# Electric Potential and Electric Field

## Equipotential Lines

### Learning Objectives

By the end of this section, you will be able to:
1. Explain equipotential lines and equipotential surfaces.
2. Describe the action of grounding an electrical appliance.
3. Compare electric field and equipotential lines.
We can represent electric potentials (voltages) pictorially, just as we drew pictures to illustrate electric fields. Of course, the two are related. Consider , which shows an isolated positive point charge and its electric field lines. Electric field lines radiate out from a positive charge and terminate on negative charges. While we use blue arrows to represent the magnitude and direction of the electric field, we use green lines to represent places where the electric potential is constant. These are called equipotential lines in two dimensions, or equipotential surfaces in three dimensions.  The term equipotential is also used as a noun, referring to an equipotential line or surface. The potential for a point charge is the same anywhere on an imaginary sphere of radius  surrounding the charge. This is true since the potential for a point charge is given by  and, thus, has the same value at any point that is a given distance  from the charge. An equipotential sphere is a circle in the two-dimensional view of . Since the electric field lines point radially away from the charge, they are perpendicular to the equipotential lines.
It is important to note that equipotential lines are always perpendicular to electric field lines. No work is required to move a charge along an equipotential, since . Thus the work is
Work is zero if force is perpendicular to motion. Force is in the same direction as , so that motion along an equipotential must be perpendicular to . More precisely, work is related to the electric field by
Note that in the above equation,

               and

              symbolize the magnitudes of the electric field strength and force, respectively. Neither  nor  nor  is zero, and so  must be 0, meaning  must be

  . In other words, motion along an equipotential is perpendicular to .
One of the rules for static electric fields and conductors is that the electric field must be perpendicular to the surface of any conductor. This implies that a conductor is an equipotential surface in static situations. There can be no voltage difference across the surface of a conductor, or charges will flow. One of the uses of this fact is that a conductor can be fixed at zero volts by connecting it to the earth with a good conductor—a process called grounding. Grounding can be a useful safety tool. For example, grounding the metal case of an electrical appliance ensures that it is at zero volts relative to the earth.
Because a conductor is an equipotential, it can replace any equipotential surface. For example, in  a charged spherical conductor can replace the point charge, and the electric field and potential surfaces outside of it will be unchanged, confirming the contention that a spherical charge distribution is equivalent to a point charge at its center.
 shows the electric field and equipotential lines for two equal and opposite charges. Given the electric field lines, the equipotential lines can be drawn simply by making them perpendicular to the electric field lines. Conversely, given the equipotential lines, as in (a), the electric field lines can be drawn by making them perpendicular to the equipotentials, as in (b).
One of the most important cases is that of the familiar parallel conducting plates shown in . Between the plates, the equipotentials are evenly spaced and parallel. The same field could be maintained by placing conducting plates at the equipotential lines at the potentials shown.
An important application of electric fields and equipotential lines involves the heart. The heart relies on electrical signals to maintain its rhythm. The movement of electrical signals causes the chambers of the heart to contract and relax. When a person has a heart attack, the movement of these electrical signals may be disturbed. An artificial pacemaker and a defibrillator can be used to initiate the rhythm of electrical signals. The equipotential lines around the heart, the thoracic region, and the axis of the heart are useful ways of monitoring the structure and functions of the heart. An electrocardiogram (ECG) measures the small electric signals being generated during the activity of the heart. More about the relationship between electric fields and the heart is discussed in Energy Stored in Capacitors.

### Test Prep for AP Courses

### Section Summary

1. An equipotential line is a line along which the electric potential is constant.
2. An equipotential surface is a three-dimensional version of equipotential lines.
3. Equipotential lines are always perpendicular to electric field lines.
4. The process by which a conductor can be fixed at zero volts by connecting it to the earth with a good conductor is called grounding.

### Conceptual Questions

### Problems & Exercises


# Electric Potential and Electric Field

## Capacitors and Dielectrics

### Learning Objectives

By the end of this section, you will be able to:
1. Describe the action of a capacitor and define capacitance.
2. Explain parallel plate capacitors and their capacitances.
3. Discuss the process of increasing the capacitance of a dielectric.
4. Determine capacitance given charge and voltage.
A capacitor is a device used to store electric charge. Capacitors have applications ranging from filtering static out of radio reception to energy storage in heart defibrillators. Typically, commercial capacitors have two conducting parts close to one another, but not touching, such as those in . (Most of the time an insulator is used between the two plates to provide separation—see the discussion on dielectrics below.) When battery terminals are connected to an initially uncharged capacitor, equal amounts of positive and negative charge,  and , are separated into its two plates. The capacitor remains neutral overall, but we refer to it as storing a charge  in this circumstance.
The amount of charge  a capacitor can store depends on two major factors—the voltage applied and the capacitor’s physical characteristics, such as its size.
A system composed of two identical, parallel conducting plates separated by a distance, as in , is called a parallel plate capacitor. It is easy to see the relationship between the voltage and the stored charge for a parallel plate capacitor, as shown in . Each electric field line starts on an individual positive charge and ends on a negative one, so that there will be more field lines if there is more charge. (Drawing a single field line per charge is a convenience, only. We can draw many field lines for each charge, but the total number is proportional to the number of charges.) The electric field strength is, thus, directly proportional to .
The field is proportional to the charge:
where the symbol  means “proportional to.” From the discussion in Electric Potential in a Uniform Electric Field, we know that the voltage across parallel plates is . Thus,
It follows, then, that , and conversely,
This is true in general: The greater the voltage applied to any capacitor, the greater the charge stored in it.
Different capacitors will store different amounts of charge for the same applied voltage, depending on their physical characteristics. We define their capacitance  to be such that the charge  stored in a capacitor is proportional to . The charge stored in a capacitor is given by
This equation expresses the two major factors affecting the amount of charge stored. Those factors are the physical characteristics of the capacitor, , and the voltage, . Rearranging the equation, we see that capacitance  or
The unit of capacitance is the farad (F), named for Michael Faraday (1791–1867), an English scientist who contributed to the fields of electromagnetism and electrochemistry. Since capacitance is charge per unit voltage, we see that a farad is a coulomb per volt, or
A 1-farad capacitor would be able to store 1 coulomb (a very large amount of charge) with the application of only 1 volt. One farad is, thus, a very large capacitance. Typical capacitors range from fractions of a picofarad  to millifarads .
 shows some common capacitors. Capacitors are primarily made of ceramic, glass, or plastic, depending upon purpose and size. Insulating materials, called dielectrics, are commonly used in their construction, as discussed below.

### Parallel Plate Capacitor

The parallel plate capacitor shown in  has two identical conducting plates, each having a surface area , separated by a distance  (with no material between the plates). When a voltage  is applied to the capacitor, it stores a charge , as shown. We can see how its capacitance depends on  and  by considering the characteristics of the Coulomb force. We know that like charges repel, unlike charges attract, and the force between charges decreases with distance. So it seems quite reasonable that the bigger the plates are, the more charge they can store—because the charges can spread out more. Thus  should be greater for larger . Similarly, the closer the plates are together, the greater the attraction of the opposite charges on them. So  should be greater for smaller .
It can be shown that for a parallel plate capacitor there are only two factors ( and ) that affect its capacitance . The capacitance of a parallel plate capacitor in equation form is given by

is the area of one plate in square meters, and

is the distance between the plates in meters. The constant

is the permittivity of free space; its numerical value in SI units is

  .
The units of F/m are equivalent to . The small numerical value of  is related to the large size of the farad. A parallel plate capacitor must have a large area to have a capacitance approaching a farad. (Note that the above equation is valid when the parallel plates are separated by air or free space. When another material is placed between the plates, the equation is modified, as discussed below.)
Another interesting biological example dealing with electric potential is found in the cell’s plasma membrane. The membrane sets a cell off from its surroundings and also allows ions to selectively pass in and out of the cell. There is a potential difference across the membrane of about

  . This is due to the mainly negatively charged ions in the cell and the predominance of positively charged sodium (

  ) ions outside. Things change when a nerve cell is stimulated.

   ions are allowed to pass through the membrane into the cell, producing a positive membrane potential—the nerve signal. The cell membrane is about 7 to 10 nm thick. An approximate value of the electric field across it is given by

This electric field is enough to cause a breakdown in air.

### Dielectric

The previous example highlights the difficulty of storing a large amount of charge in capacitors. If  is made smaller to produce a larger capacitance, then the maximum voltage must be reduced proportionally to avoid breakdown (since ). An important solution to this difficulty is to put an insulating material, called a dielectric, between the plates of a capacitor and allow

     to be as small as possible. Not only does the smaller  make the capacitance greater, but many insulators can withstand greater electric fields than air before breaking down.
There is another benefit to using a dielectric in a capacitor. Depending on the material used, the capacitance is greater than that given by the equation  by a factor , called the dielectric constant. A parallel plate capacitor with a dielectric between its plates has a capacitance given by
Values of the dielectric constant  for various materials are given in . Note that  for vacuum is exactly 1, and so the above equation is valid in that case, too. If a dielectric is used, perhaps by placing Teflon between the plates of the capacitor in , then the capacitance is greater by the factor , which for Teflon is 2.1.
Note also that the dielectric constant for air is very close to 1, so that air-filled capacitors act much like those with vacuum between their plates except that the air can become conductive if the electric field strength  becomes too great. (Recall that  for a parallel plate capacitor.) Also shown in  are maximum electric field strengths in V/m, called dielectric strengths, for several materials. These are the fields above which the material begins to break down and conduct. The dielectric strength imposes a limit on the voltage that can be applied for a given plate separation. For instance, in , the separation is 1.00 mm, and so the voltage limit for air is
However, the limit for a 1.00 mm separation filled with Teflon is 60,000 V, since the dielectric strength of Teflon is  V/m. So the same capacitor filled with Teflon has a greater capacitance and can be subjected to a much greater voltage. Using the capacitance we calculated in the above example for the air-filled parallel plate capacitor, we find that the Teflon-filled capacitor can store a maximum charge of
This is 42 times the charge of the same air-filled capacitor.
Microscopically, how does a dielectric increase capacitance? Polarization of the insulator is responsible. The more easily it is polarized, the greater its dielectric constant . Water, for example, is a polar molecule because one end of the molecule has a slight positive charge and the other end has a slight negative charge. The polarity of water causes it to have a relatively large dielectric constant of 80. The effect of polarization can be best explained in terms of the characteristics of the Coulomb force.  shows the separation of charge schematically in the molecules of a dielectric material placed between the charged plates of a capacitor. The Coulomb force between the closest ends of the molecules and the charge on the plates is attractive and very strong, since they are very close together. This attracts more charge onto the plates than if the space were empty and the opposite charges were a distance  away.
Another way to understand how a dielectric increases capacitance is to consider its effect on the electric field inside the capacitor. (b) shows the electric field lines with a dielectric in place. Since the field lines end on charges in the dielectric, there are fewer of them going from one side of the capacitor to the other. So the electric field strength is less than if there were a vacuum between the plates, even though the same charge is on the plates. The voltage between the plates is , so it too is reduced by the dielectric. Thus there is a smaller voltage  for the same charge ; since , the capacitance  is greater.
The dielectric constant is generally defined to be , or the ratio of the electric field in a vacuum to that in the dielectric material, and is intimately related to the polarizability of the material.
We will find in Atomic Physics that the orbits of electrons are more properly viewed as electron clouds with the density of the cloud related to the probability of finding an electron in that location (as opposed to the definite locations and paths of planets in their orbits around the Sun). This cloud is shifted by the Coulomb force so that the atom on average has a separation of charge. Although the atom remains neutral, it can now be the source of a Coulomb force, since a charge brought near the atom will be closer to one type of charge than the other.
Some molecules, such as those of water, have an inherent separation of charge and are thus called polar molecules.  illustrates the separation of charge in a water molecule, which has two hydrogen atoms and one oxygen atom . The water molecule is not symmetric—the hydrogen atoms are repelled to one side, giving the molecule a boomerang shape. The electrons in a water molecule are more concentrated around the more highly charged oxygen nucleus than around the hydrogen nuclei. This makes the oxygen end of the molecule slightly negative and leaves the hydrogen ends slightly positive. The inherent separation of charge in polar molecules makes it easier to align them with external fields and charges. Polar molecules therefore exhibit greater polarization effects and have greater dielectric constants. Those who study chemistry will find that the polar nature of water has many effects. For example, water molecules gather ions much more effectively because they have an electric field and a separation of charge to attract charges of both signs. Also, as brought out in the previous chapter, polar water provides a shield or screening of the electric fields in the highly charged molecules of interest in biological systems.

### Test Prep for AP Courses

### Section Summary

1. A capacitor is a device used to store charge.
2. The amount of charge  a capacitor can store depends on two major factors—the voltage applied and the capacitor’s physical characteristics, such as its size.
3. The capacitance  is the amount of charge stored per volt, or

4. The capacitance of a parallel plate capacitor is , when the plates are separated by air or free space.

   is called the permittivity of free space.
5. A parallel plate capacitor with a dielectric between its plates has a capacitance given by

where

   is the dielectric constant of the material.
6. The maximum electric field strength above which an insulating material begins to break down and conduct is called dielectric strength.

### Conceptual Questions

### Problems & Exercises


# Electric Potential and Electric Field

## Capacitors in Series and Parallel

### Learning Objectives

By the end of this section, you will be able to:
1. Derive expressions for total capacitance in series and in parallel.
2. Identify series and parallel parts in the combination of connection of capacitors.
3. Calculate the effective capacitance in series and parallel given individual capacitances.
Several capacitors may be connected together in a variety of applications. Multiple connections of capacitors act like a single equivalent capacitor. The total capacitance of this equivalent single capacitor depends both on the individual capacitors and how they are connected. There are two simple and common types of connections, called series and parallel, for which we can easily calculate the total capacitance. Certain more complicated connections can also be related to combinations of series and parallel.

### Capacitance in Series

(a) shows a series connection of three capacitors with a voltage applied. As for any capacitor, the capacitance of the combination is related to charge and voltage by .
Note in  that opposite charges of magnitude  flow to either side of the originally uncharged combination of capacitors when the voltage  is applied. Conservation of charge requires that equal-magnitude charges be created on the plates of the individual capacitors, since charge is only being separated in these originally neutral devices. The end result is that the combination resembles a single capacitor with an effective plate separation greater than that of the individual capacitors alone. (See (b).) Larger plate separation means smaller capacitance. It is a general feature of series connections of capacitors that the total capacitance is less than any of the individual capacitances.
We can find an expression for the total capacitance by considering the voltage across the individual capacitors shown in . Solving  for  gives . The voltages across the individual capacitors are thus , , and . The total voltage is the sum of the individual voltages:
Now, calling the total capacitance  for series capacitance, consider that
Entering the expressions for , , and , we get
Canceling the s, we obtain the equation for the total capacitance in series  to be
where “...” indicates that the expression is valid for any number of capacitors connected in series. An expression of this form always results in a total capacitance  that is less than any of the individual capacitances , , ..., as the next example illustrates.

### Capacitors in Parallel

(a) shows a parallel connection of three capacitors with a voltage applied. Here the total capacitance is easier to find than in the series case. To find the equivalent total capacitance , we first note that the voltage across each capacitor is , the same as that of the source, since they are connected directly to it through a conductor. (Conductors are equipotentials, and so the voltage across the capacitors is the same as that across the voltage source.) Thus the capacitors have the same charges on them as they would have if connected individually to the voltage source. The total charge  is the sum of the individual charges:
Using the relationship , we see that the total charge is , and the individual charges are , , and . Entering these into the previous equation gives
Canceling  from the equation, we obtain the equation for the total capacitance in parallel
     :

Total capacitance in parallel is simply the sum of the individual capacitances. (Again the “...” indicates the expression is valid for any number of capacitors connected in parallel.) So, for example, if the capacitors in the example above were connected in parallel, their capacitance would be
The equivalent capacitor for a parallel connection has an effectively larger plate area and, thus, a larger capacitance, as illustrated in (b).
More complicated connections of capacitors can sometimes be combinations of series and parallel. (See .) To find the total capacitance of such combinations, we identify series and parallel parts, compute their capacitances, and then find the total.

### Section Summary

1. Total capacitance in series
2. Total capacitance in parallel
3. If a circuit contains a combination of capacitors in series and parallel, identify series and parallel parts, compute their capacitances, and then find the total.

### Conceptual Questions

### Problems & Exercises


# Electric Potential and Electric Field

## Energy Stored in Capacitors

### Learning Objectives

By the end of this section, you will be able to:
1. List some uses of capacitors.
2. Express in equation form the energy stored in a capacitor.
3. Explain the function of a defibrillator.
Most of us have seen dramatizations in which medical personnel use a defibrillator to pass an electric current through a patient’s heart to get it to beat normally. (Review .) Often realistic in detail, the person applying the shock directs another person to “make it 400 joules this time.” The energy delivered by the defibrillator is stored in a capacitor and can be adjusted to fit the situation. SI units of joules are often employed. Less dramatic is the use of capacitors in microelectronics, such as certain handheld calculators, to supply energy when batteries are charged. (See .) Capacitors are also used to supply energy for flash lamps on cameras.
Energy stored in a capacitor is electrical potential energy, and it is thus related to the charge  and voltage  on the capacitor. We must be careful when applying the equation for electrical potential energy  to a capacitor. Remember that  is the potential energy of a charge  going through a voltage . But the capacitor starts with zero voltage and gradually comes up to its full voltage as it is charged. The first charge placed on a capacitor experiences a change in voltage , since the capacitor has zero voltage when uncharged. The final charge placed on a capacitor experiences , since the capacitor now has its full voltage  on it. The average voltage on the capacitor during the charging process is , and so the average voltage experienced by the full charge  is . Thus the energy stored in a capacitor, , is
where  is the charge on a capacitor with a voltage  applied. (Note that the energy is not , but .) Charge and voltage  are related to the capacitance

   of a capacitor by , and so the expression for  can be algebraically manipulated into three equivalent expressions:
where  is the charge and  the voltage on a capacitor . The energy is in joules for a charge in coulombs, voltage in volts, and capacitance in farads.
In a defibrillator, the delivery of a large charge in a short burst to a set of paddles across a person’s chest can be a lifesaver. The person’s heart attack might have arisen from the onset of fast, irregular beating of the heart—cardiac or ventricular fibrillation. The application of a large shock of electrical energy can terminate the arrhythmia and allow the body’s pacemaker to resume normal patterns. Today it is common for ambulances to carry a defibrillator, which also uses an electrocardiogram to analyze the patient’s heartbeat pattern. Automated external defibrillators (AED) are found in many public places (). These are designed to be used by lay persons. The device automatically diagnoses the patient’s heart condition and then applies the shock with appropriate energy and waveform. CPR is recommended in many cases before use of an AED.

### Test Prep for AP Courses

### Section Summary

1. Capacitors are used in a variety of devices, including defibrillators, microelectronics such as calculators, and flash lamps, to supply energy.
2. The energy stored in a capacitor can be expressed in three ways:

where
 is the charge,  is the voltage, and  is the capacitance of the capacitor. The energy is in joules when the charge is in coulombs, voltage is in volts, and capacitance is in farads.

### Conceptual Questions

### Problems & Exercises


# Electric Current, Resistance, and Ohm's Law

## Introduction to Electric Current, Resistance, and Ohm's Law

The flicker of numbers on a handheld calculator, nerve impulses carrying signals of vision to the brain, an ultrasound device sending a signal to a computer screen, the brain sending a message for a baby to twitch its toes, an electric train pulling its load over a mountain pass, a hydroelectric plant sending energy to metropolitan and rural users—these and many other examples of electricity involve electric current, the movement of charge. Humankind has indeed harnessed electricity, the basis of technology, to improve our quality of life. Whereas the previous two chapters concentrated on static electricity and the fundamental force underlying its behavior, the next few chapters will be devoted to electric and magnetic phenomena involving current. In addition to exploring applications of electricity, we shall gain new insights into nature—in particular, the fact that all magnetism results from electric current.


# Electric Current, Resistance, and Ohm's Law

## Current

### Learning Objectives

By the end of this section, you will be able to:
1. Define electric current, ampere, and drift velocity
2. Describe the direction of charge flow in conventional current.
3. Use drift velocity to calculate current and vice versa.

### Electric Current

Electric current is defined to be the rate at which charge flows. A large current, such as that used to start a truck engine, moves a large amount of charge in a small time, whereas a small current, such as that used to operate a hand-held calculator, moves a small amount of charge over a long period of time. In equation form, electric current  is defined to be
where  is the amount of charge passing through a given area in time . (As in previous chapters, initial time is often taken to be zero, in which case .) (See .) The SI unit for current is the ampere (A), named for the French physicist André-Marie Ampère (1775–1836). Since , we see that an ampere is one coulomb per second:
Not only are fuses and circuit breakers rated in amperes (or amps), so are many electrical appliances.
 shows a simple circuit and the standard schematic representation of a battery, conducting path, and load (a resistor). Schematics are very useful in visualizing the main features of a circuit. A single schematic can represent a wide variety of situations. The schematic in  (b), for example, can represent anything from a truck battery connected to a headlight lighting the street in front of the truck to a small battery connected to a penlight lighting a keyhole in a door. Such schematics are useful because the analysis is the same for a wide variety of situations. We need to understand a few schematics to apply the concepts and analysis to many more situations.
Note that the direction of current flow in  is from positive to negative. The direction of conventional current is the direction that positive charge would flow. Depending on the situation, positive charges, negative charges, or both may move. In metal wires, for example, current is carried by electrons—that is, negative charges move. In ionic solutions, such as salt water, both positive and negative charges move. This is also true in nerve cells. A Van de Graaff generator used for nuclear research can produce a current of pure positive charges, such as protons.  illustrates the movement of charged particles that compose a current. The fact that conventional current is taken to be in the direction that positive charge would flow can be traced back to American politician and scientist Benjamin Franklin in the 1700s. He named the type of charge associated with electrons negative, long before they were known to carry current in so many situations. Franklin, in fact, was totally unaware of the small-scale structure of electricity.
It is important to realize that there is an electric field in conductors responsible for producing the current, as illustrated in . Unlike static electricity, where a conductor in equilibrium cannot have an electric field in it, conductors carrying a current have an electric field and are not in static equilibrium. An electric field is needed to supply energy to move the charges.

### Drift Velocity

Electrical signals are known to move very rapidly. Telephone conversations carried by currents in wires cover large distances without noticeable delays. Lights come on as soon as a switch is flicked. Most electrical signals carried by currents travel at speeds on the order of , a significant fraction of the speed of light. Interestingly, the individual charges that make up the current move much more slowly on average, typically drifting at speeds on the order of . How do we reconcile these two speeds, and what does it tell us about standard conductors?
The high speed of electrical signals results from the fact that the force between charges acts rapidly at a distance. Thus, when a free charge is forced into a wire, as in , the incoming charge pushes other charges ahead of it, which in turn push on charges farther down the line. The density of charge in a system cannot easily be increased, and so the signal is passed on rapidly. The resulting electrical shock wave moves through the system at nearly the speed of light. To be precise, this rapidly moving signal or shock wave is a rapidly propagating change in electric field.
Good conductors have large numbers of free charges in them. In metals, the free charges are free electrons.  shows how free electrons move through an ordinary conductor. The distance that an individual electron can move between collisions with atoms or other electrons is quite small. The electron paths thus appear nearly random, like the motion of atoms in a gas. But there is an electric field in the conductor that causes the electrons to drift in the direction shown (opposite to the field, since they are negative). The drift velocity  is the average velocity of the free charges. Drift velocity is quite small, since there are so many free charges. If we have an estimate of the density of free electrons in a conductor, we can calculate the drift velocity for a given current. The larger the density, the lower the velocity required for a given current.
The free-electron collisions transfer energy to the atoms of the conductor. The electric field does work in moving the electrons through a distance, but that work does not increase the kinetic energy (nor speed, therefore) of the electrons. The work is transferred to the conductor’s atoms, possibly increasing temperature. Thus a continuous power input is required to keep a current flowing. An exception, of course, is found in superconductors, for reasons we shall explore in a later chapter. Superconductors can have a steady current without a continual supply of energy—a great energy savings. In contrast, the supply of energy can be useful, such as in a lightbulb filament. The supply of energy is necessary to increase the temperature of the tungsten filament, so that the filament glows.
We can obtain an expression for the relationship between current and drift velocity by considering the number of free charges in a segment of wire, as illustrated in . The number of free charges per unit volume is given the symbol  and depends on the material. The shaded segment has a volume , so that the number of free charges in it is . The charge  in this segment is thus , where  is the amount of charge on each carrier. (Recall that for electrons,  is .) Current is charge moved per unit time; thus, if all the original charges move out of this segment in time , the current is
Note that  is the magnitude of the drift velocity,

  , since the charges move an average distance  in a time . Rearranging terms gives
where  is the current through a wire of cross-sectional area  made of a material with a free charge density . The carriers of the current each have charge  and move with a drift velocity of magnitude .
Note that simple drift velocity is not the entire story. The speed of an electron is much greater than its drift velocity. In addition, not all of the electrons in a conductor can move freely, and those that do might move somewhat faster or slower than the drift velocity. So what do we mean by free electrons? Atoms in a metallic conductor are packed in the form of a lattice structure. Some electrons are far enough away from the atomic nuclei that they do not experience the attraction of the nuclei as much as the inner electrons do. These are the free electrons. They are not bound to a single atom but can instead move freely among the atoms in a “sea” of electrons. These free electrons respond by accelerating when an electric field is applied. Of course as they move they collide with the atoms in the lattice and other electrons, generating thermal energy, and the conductor gets warmer. In an insulator, the organization of the atoms and the structure do not allow for such free electrons.

### Test Prep for AP Courses

### Section Summary

1. Electric current  is the rate at which charge flows, given by

    where  is the amount of charge passing through an area in time .
2. The direction of conventional current is taken as the direction in which positive charge moves.
3. The SI unit for current is the ampere (A), where
4. Current is the flow of free charges, such as electrons and ions.
5. Drift velocity  is the average speed at which these charges move.
6. Current  is proportional to drift velocity , as expressed in the relationship . Here,  is the current through a wire of cross-sectional area . The wire’s material has a free-charge density , and each carrier has charge  and a drift velocity .
7. Electrical signals travel at speeds about  times greater than the drift velocity of free electrons.

### Conceptual Questions

### Problems & Exercises


# Electric Current, Resistance, and Ohm's Law

## Ohm’s Law: Resistance and Simple Circuits

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the origin of Ohm’s law.
2. Calculate voltages, currents, or resistances with Ohm’s law.
3. Explain what an ohmic material is.
4. Describe a simple circuit.
What drives current? We can think of various devices—such as batteries, generators, wall outlets, and so on—which are necessary to maintain a current. All such devices create a potential difference and are loosely referred to as voltage sources. When a voltage source is connected to a conductor, it applies a potential difference  that creates an electric field. The electric field in turn exerts force on charges, causing current.

### Ohm’s Law

The current that flows through most substances is directly proportional to the voltage  applied to it. The German physicist Georg Simon Ohm (1787–1854) was the first to demonstrate experimentally that the current in a metal wire is directly proportional to the voltage applied:
This important relationship is known as Ohm’s law. It can be viewed as a cause-and-effect relationship, with voltage the cause and current the effect. This is an empirical law like that for friction—an experimentally observed phenomenon. Such a linear relationship doesn’t always occur.

### Resistance and Simple Circuits

If voltage drives current, what impedes it? The electric property that impedes current (crudely similar to friction and air resistance) is called resistance . Collisions of moving charges with atoms and molecules in a substance transfer energy to the substance and limit current. Resistance is defined as inversely proportional to current, or
Thus, for example, current is cut in half if resistance doubles. Combining the relationships of current to voltage and current to resistance gives
This relationship is also called Ohm’s law. Ohm’s law in this form really defines resistance for certain materials. Ohm’s law (like Hooke’s law) is not universally valid. The many substances for which Ohm’s law holds are called ohmic. These include good conductors like copper and aluminum, and some poor conductors under certain circumstances. Ohmic materials have a resistance  that is independent of voltage  and current . An object that has simple resistance is called a resistor, even if its resistance is small. The unit for resistance is an ohm and is given the symbol  (upper case Greek omega). Rearranging  gives , and so the units of resistance are 1 ohm = 1 volt per ampere:
 shows the schematic for a simple circuit. A simple circuit has a single voltage source and a single resistor. The wires connecting the voltage source to the resistor can be assumed to have negligible resistance, or their resistance can be included in .
Resistances range over many orders of magnitude. Some ceramic insulators, such as those used to support power lines, have resistances of  or more. A dry person may have a hand-to-foot resistance of , whereas the resistance of the human heart is about . A meter-long piece of large-diameter copper wire may have a resistance of , and superconductors have no resistance at all (they are non-ohmic). Resistance is related to the shape of an object and the material of which it is composed, as will be seen in Resistance and Resistivity.
Additional insight is gained by solving  for  yielding
This expression for  can be interpreted as the voltage drop across a resistor produced by the flow of current . The phrase  drop is often used for this voltage. For instance, the headlight in  has an  drop of 12.0 V. If voltage is measured at various points in a circuit, it will be seen to increase at the voltage source and decrease at the resistor. Voltage is similar to fluid pressure. The voltage source is like a pump, creating a pressure difference, causing current—the flow of charge. The resistor is like a pipe that reduces pressure and limits flow because of its resistance. Conservation of energy has important consequences here. The voltage source supplies energy (causing an electric field and a current), and the resistor converts it to another form (such as thermal energy). In a simple circuit (one with a single simple resistor), the voltage supplied by the source equals the voltage drop across the resistor, since , and the same  flows through each. Thus the energy supplied by the voltage source and the energy converted by the resistor are equal. (See .)

### Test Prep for AP Courses

### Section Summary

1. A simple circuit is one in which there is a single voltage source and a single resistance.
2. One statement of Ohm’s law gives the relationship between current

  ,
voltage

  ,
and resistance

in a simple circuit to be

3. Resistance has units of ohms (

  ), related to volts and amperes by .
4. There is a voltage or  drop across a resistor, caused by the current flowing through it, given by .

### Conceptual Questions

### Problems & Exercises


# Electric Current, Resistance, and Ohm's Law

## Resistance and Resistivity

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the concept of resistivity.
2. Use resistivity to calculate the resistance of specified configurations of material.
3. Use the thermal coefficient of resistivity to calculate the change of resistance with temperature.

### Material and Shape Dependence of Resistance

The resistance of an object depends on its shape and the material of which it is composed. The cylindrical resistor in  is easy to analyze, and, by so doing, we can gain insight into the resistance of more complicated shapes. As you might expect, the cylinder’s electric resistance  is directly proportional to its length , similar to the resistance of a pipe to fluid flow. The longer the cylinder, the more collisions charges will make with its atoms. The greater the diameter of the cylinder, the more current it can carry (again similar to the flow of fluid through a pipe). In fact,  is inversely proportional to the cylinder’s cross-sectional area .
For a given shape, the resistance depends on the material of which the object is composed. Different materials offer different resistance to the flow of charge. We define the resistivity  of a substance so that the resistance  of an object is directly proportional to . Resistivity  is an intrinsic property of a material, independent of its shape or size. The resistance  of a uniform cylinder of length , of cross-sectional area , and made of a material with resistivity , is
 gives representative values of . The materials listed in the table are separated into categories of conductors, semiconductors, and insulators, based on broad groupings of resistivities. Conductors have the smallest resistivities, and insulators have the largest; semiconductors have intermediate resistivities. Conductors have varying but large free charge densities, whereas most charges in insulators are bound to atoms and are not free to move. Semiconductors are intermediate, having far fewer free charges than conductors, but having properties that make the number of free charges depend strongly on the type and amount of impurities in the semiconductor. These unique properties of semiconductors are put to use in modern electronics, as will be explored in later chapters.

### Temperature Variation of Resistance

The resistivity of all materials depends on temperature. Some even become superconductors (zero resistivity) at very low temperatures. (See .) Conversely, the resistivity of conductors increases with increasing temperature. Since the atoms vibrate more rapidly and over larger distances at higher temperatures, the electrons moving through a metal make more collisions, effectively making the resistivity higher. Over relatively small temperature changes (about  or less), resistivity  varies with temperature change  as expressed in the following equation
where  is the original resistivity and  is the temperature coefficient of resistivity. (See the values of  in  below.) For larger temperature changes,  may vary or a nonlinear equation may be needed to find . Note that  is positive for metals, meaning their resistivity increases with temperature. Some alloys have been developed specifically to have a small temperature dependence. Manganin (which is made of copper, manganese and nickel), for example, has  close to zero (to three digits on the scale in ), and so its resistivity varies only slightly with temperature. This is useful for making a temperature-independent resistance standard, for example.
Note also that  is negative for the semiconductors listed in , meaning that their resistivity decreases with increasing temperature. They become better conductors at higher temperature, because increased thermal agitation increases the number of free charges available to carry current. This property of decreasing  with temperature is also related to the type and amount of impurities present in the semiconductors.
The resistance of an object also depends on temperature, since  is directly proportional to . For a cylinder we know , and so, if  and  do not change greatly with temperature,  will have the same temperature dependence as . (Examination of the coefficients of linear expansion shows them to be about two orders of magnitude less than typical temperature coefficients of resistivity, and so the effect of temperature on  and  is about two orders of magnitude less than on .) Thus,
is the temperature dependence of the resistance of an object, where  is the original resistance and  is the resistance after a temperature change . Numerous thermometers are based on the effect of temperature on resistance. (See .) One of the most common is the thermistor, a semiconductor crystal with a strong temperature dependence, the resistance of which is measured to obtain its temperature. The device is small, so that it quickly comes into thermal equilibrium with the part of a person it touches.

### Test Prep for AP Courses

### Section Summary

1. The resistance  of a cylinder of length  and cross-sectional area  is , where  is the resistivity of the material.
2. Values of  in  show that materials fall into three groups—conductors, semiconductors, and insulators.
3. Temperature affects resistivity; for relatively small temperature changes , resistivity is , where  is the original resistivity and

    is the temperature coefficient of resistivity.
4.  gives values for , the temperature coefficient of resistivity.
5. The resistance  of an object also varies with temperature: , where  is the original resistance, and

    is the resistance after the temperature change.

### Conceptual Questions

### Problems & Exercises


# Electric Current, Resistance, and Ohm's Law

## Electric Power and Energy

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate the power dissipated by a resistor and power supplied by a power supply.
2. Calculate the cost of electricity under various circumstances.

### Power in Electric Circuits

Power is associated by many people with electricity. Knowing that power is the rate of energy use or energy conversion, what is the expression for electric power? Power transmission lines might come to mind. We also think of lightbulbs in terms of their power ratings in watts. Let us compare a 25-W bulb with a 60-W bulb. (See (a).) Since both operate on the same voltage, the 60-W bulb must draw more current to have a greater power rating. Thus the 60-W bulb’s resistance must be lower than that of a 25-W bulb. If we increase voltage, we also increase power. For example, when a 25-W bulb that is designed to operate on 120 V is connected to 240 V, it briefly glows very brightly and then burns out. Precisely how are voltage, current, and resistance related to electric power?
Electric energy depends on both the voltage involved and the charge moved. This is expressed most simply as , where  is the charge moved and  is the voltage (or more precisely, the potential difference the charge moves through). Power is the rate at which energy is moved, and so electric power is
Recognizing that current is  (note that  here), the expression for power becomes
Electric power ( ) is simply the product of current times voltage. Power has familiar units of watts. Since the SI unit for potential energy (PE) is the joule, power has units of joules per second, or watts. Thus, . For example, cars often have one or more auxiliary power outlets with which you can charge a cell phone or other electronic devices. These outlets may be rated at 20 A, so that the circuit can deliver a maximum power
. In some applications, electric power may be expressed as volt-amperes or even kilovolt-amperes (
).
To see the relationship of power to resistance, we combine Ohm’s law with . Substituting  gives . Similarly, substituting  gives . Three expressions for electric power are listed together here for convenience:
Note that the first equation is always valid, whereas the other two can be used only for resistors. In a simple circuit, with one voltage source and a single resistor, the power supplied by the voltage source and that dissipated by the resistor are identical. (In more complicated circuits,  can be the power dissipated by a single device and not the total power in the circuit.)
Different insights can be gained from the three different expressions for electric power. For example,  implies that the lower the resistance connected to a given voltage source, the greater the power delivered. Furthermore, since voltage is squared in , the effect of applying a higher voltage is perhaps greater than expected. Thus, when the voltage is doubled to a 25-W bulb, its power nearly quadruples to about 100 W, burning it out. If the bulb’s resistance remained constant, its power would be exactly 100 W, but at the higher temperature its resistance is higher, too.

### The Cost of Electricity

The more electric appliances you use and the longer they are left on, the higher your electric bill. This familiar fact is based on the relationship between energy and power. You pay for the energy used. Since , we see that
is the energy used by a device using power  for a time interval . For example, the more lightbulbs burning, the greater  used; the longer they are on, the greater  is. The energy unit on electric bills is the kilowatt-hour (), consistent with the relationship . It is easy to estimate the cost of operating electric appliances if you have some idea of their power consumption rate in watts or kilowatts, the time they are on in hours, and the cost per kilowatt-hour for your electric utility. Kilowatt-hours, like all other specialized energy units such as food calories, can be converted to joules. You can prove to yourself that .
The electrical energy ( ) used can be reduced either by reducing the time of use or by reducing the power consumption of that appliance or fixture. This will not only reduce the cost, but it will also result in a reduced impact on the environment. Improvements to lighting are some of the fastest ways to reduce the electrical energy used in a home or business. About 20% of a home’s use of energy goes to lighting, while the number for commercial establishments is closer to 40%. Fluorescent lights are about four times more efficient than incandescent lights—this is true for both the long tubes and the compact fluorescent lights (CFL). (See (b).) Thus, a 60-W incandescent bulb can be replaced by a 15-W CFL, which has the same brightness and color. CFLs have a bent tube inside a globe or a spiral-shaped tube, all connected to a standard screw-in base that fits standard incandescent light sockets. (Original problems with color, flicker, shape, and high initial investment for CFLs have been addressed in recent years.) The heat transfer from these CFLs is less, and they last up to 10 times longer. The significance of an investment in such bulbs is addressed in the next example. New white LED lights (which are clusters of small LED bulbs) are even more efficient (twice that of CFLs) and last 5 times longer than CFLs. However, their cost is still high.

### Test Prep for AP Courses

### Section Summary

1. Electric power  is the rate (in watts) that energy is supplied by a source or dissipated by a device.
2. Three expressions for electrical power are

    and

3. The energy used by a device with a power  over a time  is .

### Conceptual Questions

### Problem Exercises


# Electric Current, Resistance, and Ohm's Law

## Alternating Current versus Direct Current

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the differences and similarities between AC and DC current.
2. Calculate rms voltage, current, and average power.
3. Explain why AC current is used for power transmission.

### Alternating Current

Most of the examples dealt with so far, and particularly those utilizing batteries, have constant voltage sources. Once the current is established, it is thus also a constant. Direct current (DC) is the flow of electric charge in only one direction. It is the steady state of a constant-voltage circuit. Most well-known applications, however, use a time-varying voltage source. Alternating current (AC) is the flow of electric charge that periodically reverses direction. If the source varies periodically, particularly sinusoidally, the circuit is known as an alternating current circuit. Examples include the commercial and residential power that serves so many of our needs.  shows graphs of voltage and current versus time for typical DC and AC power. The AC voltages and frequencies commonly used in homes and businesses vary around the world.
 shows a schematic of a simple circuit with an AC voltage source. The voltage between the terminals fluctuates as shown, with the AC voltage given by
where  is the voltage at time ,  is the peak voltage, and  is the frequency in hertz. For this simple resistance circuit, , and so the AC current is
where  is the current at time , and  is the peak current. For this example, the voltage and current are said to be in phase, as seen in (b).
Current in the resistor alternates back and forth just like the driving voltage, since . If the resistor is a fluorescent light bulb, for example, it brightens and dims 120 times per second as the current repeatedly goes through zero. A 120-Hz flicker is too rapid for your eyes to detect, but if you wave your hand back and forth between your face and a fluorescent light, you will see a stroboscopic effect evidencing AC. The fact that the light output fluctuates means that the power is fluctuating. The power supplied is . Using the expressions for  and  above, we see that the time dependence of power is , as shown in .
We are most often concerned with average power rather than its fluctuations—that 60-W light bulb in your desk lamp has an average power consumption of 60 W, for example. As illustrated in , the average power  is
This is evident from the graph, since the areas above and below the  line are equal, but it can also be proven using trigonometric identities. Similarly, we define an average or rms current  and average or rms voltage  to be, respectively,
and
where rms stands for root mean square, a particular kind of average. In general, to obtain a root mean square, the particular quantity is squared, its mean (or average) is found, and the square root is taken. This is useful for AC, since the average value is zero. Now,
which gives
as stated above. It is standard practice to quote , , and  rather than the peak values. For example, most household electricity is 120 V AC, which means that  is 120 V. The common 10-A circuit breaker will interrupt a sustained  greater than 10 A. Your 1.0-kW microwave oven consumes , and so on. You can think of these rms and average values as the equivalent DC values for a simple resistive circuit.
To summarize, when dealing with AC, Ohm’s law and the equations for power are completely analogous to those for DC, but rms and average values are used for AC. Thus, for AC, Ohm’s law is written
The various expressions for AC power  are
and

### Why Use AC for Power Distribution?

Most large power-distribution systems are AC. Moreover, the power is transmitted at much higher voltages than the 120-V AC (240 V in most parts of the world) we use in homes and on the job. Economies of scale make it cheaper to build a few very large electric power-generation plants than to build numerous small ones. This necessitates sending power long distances, and it is obviously important that energy losses en route be minimized. High voltages can be transmitted with much smaller power losses than low voltages, as we shall see. (See .) For safety reasons, the voltage at the user is reduced to familiar values. The crucial factor is that it is much easier to increase and decrease AC voltages than DC, so AC is used in most large power distribution systems.
It is widely recognized that high voltages pose greater hazards than low voltages. But, in fact, some high voltages, such as those associated with common static electricity, can be harmless. So it is not voltage alone that determines a hazard. It is not so widely recognized that AC shocks are often more harmful than similar DC shocks. Thomas Edison thought that AC shocks were more harmful and set up a DC power-distribution system in New York City in the late 1800s. There were bitter fights, in particular between Edison and George Westinghouse and Nikola Tesla, who were advocating the use of AC in early power-distribution systems. AC has prevailed largely due to transformers and lower power losses with high-voltage transmission.

### Section Summary

1. Direct current  (DC) is the flow of electric current in only one direction. It refers to systems where the source voltage is constant.
2. The voltage source of an alternating current (AC) system puts out , where  is the voltage at time ,  is the peak voltage, and  is the frequency in hertz.
3. In a simple circuit,  and AC current is , where  is the current at time , and  is the peak current.
4. The average AC power  is  .
5. Average (rms) current  and average (rms) voltage  are  and , where rms stands for root mean square.
6. Thus, .
7. Ohm’s law for AC is .
8. Expressions for the average power of an AC circuit  are ,

, and

, analogous to the expressions for DC circuits.

### Conceptual Questions

### Problem Exercises


# Electric Current, Resistance, and Ohm's Law

## Electric Hazards and the Human Body

### Learning Objectives

By the end of this section, you will be able to:
1. Define thermal hazard, shock hazard, and short circuit.
2. Explain what effects various levels of current have on the human body.
There are two known hazards of electricity—thermal and shock. A thermal hazard is one where excessive electric power causes undesired thermal effects, such as starting a fire in the wall of a house. A shock hazard occurs when electric current passes through a person. Shocks range in severity from painful, but otherwise harmless, to heart-stopping lethality. This section considers these hazards and the various factors affecting them in a quantitative manner. Electrical Safety: Systems and Devices will consider systems and devices for preventing electrical hazards.

### Thermal Hazards

Electric power causes undesired heating effects whenever electric energy is converted to thermal energy at a rate faster than it can be safely dissipated. A classic example of this is the short circuit, a low-resistance path between terminals of a voltage source. An example of a short circuit is shown in . Insulation on wires leading to an appliance has worn through, allowing the two wires to come into contact. Such an undesired contact with a high voltage is called a short. Since the resistance of the short, , is very small, the power dissipated in the short, , is very large. For example, if  is 120 V and  is , then the power is 144 kW, much greater than that used by a typical household appliance. Thermal energy delivered at this rate will very quickly raise the temperature of surrounding materials, melting or perhaps igniting them.
One particularly insidious aspect of a short circuit is that its resistance may actually be decreased due to the increase in temperature. This can happen if the short creates ionization. These charged atoms and molecules are free to move and, thus, lower the resistance . Since , the power dissipated in the short rises, possibly causing more ionization, more power, and so on. High voltages, such as the 480-V AC used in some industrial applications, lend themselves to this hazard, because higher voltages create higher initial power production in a short.
Another serious, but less dramatic, thermal hazard occurs when wires supplying power to a user are overloaded with too great a current. As discussed in the previous section, the power dissipated in the supply wires is , where  is the resistance of the wires and  the current flowing through them. If either  or  is too large, the wires overheat. For example, a worn appliance cord (with some of its braided wires broken) may have  rather than the  it should be. If 10.0 A of current passes through the cord, then  is dissipated in the cord—much more than is safe. Similarly, if a wire with a  resistance is meant to carry a few amps, but is instead carrying 100 A, it will severely overheat. The power dissipated in the wire will in that case be . Fuses and circuit breakers are used to limit excessive currents. (See  and .) Each device opens the circuit automatically when a sustained current exceeds safe limits.
Fuses and circuit breakers for typical household voltages and currents are relatively simple to produce, but those for large voltages and currents experience special problems. For example, when a circuit breaker tries to interrupt the flow of high-voltage electricity, a spark can jump across its points that ionizes the air in the gap and allows the current to continue flowing. Large circuit breakers found in power-distribution systems employ insulating gas and even use jets of gas to blow out such sparks. Here AC is safer than DC, since AC current goes through zero 120 times per second, giving a quick opportunity to extinguish these arcs.

### Shock Hazards

Electrical currents through people produce tremendously varied effects. An electrical current can be used to block back pain. The possibility of using electrical current to stimulate muscle action in paralyzed limbs, perhaps allowing paraplegics to walk, is under study. TV dramatizations in which electrical shocks are used to bring a heart attack victim out of ventricular fibrillation (a massively irregular, often fatal, beating of the heart) are more than common. Yet most electrical shock fatalities occur because a current put the heart into fibrillation. A pacemaker uses electrical shocks to stimulate the heart to beat properly. Some fatal shocks do not produce burns, but warts can be safely burned off with electric current (though freezing using liquid nitrogen is now more common). Of course, there are consistent explanations for these disparate effects. The major factors upon which the effects of electrical shock depend are
1. The amount of current
2. The path taken by the current
3. The duration of the shock
4. The frequency  of the current ( for DC)
 gives the effects of electrical shocks as a function of current for a typical accidental shock. The effects are for a shock that passes through the trunk of the body, has a duration of 1 s, and is caused by 60-Hz power.
Our bodies are relatively good conductors due to the water in our bodies. Given that larger currents will flow through sections with lower resistance (to be further discussed in the next chapter), electric currents preferentially flow through paths in the human body that have a minimum resistance in a direct path to earth. The earth is a natural electron sink. Wearing insulating shoes, a requirement in many professions, prohibits a pathway for electrons by providing a large resistance in that path. Whenever working with high-power tools (drills), or in risky situations, ensure that you do not provide a pathway for current flow (especially through the heart).
Very small currents pass harmlessly and unfelt through the body. This happens to you regularly without your knowledge. The threshold of sensation is only 1 mA and, although unpleasant, shocks are apparently harmless for currents less than 5 mA. A great number of safety rules take the 5-mA value for the maximum allowed shock. At 10 to 20 mA and above, the current can stimulate sustained muscular contractions much as regular nerve impulses do. People sometimes say they were knocked across the room by a shock, but what really happened was that certain muscles contracted, propelling them in a manner not of their own choosing. (See (a).) More frightening, and potentially more dangerous, is the “can’t let go” effect illustrated in (b). The muscles that close the fingers are stronger than those that open them, so the hand closes involuntarily on the wire shocking it. This can prolong the shock indefinitely. It can also be a danger to a person trying to rescue the victim, because the rescuer’s hand may close about the victim’s wrist. Usually the best way to help the victim is to give the fist a hard knock/blow/jar with an insulator or to throw an insulator at the fist. Modern electric fences, used in animal enclosures, are now pulsed on and off to allow people who touch them to get free, rendering them less lethal than in the past.
Greater currents may affect the heart. Its electrical patterns can be disrupted, so that it beats irregularly and ineffectively in a condition called “ventricular fibrillation.” This condition often lingers after the shock and is fatal due to a lack of blood circulation. The threshold for ventricular fibrillation is between 100 and 300 mA. At about 300 mA and above, the shock can cause burns, depending on the concentration of current—the more concentrated, the greater the likelihood of burns.
Very large currents cause the heart and diaphragm to contract for the duration of the shock. Both the heart and breathing stop. Interestingly, both often return to normal following the shock. The electrical patterns on the heart are completely erased in a manner that the heart can start afresh with normal beating, as opposed to the permanent disruption caused by smaller currents that can put the heart into ventricular fibrillation. The latter is something like scribbling on a blackboard, whereas the former completely erases it. TV dramatizations of electric shock used to bring a heart attack victim out of ventricular fibrillation also show large paddles. These are used to spread out current passed through the victim to reduce the likelihood of burns.
Current is the major factor determining shock severity (given that other conditions such as path, duration, and frequency are fixed, such as in the table and preceding discussion). A larger voltage is more hazardous, but since , the severity of the shock depends on the combination of voltage and resistance. For example, a person with dry skin has a resistance of about . If he comes into contact with 120-V AC, a current  passes harmlessly through him. The same person soaking wet may have a resistance of  and the same 120 V will produce a current of 12 mA—above the “can’t let go” threshold and potentially dangerous.
Most of the body’s resistance is in its dry skin. When wet, salts go into ion form, lowering the resistance significantly. The interior of the body has a much lower resistance than dry skin because of all the ionic solutions and fluids it contains. If skin resistance is bypassed, such as by an intravenous infusion, a catheter, or exposed pacemaker leads, a person is rendered microshock sensitive. In this condition, currents about 1/1000 those listed in  produce similar effects. During open-heart surgery, currents as small as  can be used to still the heart. Stringent electrical safety requirements in hospitals, particularly in surgery and intensive care, are related to the doubly disadvantaged microshock-sensitive patient. The break in the skin has reduced his resistance, and so the same voltage causes a greater current, and a much smaller current has a greater effect.
Factors other than current that affect the severity of a shock are its path, duration, and AC frequency. Path has obvious consequences. For example, the heart is unaffected by an electric shock through the brain, such as may be used to treat manic depression. And it is a general truth that the longer the duration of a shock, the greater its effects.  presents a graph that illustrates the effects of frequency on a shock. The curves show the minimum current for two different effects, as a function of frequency. The lower the current needed, the more sensitive the body is at that frequency. Ironically, the body is most sensitive to frequencies near the 50- or 60-Hz frequencies in common use. The body is slightly less sensitive for DC (), mildly confirming Edison’s claims that AC presents a greater hazard. At higher and higher frequencies, the body becomes progressively less sensitive to any effects that involve nerves. This is related to the maximum rates at which nerves can fire or be stimulated. At very high frequencies, electrical current travels only on the surface of a person. Thus a wart can be burned off with very high frequency current without causing the heart to stop. (Do not try this at home with 60-Hz AC!) Some of the spectacular demonstrations of electricity, in which high-voltage arcs are passed through the air and over people’s bodies, employ high frequencies and low currents. (See .) Electrical safety devices and techniques are discussed in detail in Electrical Safety: Systems and Devices.

### Section Summary

1. The two types of electric hazards are thermal (excessive power) and shock (current through a person).
2. Shock severity is determined by current, path, duration, and AC frequency.
3.  lists shock hazards as a function of current.
4.  graphs the threshold current for two hazards as a function of frequency.

### Conceptual Questions

### Problem Exercises


# Electric Current, Resistance, and Ohm's Law

## Nerve Conduction–Electrocardiograms

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the process by which electric signals are transmitted along a neuron.
2. Explain the effects myelin sheaths have on signal propagation.
3. Explain what the features of an ECG signal indicate.

### Nerve Conduction

Electric currents in the vastly complex system of billions of nerves in our body allow us to sense the world, control parts of our body, and think. These are representative of the three major functions of nerves. First, nerves carry messages from our sensory organs and others to the central nervous system, consisting of the brain and spinal cord. Second, nerves carry messages from the central nervous system to muscles and other organs. Third, nerves transmit and process signals within the central nervous system. The sheer number of nerve cells and the incredibly greater number of connections between them makes this system the subtle wonder that it is. Nerve conduction is a general term for electrical signals carried by nerve cells. It is one aspect of bioelectricity, or electrical effects in and created by biological systems.
Nerve cells, properly called neurons, look different from other cells—they have tendrils, some of them many centimeters long, connecting them with other cells. (See .) Signals arrive at the cell body across synapses or through dendrites, stimulating the neuron to generate its own signal, sent along its long axon to other nerve or muscle cells. Signals may arrive from many other locations and be transmitted to yet others, conditioning the synapses by use, giving the system its complexity and its ability to learn.
The method by which these electric currents are generated and transmitted is more complex than the simple movement of free charges in a conductor, but it can be understood with principles already discussed in this text. The most important of these are the Coulomb force and diffusion.
 illustrates how a voltage (potential difference) is created across the cell membrane of a neuron in its resting state. This thin membrane separates electrically neutral fluids having differing concentrations of ions, the most important varieties being , , and  (these are sodium, potassium, and chlorine ions with single plus or minus charges as indicated). As discussed in Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes, free ions will diffuse from a region of high concentration to one of low concentration. But the cell membrane is semipermeable, meaning that some ions may cross it while others cannot. In its resting state, the cell membrane is permeable to  and , and impermeable to . Diffusion of  and  thus creates the layers of positive and negative charge on the outside and inside of the membrane. The Coulomb force prevents the ions from diffusing across in their entirety. Once the charge layer has built up, the repulsion of like charges prevents more from moving across, and the attraction of unlike charges prevents more from leaving either side. The result is two layers of charge right on the membrane, with diffusion being balanced by the Coulomb force. A tiny fraction of the charges move across and the fluids remain neutral (other ions are present), while a separation of charge and a voltage have been created across the membrane.
The separation of charge creates a potential difference of 70 to 90 mV across the cell membrane. While this is a small voltage, the resulting electric field () across the only 8-nm-thick membrane is immense (on the order of 11 MV/m!) and has fundamental effects on its structure and permeability. Now, if the exterior of a neuron is taken to be at 0 V, then the interior has a resting potential of about –90 mV. Such voltages are created across the membranes of almost all types of animal cells but are largest in nerve and muscle cells. In fact, fully 25% of the energy used by cells goes toward creating and maintaining these potentials.
Electric currents along the cell membrane are created by any stimulus that changes the membrane’s permeability. The membrane thus temporarily becomes permeable to , which then rushes in, driven both by diffusion and the Coulomb force. This inrush of  first neutralizes the inside membrane, or depolarizes it, and then makes it slightly positive. The depolarization causes the membrane to again become impermeable to , and the movement of  quickly returns the cell to its resting potential, or repolarizes it. This sequence of events results in a voltage pulse, called the action potential. (See .) Only small fractions of the ions move, so that the cell can fire many hundreds of times without depleting the excess concentrations of  and . Eventually, the cell must replenish these ions to maintain the concentration differences that create bioelectricity. This sodium-potassium pump is an example of active transport, wherein cell energy is used to move ions across membranes against diffusion gradients and the Coulomb force.
The action potential is a voltage pulse at one location on a cell membrane. How does it get transmitted along the cell membrane, and in particular down an axon, as a nerve impulse? The answer is that the changing voltage and electric fields affect the permeability of the adjacent cell membrane, so that the same process takes place there. The adjacent membrane depolarizes, affecting the membrane further down, and so on, as illustrated in . Thus the action potential stimulated at one location triggers a nerve impulse that moves slowly (about 1 m/s) along the cell membrane.
Some axons, like that in , are sheathed with myelin, consisting of fat-containing cells.  shows an enlarged view of an axon having myelin sheaths characteristically separated by unmyelinated gaps (called nodes of Ranvier). This arrangement gives the axon a number of interesting properties. Since myelin is an insulator, it prevents signals from jumping between adjacent nerves (cross talk). Additionally, the myelinated regions transmit electrical signals at a very high speed, as an ordinary conductor or resistor would. There is no action potential in the myelinated regions, so that no cell energy is used in them. There is an  signal loss in the myelin, but the signal is regenerated in the gaps, where the voltage pulse triggers the action potential at full voltage. So a myelinated axon transmits a nerve impulse faster, with less energy consumption, and is better protected from cross talk than an unmyelinated one. Not all axons are myelinated, so that cross talk and slow signal transmission are a characteristic of the normal operation of these axons, another variable in the nervous system.
The degeneration or destruction of the myelin sheaths that surround the nerve fibers impairs signal transmission and can lead to numerous neurological effects. One of the most prominent of these diseases comes from the body’s own immune system attacking the myelin in the central nervous system—multiple sclerosis. MS symptoms include fatigue, vision problems, weakness of arms and legs, loss of balance, and tingling or numbness in one’s extremities (neuropathy). It is more apt to strike younger adults, especially females. Causes might come from infection, environmental or geographic affects, or genetics. At the moment there is no known cure for MS.
Most animal cells can fire or create their own action potential. Muscle cells contract when they fire and are often induced to do so by a nerve impulse. In fact, nerve and muscle cells are physiologically similar, and there are even hybrid cells, such as in the heart, that have characteristics of both nerves and muscles. Some animals, like the infamous electric eel (see ), use muscles ganged so that their voltages add in order to create a shock great enough to stun prey.

### Electrocardiograms

Just as nerve impulses are transmitted by depolarization and repolarization of adjacent membrane, the depolarization that causes muscle contraction can also stimulate adjacent muscle cells to depolarize (fire) and contract. Thus, a depolarization wave can be sent across the heart, coordinating its rhythmic contractions and enabling it to perform its vital function of propelling blood through the circulatory system.  is a simplified graphic of a depolarization wave spreading across the heart from the sinoarterial (SA) node, the heart’s natural pacemaker.
An electrocardiogram (ECG) is a record of the voltages created by the wave of depolarization and subsequent repolarization in the heart. (They are also abbreviated EKG.) Voltages between pairs of electrodes placed on the chest are vector components of the voltage wave on the heart. Standard ECGs have 12 or more electrodes, but only three are shown in  for clarity. Decades ago, three-electrode ECGs were performed by placing electrodes on the left and right arms and the left leg. The voltage between the right arm and the left leg is called the lead II potential and is the most often graphed. We shall examine the lead II potential as an indicator of heart-muscle function and see that it is coordinated with arterial blood pressure as well.
Heart function and its four-chamber action are explored in Viscosity and Laminar Flow; Poiseuille’s Law. Basically, the right and left atria receive blood from the body and lungs, respectively, and pump the blood into the ventricles. The right and left ventricles, in turn, pump blood through the lungs and the rest of the body, respectively. Depolarization of the heart muscle causes it to contract. After contraction it is repolarized to ready it for the next beat. The ECG measures components of depolarization and repolarization of the heart muscle and can yield significant information on the functioning and malfunctioning of the heart.
 shows an ECG of the lead II potential and a graph of the corresponding arterial blood pressure. The major features are labeled P, Q, R, S, and T. The P wave is generated by the depolarization and contraction of the atria as they pump blood into the ventricles. The QRS complex is created by the depolarization of the ventricles as they pump blood to the lungs and body. Since the shape of the heart and the path of the depolarization wave are not simple, the QRS complex has this typical shape and time span. The lead II QRS signal also masks the repolarization of the atria, which occur at the same time. Finally, the T wave is generated by the repolarization of the ventricles and is followed by the next P wave in the next heartbeat. Arterial blood pressure varies with each part of the heartbeat, with systolic (maximum) pressure occurring closely after the QRS complex, which signals contraction of the ventricles.
Taken together, the 12 leads of a state-of-the-art ECG can yield a wealth of information about the heart. For example, regions of damaged heart tissue, called infarcts, reflect electrical waves and are apparent in one or more lead potentials. Subtle changes due to slight or gradual damage to the heart are most readily detected by comparing a recent ECG to an older one. This is particularly the case since individual heart shape, size, and orientation can cause variations in ECGs from one individual to another. ECG technology has advanced to the point where a portable ECG monitor can be incorporated into wearable devices and other small objects. See .

### Section Summary

1. Electric potentials in neurons and other cells are created by ionic concentration differences across semipermeable membranes.
2. Stimuli change the permeability and create action potentials that propagate along neurons.
3. Myelin sheaths speed this process and reduce the needed energy input.
4. This process in the heart can be measured with an electrocardiogram (ECG).

### Conceptual Questions

### Problems & Exercises


# Circuits and DC Instruments

## Introduction to Circuits and DC Instruments

Electric circuits are commonplace. Some are simple, such as those in flashlights. Others, such as those used in supercomputers, are extremely complex.
This collection of modules takes the topic of electric circuits a step beyond simple circuits. When the circuit is purely resistive, everything in this module applies to both DC and AC. Matters become more complex when capacitance is involved. We do consider what happens when capacitors are connected to DC voltage sources, but the interaction of capacitors and other nonresistive devices with AC is left for a later chapter. Finally, a number of important DC instruments, such as meters that measure voltage and current, are covered in this chapter.


# Circuits and DC Instruments

## Resistors in Series and Parallel

### Learning Objectives

By the end of this section, you will be able to:
1. Draw a circuit with resistors in parallel and in series.
2. Calculate the voltage drop of a current across a resistor using Ohm’s law.
3. Contrast the way total resistance is calculated for resistors in series and in parallel.
4. Explain why total resistance of a parallel circuit is less than the smallest resistance of any of the resistors in that circuit.
5. Calculate total resistance of a circuit that contains a mixture of resistors connected in series and in parallel.
Most circuits have more than one component, called a resistor that limits the flow of charge in the circuit. A measure of this limit on charge flow is called resistance. The simplest combinations of resistors are the series and parallel connections illustrated in . The total resistance of a combination of resistors depends on both their individual values and how they are connected.

### Resistors in Series

When are resistors in series? Resistors are in series whenever the flow of charge, called the current, must flow through devices sequentially. For example, if current flows through a person holding a screwdriver and into the Earth, then  in (a) could be the resistance of the screwdriver’s shaft,  the resistance of its handle,  the person’s body resistance, and  the resistance of her shoes.
 shows resistors in series connected to a voltage source. It seems reasonable that the total resistance is the sum of the individual resistances, considering that the current has to pass through each resistor in sequence. (This fact would be an advantage to a person wishing to avoid an electrical shock, who could reduce the current by wearing high-resistance rubber-soled shoes. It could be a disadvantage if one of the resistances were a faulty high-resistance cord to an appliance that would reduce the operating current.)
To verify that resistances in series do indeed add, let us consider the loss of electrical power, called a voltage drop, in each resistor in .
According to Ohm’s law, the voltage drop, , across a resistor when a current flows through it is calculated using the equation , where  equals the current in amps (A) and  is the resistance in ohms . Another way to think of this is that  is the voltage necessary to make a current  flow through a resistance .
So the voltage drop across  is , that across  is , and that across  is . The sum of these voltages equals the voltage output of the source; that is,
This equation is based on the conservation of energy and conservation of charge. Electrical potential energy can be described by the equation , where  is the electric charge and  is the voltage. Thus the energy supplied by the source is , while that dissipated by the resistors is
These energies must be equal, because there is no other source and no other destination for energy in the circuit. Thus, . The charge  cancels, yielding , as stated. (Note that the same amount of charge passes through the battery and each resistor in a given amount of time, since there is no capacitance to store charge, there is no place for charge to leak, and charge is conserved.)
Now substituting the values for the individual voltages gives
Note that for the equivalent single series resistance , we have
This implies that the total or equivalent series resistance  of three resistors is .
This logic is valid in general for any number of resistors in series; thus, the total resistance  of a series connection is
as proposed. Since all of the current must pass through each resistor, it experiences the resistance of each, and resistances in series simply add up.

### Resistors in Parallel

 shows resistors in parallel, wired to a voltage source. Resistors are in parallel when each resistor is connected directly to the voltage source by connecting wires having negligible resistance. Each resistor thus has the full voltage of the source applied to it.
Each resistor draws the same current it would if it alone were connected to the voltage source (provided the voltage source is not overloaded). For example, an automobile’s headlights, radio, and so on, are wired in parallel, so that they utilize the full voltage of the source and can operate completely independently. The same is true in your house, or any building. (See (b).)
To find an expression for the equivalent parallel resistance , let us consider the currents that flow and how they are related to resistance. Since each resistor in the circuit has the full voltage, the currents flowing through the individual resistors are , , and . Conservation of charge implies that the total current  produced by the source is the sum of these currents:
Substituting the expressions for the individual currents gives
Note that Ohm’s law for the equivalent single resistance gives
The terms inside the parentheses in the last two equations must be equal. Generalizing to any number of resistors, the total resistance  of a parallel connection is related to the individual resistances by
This relationship results in a total resistance  that is less than the smallest of the individual resistances. (This is seen in the next example.) When resistors are connected in parallel, more current flows from the source than would flow for any of them individually, and so the total resistance is lower.

### Combinations of Series and Parallel

More complex connections of resistors are sometimes just combinations of series and parallel. These are commonly encountered, especially when wire resistance is considered. In that case, wire resistance is in series with other resistances that are in parallel.
Combinations of series and parallel can be reduced to a single equivalent resistance using the technique illustrated in . Various parts are identified as either series or parallel, reduced to their equivalents, and further reduced until a single resistance is left. The process is more time consuming than difficult.
The simplest combination of series and parallel resistance, shown in , is also the most instructive, since it is found in many applications. For example,  could be the resistance of wires from a car battery to its electrical devices, which are in parallel.  and  could be the starter motor and a passenger compartment light. We have previously assumed that wire resistance is negligible, but, when it is not, it has important effects, as the next example indicates.

### Practical Implications

One implication of this last example is that resistance in wires reduces the current and power delivered to a resistor. If wire resistance is relatively large, as in a worn (or a very long) extension cord, then this loss can be significant. If a large current is drawn, the  drop in the wires can also be significant.
For example, when you are rummaging in the refrigerator and the motor comes on, the refrigerator light dims momentarily. Similarly, you can see the passenger compartment light dim when you start the engine of your car (although this may be due to resistance inside the battery itself).
What is happening in these high-current situations is illustrated in . The device represented by  has a very low resistance, and so when it is switched on, a large current flows. This increased current causes a larger  drop in the wires represented by , reducing the voltage across the light bulb (which is ), which then dims noticeably.

### Test Prep for AP Courses

### Section Summary

1. The total resistance of an electrical circuit with resistors wired in a series is the sum of the individual resistances:
2. Each resistor in a series circuit has the same amount of current flowing through it.
3. The voltage drop, or power dissipation, across each individual resistor in a series is different, and their combined total adds up to the power source input.
4. The total resistance of an electrical circuit with resistors wired in parallel is less than the lowest resistance of any of the components and can be determined using the formula:

5. Each resistor in a parallel circuit has the same full voltage of the source applied to it.
6. The current flowing through each resistor in a parallel circuit is different, depending on the resistance.
7. If a more complex connection of resistors is a combination of series and parallel, it can be reduced to a single equivalent resistance by identifying its various parts as series or parallel, reducing each to its equivalent, and continuing until a single resistance is eventually reached.

### Conceptual Questions

### Problem Exercises

Note: Data taken from figures can be assumed to be accurate to three significant digits.