charge the capacitor that powers the flash. The light flash discharges the capacitor in a tiny fraction of a second. Why does charging take longer than discharging? This question and a number of other phenomena that involve charging and discharging capacitors are discussed in this module.

### RC Circuits

An  is one containing a resistor R and a capacitor C. The capacitor is an electrical component that stores electric charge.
 shows a simple RC circuit that employs a DC (direct current) voltage source. The capacitor is initially uncharged. As soon as the switch is closed, current flows to and from the initially uncharged capacitor. As charge increases on the capacitor plates, there is increasing opposition to the flow of charge by the repulsion of like charges on each plate.
In terms of voltage, this is because voltage across the capacitor is given by , where  is the amount of charge stored on each plate and  is the capacitance. This voltage opposes the battery, growing from zero to the maximum emf when fully charged. The current thus decreases from its initial value of  to zero as the voltage on the capacitor reaches the same value as the emf. When there is no current, there is no  drop, and so the voltage on the capacitor must then equal the emf of the voltage source. This can also be explained with Kirchhoff’s second rule (the loop rule), discussed in Kirchhoff’s Rules, which says that the algebraic sum of changes in potential around any closed loop must be zero.
The initial current is , because all of the  drop is in the resistance. Therefore, the smaller the resistance, the faster a given capacitor will be charged. Note that the internal resistance of the voltage source is included in , as are the resistances of the capacitor and the connecting wires. In the flash camera scenario above, when the batteries powering the camera begin to wear out, their internal resistance rises, reducing the current and lengthening the time it takes to get ready for the next flash.
Voltage on the capacitor is initially zero and rises rapidly at first, since the initial current is a maximum. (b) shows a graph of capacitor voltage versus time () starting when the switch is closed at . The voltage approaches emf asymptotically, since the closer it gets to emf the less current flows. The equation for voltage versus time when charging a capacitor  through a resistor , derived using calculus, is
where  is the voltage across the capacitor, emf is equal to the emf of the DC voltage source, and the exponential e = 2.718 … is the base of the natural logarithm. Note that the units of  are seconds. We define
where  (the Greek letter tau) is called the time constant for an  circuit. As noted before, a small resistance  allows the capacitor to charge faster. This is reasonable, since a larger current flows through a smaller resistance. It is also reasonable that the smaller the capacitor , the less time needed to charge it. Both factors are contained in .
More quantitatively, consider what happens when . Then the voltage on the capacitor is
This means that in the time , the voltage rises to 0.632 of its final value. The voltage will rise 0.632 of the remainder in the next time . It is a characteristic of the exponential function that the final value is never reached, but 0.632 of the remainder to that value is achieved in every time, . In just a few multiples of the time constant , then, the final value is very nearly achieved, as the graph in (b) illustrates.

### Discharging a Capacitor

Discharging a capacitor through a resistor proceeds in a similar fashion, as  illustrates. Initially, the current is , driven by the initial voltage  on the capacitor. As the voltage decreases, the current and hence the rate of discharge decreases, implying another exponential formula for . Using calculus, the voltage  on a capacitor  being discharged through a resistor  is found to be
The graph in (b) is an example of this exponential decay. Again, the time constant is . A small resistance  allows the capacitor to discharge in a small time, since the current is larger. Similarly, a small capacitance requires less time to discharge, since less charge is stored. In the first time interval  after the switch is closed, the voltage falls to 0.368 of its initial value, since .
During each successive time , the voltage falls to 0.368 of its preceding value. In a few multiples of , the voltage becomes very close to zero, as indicated by the graph in (b).
Now we can explain why the flash camera in our scenario takes so much longer to charge than discharge; the resistance while charging is significantly greater than while discharging. The internal resistance of the battery accounts for most of the resistance while charging. As the battery ages, the increasing internal resistance makes the charging process even slower. (You may have noticed this.)
The flash discharge is through a low-resistance ionized gas in the flash tube and proceeds very rapidly. Flash photographs, such as in , can capture a brief instant of a rapid motion because the flash can be less than a microsecond in duration. Such flashes can be made extremely intense.
During World War II, nighttime reconnaissance photographs were made from the air with a single flash illuminating more than a square kilometer of enemy territory. The brevity of the flash eliminated blurring due to the surveillance aircraft’s motion. Today, an important use of intense flash lamps is to pump energy into a laser. The short intense flash can rapidly energize a laser and allow it to reemit the energy in another form.

### RC Circuits for Timing

 circuits are commonly used for timing purposes. A mundane example of this is found in the ubiquitous intermittent wiper systems of modern cars. The time between wipes is varied by adjusting the resistance in an  circuit. Another example of an  circuit is found in novelty jewelry, Halloween costumes, and various toys that have battery-powered flashing lights. (See  for a timing circuit.)
A more crucial use of  circuits for timing purposes is in the artificial pacemaker, used to control heart rate. The heart rate is normally controlled by electrical signals generated by the sino-atrial (SA) node, which is on the wall of the right atrium chamber. This causes the muscles to contract and pump blood. Sometimes the heart rhythm is abnormal and the heartbeat is too high or too low.
The artificial pacemaker is inserted near the heart to provide electrical signals to the heart when needed with the appropriate time constant. Pacemakers have sensors that detect body motion and breathing to increase the heart rate during exercise to meet the body’s increased needs for blood and oxygen.

### Test Prep for AP Courses

### Section Summary

1. An  circuit is one that has both a resistor and a capacitor.
2. The time constant  for an  circuit is .
3. When an initially uncharged ( at ) capacitor in series with a resistor is charged by a DC voltage source, the voltage rises, asymptotically approaching the emf of the voltage source; as a function of time,

4. Within the span of each time constant , the voltage rises by 0.632 of the remaining value, approaching the final voltage asymptotically.
5. If a capacitor with an initial voltage  is discharged through a resistor starting at , then its voltage decreases exponentially as given by

6. In each time constant , the voltage falls by 0.368 of its  remaining initial value, approaching zero asymptotically.

### Conceptual questions

### Problem Exercises


# Magnetism

## Introduction to Magnetism

One evening, an Alaskan sticks a note to his refrigerator with a small magnet. Through the kitchen window, the Aurora Borealis glows in the night sky. This grand spectacle is shaped by the same force that holds the note to the refrigerator.
People have been aware of magnets and magnetism for thousands of years. The earliest records date to well before the time of Christ, particularly in a region of Asia Minor called Magnesia (the name of this region is the source of words like magnetic). Magnetic rocks found in Magnesia, which is now part of western Turkey, stimulated interest during ancient times. A practical application for magnets was found later, when they were employed as navigational compasses. The use of magnets in compasses resulted not only in improved long-distance sailing, but also in the names of “north” and “south” being given to the two types of magnetic poles.
Today magnetism plays many important roles in our lives. Physicists’ understanding of magnetism has enabled the development of technologies that affect our everyday lives. The iPod in your purse or backpack, for example, wouldn’t have been possible without the applications of magnetism and electricity on a small scale.
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. The 2007 Nobel Prize in Physics went to Albert Fert from France and Peter Grunberg from Germany for this discovery of  giant magnetoresistance  and its applications to computer memory.

All electric motors, with uses as diverse as powering refrigerators, starting cars, and moving elevators, contain magnets. Generators, whether producing hydroelectric power or running bicycle lights, use magnetic fields. Recycling facilities employ magnets to separate iron from other refuse. Hundreds of millions of dollars are spent annually on magnetic containment of fusion as a future energy source. 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. The list of applications also includes computer hard drives, tape recording, detection of inhaled asbestos, and levitation of high-speed trains. Magnetism is used to explain atomic energy levels, cosmic rays, and charged particles trapped in the Van Allen belts. Once again, we will find all these disparate phenomena are linked by a small number of underlying physical principles.


# 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

## Introduction to Electromagnetic Induction, AC Circuits and Electrical Technologies

Nature’s displays of symmetry are beautiful and alluring. A butterfly’s wings exhibit an appealing symmetry in a complex system. (See .) 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.
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 here, then perhaps a magnetic field can create a current. The Hall effect is a voltage caused by a magnetic force. That voltage could drive a current. Historically, it was very shortly after Oersted discovered 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 force) 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. Not so familiar perhaps, but important nevertheless, is that the behavior of AC circuits depends strongly on the effect of magnetic fields on currents.