# Electric Charge and Electric Field

## Conductors and Insulators

### Learning Objectives

By the end of this section, you will be able to:
1. Define conductor and insulator, explain the difference, and give examples of each.
2. Describe three methods for charging an object.
3. Explain what happens to an electric force as you move farther from the source.
4. Define polarization.
Some substances, such as metals and salty water, allow charges to move through them with relative ease. Some of the electrons in metals and similar conductors are not bound to individual atoms or sites in the material. These free electrons can move through the material much as air moves through loose sand. Any substance that has free electrons and allows charge to move relatively freely through it is called a conductor. The moving electrons may collide with fixed atoms and molecules, losing some energy, but they can move in a conductor. Superconductors allow the movement of charge without any loss of energy. Salty water and other similar conducting materials contain free ions that can move through them. An ion is an atom or molecule having a positive or negative (nonzero) total charge. In other words, the total number of electrons is not equal to the total number of protons.
Other substances, such as glass, do not allow charges to move through them. These are called insulators. Electrons and ions in insulators are bound in the structure and cannot move easily—as much as  times more slowly than in conductors. Pure water and dry table salt are insulators, for example, whereas molten salt and salty water are conductors.

### Charging by Contact

 shows an electroscope being charged by touching it with a positively charged glass rod. Because the glass rod is an insulator, it must actually touch the electroscope to transfer charge to or from it. (Note that the extra positive charges reside on the surface of the glass rod as a result of rubbing it with silk before starting the experiment.) Since only electrons move in metals, we see that they are attracted to the top of the electroscope. There, some are transferred to the positive rod by touch, leaving the electroscope with a net positive charge.
Electrostatic repulsion in the leaves of the charged electroscope separates them. The electrostatic force has a horizontal component that results in the leaves moving apart as well as a vertical component that is balanced by the gravitational force. Similarly, the electroscope can be negatively charged by contact with a negatively charged object.

### Charging by Induction

It is not necessary to transfer excess charge directly to an object in order to charge it.  shows a method of induction wherein a charge is created in a nearby object, without direct contact. Here we see two neutral metal spheres in contact with one another but insulated from the rest of the world. A positively charged rod is brought near one of them, attracting negative charge to that side, leaving the other sphere positively charged.
This is an example of induced polarization of neutral objects. Polarization is the separation of charges in an object that remains neutral. If the spheres are now separated (before the rod is pulled away), each sphere will have a net charge. Note that the object closest to the charged rod receives an opposite charge when charged by induction. Note also that no charge is removed from the charged rod, so that this process can be repeated without depleting the supply of excess charge.
Another method of charging by induction is shown in . The neutral metal sphere is polarized when a charged rod is brought near it. The sphere is then grounded, meaning that a conducting wire is run from the sphere to the ground. Since the earth is large and most ground is a good conductor, it can supply or accept excess charge easily. In this case, electrons are attracted to the sphere through a wire called the ground wire, because it supplies a conducting path to the ground. The ground connection is broken before the charged rod is removed, leaving the sphere with an excess charge opposite to that of the rod. Again, an opposite charge is achieved when charging by induction and the charged rod loses none of its excess charge.
Neutral objects can be attracted to any charged object. The pieces of straw attracted to polished amber are neutral, for example. If you run a plastic comb through your hair, the charged comb can pick up neutral pieces of paper.  shows how the polarization of atoms and molecules in neutral objects results in their attraction to a charged object.
When a charged rod is brought near a neutral substance, an insulator in this case, the distribution of charge in atoms and molecules is shifted slightly. Opposite charge is attracted nearer the external charged rod, while like charge is repelled. Since the electrostatic force decreases with distance, the repulsion of like charges is weaker than the attraction of unlike charges, and so there is a net attraction. Thus a positively charged glass rod attracts neutral pieces of paper, as will a negatively charged rubber rod. Some molecules, like water, are polar molecules. Polar molecules have a natural or inherent separation of charge, although they are neutral overall. Polar molecules are particularly affected by other charged objects and show greater polarization effects than molecules with naturally uniform charge distributions.

### Test Prep for AP Courses

### Section Summary

1. Polarization is the separation of positive and negative charges in a neutral object.
2. A conductor is a substance that allows charge to flow freely through its atomic structure.
3. An insulator holds charge within its atomic structure.
4. Objects with like charges repel each other, while those with unlike charges attract each other.
5. A conducting object is said to be grounded if it is connected to the Earth through a conductor. Grounding allows transfer of charge to and from the earth’s large reservoir.
6. Objects can be charged by contact with another charged object and obtain the same sign charge.
7. If an object is temporarily grounded, it can be charged by induction, and obtains the opposite sign charge.
8. Polarized objects have their positive and negative charges concentrated in different areas, giving them a non-symmetrical charge.
9. Polar molecules have an inherent separation of charge.

### Conceptual Questions

### Problems & Exercises


# Electric Charge and Electric Field

## Conductors and Electric Fields in Static Equilibrium

### Learning Objectives

By the end of this section, you will be able to:
1. List the three properties of a conductor in electrostatic equilibrium.
2. Explain the effect of an electric field on free charges in a conductor.
3. Explain why no electric field may exist inside a conductor.
4. Describe the electric field surrounding Earth.
5. Explain what happens to an electric field applied to an irregular conductor.
6. Describe how a lightning rod works.
7. Explain how a metal car may protect passengers inside from the dangerous electric fields caused by a downed line touching the car.
Conductors contain free charges that move easily. When excess charge is placed on a conductor or the conductor is put into a static electric field, charges in the conductor quickly respond to reach a steady state called electrostatic equilibrium.
 shows the effect of an electric field on free charges in a conductor. The free charges move until the field is perpendicular to the conductor’s surface. There can be no component of the field parallel to the surface in electrostatic equilibrium, since, if there were, it would produce further movement of charge. A positive free charge is shown, but free charges can be either positive or negative and are, in fact, negative in metals. The motion of a positive charge is equivalent to the motion of a negative charge in the opposite direction.
A conductor placed in an electric field will be polarized.  shows the result of placing a neutral conductor in an originally uniform electric field. The field becomes stronger near the conductor but entirely disappears inside it.
The properties of a conductor are consistent with the situations already discussed and can be used to analyze any conductor in electrostatic equilibrium. This can lead to some interesting new insights, such as described below.
How can a very uniform electric field be created? Consider a system of two metal plates with opposite charges on them, as shown in . The properties of conductors in electrostatic equilibrium indicate that the electric field between the plates will be uniform in strength and direction. Except near the edges, the excess charges distribute themselves uniformly, producing field lines that are uniformly spaced (hence uniform in strength) and perpendicular to the surfaces (hence uniform in direction, since the plates are flat). The edge effects are less important when the plates are close together.

### Earth’s Electric Field

A near uniform electric field of approximately 150 N/C, directed downward, surrounds Earth, with the magnitude increasing slightly as we get closer to the surface. What causes the electric field? At around 100 km above the surface of Earth we have a layer of charged particles, called the ionosphere. The ionosphere is responsible for a range of phenomena including the electric field surrounding Earth. In fair weather the ionosphere is positive and the Earth largely negative, maintaining the electric field ((a)).
In storm conditions clouds form and localized electric fields can be larger and reversed in direction ((b)). The exact charge distributions depend on the local conditions, and variations of (b) are possible.
If the electric field is sufficiently large, the insulating properties of the surrounding material break down and it becomes conducting. For air this occurs at around  N/C. Air ionizes ions and electrons recombine, and we get discharge in the form of lightning sparks and corona discharge.

### Electric Fields on Uneven Surfaces

So far we have considered excess charges on a smooth, symmetrical conductor surface. What happens if a conductor has sharp corners or is pointed? Excess charges on a nonuniform conductor become concentrated at the sharpest points. Additionally, excess charge may move on or off the conductor at the sharpest points.
To see how and why this happens, consider the charged conductor in . The electrostatic repulsion of like charges is most effective in moving them apart on the flattest surface, and so they become least concentrated there. This is because the forces between identical pairs of charges at either end of the conductor are identical, but the components of the forces parallel to the surfaces are different. The component parallel to the surface is greatest on the flattest surface and, hence, more effective in moving the charge.
The same effect is produced on a conductor by an externally applied electric field, as seen in  (c). Since the field lines must be perpendicular to the surface, more of them are concentrated on the most curved parts.

### Applications of Conductors

On a very sharply curved surface, such as shown in , the charges are so concentrated at the point that the resulting electric field can be great enough to remove them from the surface. This can be useful.
Lightning rods work best when they are most pointed. The large charges created in storm clouds induce an opposite charge on a building that can result in a lightning bolt hitting the building. The induced charge is bled away continually by a lightning rod, preventing the more dramatic lightning strike.
Of course, we sometimes wish to prevent the transfer of charge rather than to facilitate it. In that case, the conductor should be very smooth and have as large a radius of curvature as possible. (See .) Smooth surfaces are used on high-voltage transmission lines, for example, to avoid leakage of charge into the air.
Another device that makes use of some of these principles is a Faraday cage. This is a metal shield that encloses a volume. All electrical charges will reside on the outside surface of this shield, and there will be no electrical field inside. A Faraday cage is used to prohibit stray electrical fields in the environment from interfering with sensitive measurements, such as the electrical signals inside a nerve cell.
During electrical storms if you are driving a car, it is best to stay inside the car as its metal body acts as a Faraday cage with zero electrical field inside. If in the vicinity of a lightning strike, its effect is felt on the outside of the car and the inside is unaffected, provided you remain totally inside. This is also true if an active (“hot”) electrical wire was broken (in a storm or an accident) and fell on your car.

### Test Prep for AP Courses

### Section Summary

1. A conductor allows free charges to move about within it.
2. The electrical forces around a conductor will cause free charges to move around inside the conductor until static equilibrium is reached.
3. Any excess charge will collect along the surface of a conductor.
4. Conductors with sharp corners or points will collect more charge at those points.
5. A lightning rod is a conductor with sharply pointed ends that collect excess charge on the building caused by an electrical storm and allow it to dissipate back into the air.
6. Electrical storms result when the electrical field of Earth’s surface in certain locations becomes more strongly charged, due to changes in the insulating effect of the air.
7. A Faraday cage acts like a shield around an object, preventing electric charge from penetrating inside.

### Conceptual Questions

### Problems & Exercises


# Electric Charge and Electric Field

## Coulomb’s Law

### Learning Objectives

By the end of this section, you will be able to:
1. State Coulomb’s law in terms of how the electrostatic force changes with the distance between two objects.
2. Calculate the electrostatic force between two charged point forces, such as electrons or protons.
3. Compare the electrostatic force to the gravitational attraction for a proton and an electron; for a human and the Earth.
Through the work of scientists in the late 18th century, the main features of the electrostatic force—the existence of two types of charge, the observation that like charges repel, unlike charges attract, and the decrease of force with distance—were eventually refined, and expressed as a mathematical formula. The mathematical formula for the electrostatic force is called Coulomb’s law after the French physicist Charles Coulomb (1736–1806), who performed experiments and first proposed a formula to calculate it.
Although the formula for Coulomb’s law is simple, it was no mean task to prove it. The experiments Coulomb did, with the primitive equipment then available, were difficult. Modern experiments have verified Coulomb’s law to great precision. For example, it has been shown that the force is inversely proportional to distance between two objects squared  to an accuracy of 1 part in . No exceptions have ever been found, even at the small distances within the atom.

As the example implies, gravitational force is completely negligible on a small scale, where the interactions of individual charged particles are important. On a large scale, such as between the Earth and a person, the reverse is true. Most objects are nearly electrically neutral, and so attractive and repulsive Coulomb forces nearly cancel. Gravitational force on a large scale dominates interactions between large objects because it is always attractive, while Coulomb forces tend to cancel.

### Test Prep for AP Courses

### Section Summary

1. Frenchman Charles Coulomb was the first to publish the mathematical equation that describes the electrostatic force between two objects.
2. Coulomb’s law gives the magnitude of the force between point charges. It is
                where
3. This Coulomb force is extremely basic, since most charges are due to point-like particles. It is responsible for all electrostatic effects and underlies most macroscopic forces.
4. The Coulomb force is extraordinarily strong compared with the gravitational force, another basic force—but unlike gravitational force it can cancel, since it can be either attractive or repulsive.
5. The electrostatic force between two subatomic particles is far greater than the gravitational force between the same two particles.

### Conceptual Questions

### Problems & Exercises


# Electric Charge and Electric Field

## Electric Field: Concept of a Field Revisited

### Learning Objectives

By the end of this section, you will be able to:
1. Describe a force field and calculate the strength of an electric field due to a point charge.
2. Calculate the force exerted on a test charge by an electric field.
3. Explain the relationship between electrical force (F) on a test charge and electrical field strength (E).
Contact forces, such as between a baseball and a bat, are explained on the small scale by the interaction of the charges in atoms and molecules in close proximity. They interact through forces that include the Coulomb force. Action at a distance is a force between objects that are not close enough for their atoms to “touch.” That is, they are separated by more than a few atomic diameters.
For example, a charged rubber comb attracts neutral bits of paper from a distance via the Coulomb force. It is very useful to think of an object being surrounded in space by a force field. The force field carries the force to another object (called a test object) some distance away.

### Concept of a Field

A field is a way of conceptualizing and mapping the force that surrounds any object and acts on another object at a distance without apparent physical connection. For example, the gravitational field surrounding the earth (and all other masses) represents the gravitational force that would be experienced if another mass were placed at a given point within the field.
In the same way, the Coulomb force field surrounding any charge extends throughout space. Using Coulomb’s law, , its magnitude is given by the equation

, for a point charge (a particle having a charge ) acting on a test charge  at a distance  (see ). Both the magnitude and direction of the Coulomb force field depend on  and the test charge .
To simplify things, we would prefer to have a field that depends only on  and not on the test charge . The electric field is defined in such a manner that it represents only the charge creating it and is unique at every point in space. Specifically, the electric field  is defined to be the ratio of the Coulomb force to the test charge:

where  is the electrostatic force (or Coulomb force) exerted on a positive test charge
. It is understood that
 is in the same direction as
. It is also assumed that  is so small that it does not alter the charge distribution creating the electric field. The units of electric field are newtons per coulomb (N/C). If the electric field is known, then the electrostatic force on any charge  is simply obtained by multiplying charge times electric field, or

          . Consider the electric field due to a point charge . According to Coulomb’s law, the force it exerts on a test charge

 is

. Thus the magnitude of the electric field,
, for a point charge is
Since the test charge cancels, we see that
The electric field is thus seen to depend only on the charge  and the distance ; it is completely independent of the test charge .

### Test Prep for AP Courses

### Section Summary

1. The electrostatic force field surrounding a charged object extends out into space in all directions.
2. The electrostatic force exerted by a point charge on a test charge at a distance  depends on the charge of both charges, as well as the distance between the two.
3. The electric field  is defined to be
    where
4. The magnitude of the electric field  created by a point charge  is
    where

### Conceptual Questions

### Problem Exercises


# Electric Charge and Electric Field

## Electric Field Lines: Multiple Charges

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate the total force (magnitude and direction) exerted on a test charge from more than one charge
2. Describe an electric field diagram of a positive point charge; of a negative point charge with twice the magnitude of positive charge
3. Draw the electric field lines between two points of the same charge; between two points of opposite charge.
Drawings using lines to represent electric fields around charged objects are very useful in visualizing field strength and direction. Since the electric field has both magnitude and direction, it is a vector. Like all vectors, the electric field can be represented by an arrow that has length proportional to its magnitude and that points in the correct direction. (We have used arrows extensively to represent force vectors, for example.)
 shows two pictorial representations of the same electric field created by a positive point charge .  (b) shows the standard representation using continuous lines.  (a) shows numerous individual arrows with each arrow representing the force on a test charge . Field lines are essentially a map of infinitesimal force vectors.
Note that the electric field is defined for a positive test charge , so that the field lines point away from a positive charge and toward a negative charge. (See .) The electric field strength is exactly proportional to the number of field lines per unit area, since the magnitude of the electric field for a point charge is  and area is proportional to . This pictorial representation, in which field lines represent the direction and their closeness (that is, their areal density or the number of lines crossing a unit area) represents strength, is used for all fields: electrostatic, gravitational, magnetic, and others.
In many situations, there are multiple charges. The total electric field created by multiple charges is the vector sum of the individual fields created by each charge. The following example shows how to add electric field vectors.
 shows how the electric field from two point charges can be drawn by finding the total field at representative points and drawing electric field lines consistent with those points. While the electric fields from multiple charges are more complex than those of single charges, some simple features are easily noticed.
For example, the field is weaker between like charges, as shown by the lines being farther apart in that region. (This is because the fields from each charge exert opposing forces on any charge placed between them.) (See  and (a).) Furthermore, at a great distance from two like charges, the field becomes identical to the field from a single, larger charge.
(b) shows the electric field of two unlike charges. The field is stronger between the charges. In that region, the fields from each charge are in the same direction, and so their strengths add. The field of two unlike charges is weak at large distances, because the fields of the individual charges are in opposite directions and so their strengths subtract. At very large distances, the field of two unlike charges looks like that of a smaller single charge.
We use electric field lines to visualize and analyze electric fields (the lines are a pictorial tool, not a physical entity in themselves). The properties of electric field lines for any charge distribution can be summarized as follows:
1. Field lines must begin on positive charges and terminate on negative charges, or at infinity in the hypothetical case of isolated charges.
2. The number of field lines leaving a positive charge or entering a negative charge is proportional to the magnitude of the charge.
3. The strength of the field is proportional to the closeness of the field lines—more precisely, it is proportional to the number of lines per unit area perpendicular to the lines.
4. The direction of the electric field is tangent to the field line at any point in space.
5. Field lines can never cross.
The last property means that the field is unique at any point. The field line represents the direction of the field; so if they crossed, the field would have two directions at that location (an impossibility if the field is unique).

### Test Prep for AP Courses

### Section Summary

1. Drawings of electric field lines are useful visual tools. The properties of electric field lines for any charge distribution are that:
2. Field lines must begin on positive charges and terminate on negative charges, or at infinity in the hypothetical case of isolated charges.
3. The number of field lines leaving a positive charge or entering a negative charge is proportional to the magnitude of the charge.
4. The strength of the field is proportional to the closeness of the field lines—more precisely, it is proportional to the number of lines per unit area perpendicular to the lines.
5. The direction of the electric field is tangent to the field line at any point in space.
6. Field lines can never cross.

### Conceptual Questions

### Problem Exercises


# Electric Charge and Electric Field

## Electric Forces in Biology

### Learning Objectives

By the end of this section, you will be able to:
1. Describe how a water molecule is polar.
2. Explain electrostatic screening by a water molecule within a living cell.
Classical electrostatics has an important role to play in modern molecular biology. Large molecules such as proteins, nucleic acids, and so on—so important to life—are usually electrically charged. DNA itself is highly charged; it is the electrostatic force that not only holds the molecule together but gives the molecule structure and strength.  is a schematic of the DNA double helix.
The four nucleotide bases are given the symbols A (adenine), C (cytosine), G (guanine), and T (thymine). The order of the four bases varies in each strand, but the pairing between bases is always the same. C and G are always paired and A and T are always paired, which helps to preserve the order of bases in cell division (mitosis) so as to pass on the correct genetic information. Since the Coulomb force drops with distance (), the distances between the base pairs must be small enough that the electrostatic force is sufficient to hold them together.
DNA is a highly charged molecule, with about  (fundamental charge) per  m. The distance separating the two strands that make up the DNA structure is about 1 nm, while the distance separating the individual atoms within each base is about 0.3 nm.
One might wonder why electrostatic forces do not play a larger role in biology than they do if we have so many charged molecules. The reason is that the electrostatic force is “diluted” due to screening between molecules. This is due to the presence of other charges in the cell.

### Polarity of Water Molecules

The best example of this charge screening is the water molecule, represented as . Water is a strongly polar molecule. Its 10 electrons (8 from the oxygen atom and 2 from the two hydrogen atoms) tend to remain closer to the oxygen nucleus than the hydrogen nuclei. This creates two centers of equal and opposite charges—what is called a dipole, as illustrated in . The magnitude of the dipole is called the dipole moment.
These two centers of charge will terminate some of the electric field lines coming from a free charge, as on a DNA molecule. This results in a reduction in the strength of the Coulomb interaction. One might say that screening makes the Coulomb force a short range force rather than long range.

### Cell Membranes

Other ions of importance in biology that can reduce or screen Coulomb interactions are  and  and . These ions are located both inside and outside of living cells. The movement of these ions through cell membranes is crucial to the motion of nerve impulses through nerve axons.
Recent studies of electrostatics in biology seem to show that electric fields in cells can be extended over larger distances, in spite of screening, by “microtubules” within the cell. These microtubules are hollow tubes composed of proteins that guide the movement of chromosomes when cells divide, the motion of other organisms within the cell, and provide mechanisms for motion of some cells (as motors).
You are likely familiar with the role of electrical signals in nerve conduction and the importance of charges in cardiac and related activity. Changes in electrical properties are also essential in core biological processes. Ernest Everett Just, whose expertise in understanding and handling egg cells led to a number of critical experimental discoveries, investigated the role of the cell membrane in reproductive fertilization. In one key experiment, Just established that the egg membrane undergoes a depolarizing "wave of negativity" the moment it fuses with a sperm cell. This change in charge is now known as the "fast block" that ensures that only one sperm cell fuses with an egg cell and is critical for embryonic development.

### Bioelectricity and Wound Healing

Just as electrical forces drive activities in healthy cells and systems, they are also critical in damaged ones. Scientists have long known that injuries or infections are managed by the body through various responses, including increased white blood cell concentrations, swelling, and tissue repair. For example, human cells damaged by wounds heal through a complex process. But what triggers it?
Physicists and biologists working together at Vanderbilt University used an ultra-precise laser to uncover the processes organisms use to repair damage. Lead researchers Andrea Page-Degraw and Shane Hutson and study author Erica Shannon discovered that immediately upon damage, cells release calcium ions and eventually other molecules, driving an electrochemical response that initiates the healing process. Shannon notes that different types of damage lead to different chemical releases, demonstrating how organisms may initiate specific responses to best address the injury.
While far more research is required to understand the triggering and response method, other research indicates that bioelectricity is highly involved in wound healing. Several studies have indicated that precise and low-level electrical stimulation of wounds (such as those from surgeries) leads to faster healing. While the mechanisms are not fully understood, electrical stimulation is a growing area of research and practice in medicine.

### Section Summary

1. Many molecules in living organisms, such as DNA, carry a charge.
2. An uneven distribution of the positive and negative charges within a polar molecule produces a dipole.
3. The effect of a Coulomb field generated by a charged object may be reduced or blocked by other nearby charged objects.
4. Biological systems contain water, and because water molecules are polar, they have a strong effect on other molecules in living systems.

### Conceptual Question


# Electric Charge and Electric Field

## Applications of Electrostatics

### Learning Objectives

By the end of this section, you will be able to:
The study of electrostatics has proven useful in many areas. This module covers just a few of the many applications of electrostatics.
1. Name several real-world applications of the study of electrostatics.

### The Van de Graaff Generator

Van de Graaff generators (or Van de Graaffs) are not only spectacular devices used to demonstrate high voltage due to static electricity—they are also used for serious research. The first was built by Robert Van de Graaff in 1931 (based on original suggestions by Lord Kelvin) for use in nuclear physics research.  shows a schematic of a large research version. Van de Graaffs utilize both smooth and pointed surfaces, and conductors and insulators to generate large static charges and, hence, large voltages.
A very large excess charge can be deposited on the sphere, because it moves quickly to the outer surface. Practical limits arise because the large electric fields polarize and eventually ionize surrounding materials, creating free charges that neutralize excess charge or allow it to escape. Nevertheless, voltages of 15 million volts are well within practical limits.

### Xerography

Most copy machines use an electrostatic process called xerography—a word coined from the Greek words xeros for dry and graphos for writing. The heart of the process is shown in simplified form in .
A selenium-coated aluminum drum is sprayed with positive charge from points on a device called a corotron. Selenium is a substance with an interesting property—it is a photoconductor. That is, selenium is an insulator when in the dark and a conductor when exposed to light.
In the first stage of the xerography process, the conducting aluminum drum is grounded so that a negative charge is induced under the thin layer of uniformly positively charged selenium. In the second stage, the surface of the drum is exposed to the image of whatever is to be copied. Where the image is light, the selenium becomes conducting, and the positive charge is neutralized. In dark areas, the positive charge remains, and so the image has been transferred to the drum.
The third stage takes a dry black powder, called toner, and sprays it with a negative charge so that it will be attracted to the positive regions of the drum. Next, a blank piece of paper is given a greater positive charge than on the drum so that it will pull the toner from the drum. Finally, the paper and electrostatically held toner are passed through heated pressure rollers, which melt and permanently adhere the toner within the fibers of the paper.

### Laser Printers

Laser printers use the xerographic process to make high-quality images on paper, employing a laser to produce an image on the photoconducting drum as shown in . In its most common application, the laser printer receives output from a computer, and it can achieve high-quality output because of the precision with which laser light can be controlled. Many laser printers do significant information processing, such as making sophisticated letters or fonts, and may contain a computer more powerful than the one giving them the raw data to be printed.

### Ink Jet Printers and Electrostatic Painting

The ink jet printer, commonly used to print computer-generated text and graphics, also employs electrostatics. A nozzle makes a fine spray of tiny ink droplets, which are then given an electrostatic charge. (See .)
Once charged, the droplets can be directed, using pairs of charged plates, with great precision to form letters and images on paper. Ink jet printers can produce color images by using a black jet and three other jets with primary colors, usually cyan, magenta, and yellow, much as a color television produces color. (This is more difficult with xerography, requiring multiple drums and toners.)
Electrostatic painting employs electrostatic charge to spray paint onto odd-shaped surfaces. Mutual repulsion of like charges causes the paint to fly away from its source. Surface tension forms drops, which are then attracted by unlike charges to the surface to be painted. Electrostatic painting can reach those hard-to-get at places, applying an even coat in a controlled manner. If the object is a conductor, the electric field is perpendicular to the surface, tending to bring the drops in perpendicularly. Corners and points on conductors will receive extra paint. Felt can similarly be applied.

### Smoke Precipitators and Electrostatic Air Cleaning

Another important application of electrostatics is found in air cleaners, both large and small. The electrostatic part of the process places excess (usually positive) charge on smoke, dust, pollen, and other particles in the air and then passes the air through an oppositely charged grid that attracts and retains the charged particles. (See .)
Large electrostatic precipitators are used industrially to remove over 99% of the particles from stack gas emissions associated with the burning of coal and oil. Home precipitators, often in conjunction with the home heating and air conditioning system, are very effective in removing polluting particles, irritants, and allergens.

### Integrated Concepts

The Integrated Concepts exercises for this module involve concepts such as electric charges, electric fields, and several other topics. Physics is most interesting when applied to general situations involving more than a narrow set of physical principles. The electric field exerts force on charges, for example, and hence the relevance of Dynamics: Force and Newton’s Laws of Motion. The following topics are involved in some or all of the problems labeled “Integrated Concepts”:

The following worked example illustrates how this strategy is applied to an Integrated Concept problem:

### Section Summary

1. Electrostatics is the study of electric fields in static equilibrium.
2. In addition to research using equipment such as a Van de Graaff generator, many practical applications of electrostatics exist, including photocopiers, laser printers, ink-jet printers and electrostatic air filters.

### Problems & Exercises


# Electric Potential and Electric Field

## Connection for AP® Courses

In Electric Charge and Electric Field, we just scratched the surface (or at least rubbed it) of electrical phenomena. Two of the most familiar aspects of electricity are its energy and voltage. We know, for example, that great amounts of electrical energy can be stored in batteries, are transmitted cross-country through power lines, and may jump from clouds to explode the sap of trees. In a similar manner, at molecular levels, ions cross cell membranes and transfer information. We also know about voltages associated with electricity. Batteries are typically a few volts, the outlets in your home produce 120 volts, and power lines can be as high as hundreds of thousands of volts. But energy and voltage are not the same thing. A motorcycle battery, for example, is small and would not be very successful in replacing the much larger battery in a car, yet each has the same voltage. In this chapter, we shall examine the relationship between voltage and electrical energy and begin to explore some of the many applications of electricity. We do so by introducing the concept of electric potential and describing the relationship between electric field and electric potential.
This chapter presents the concept of equipotential lines (lines of equal potential) as a way to visualize the electric field (Enduring Understanding 2.E, Essential Knowledge 2.E.2). An analogy between the isolines on topographic maps for gravitational field and equipotential lines for the electric field is used to develop a conceptual understanding of equipotential lines (Essential Knowledge 2.E.1). The relationship between the magnitude of an electric field, change in electric potential, and displacement is stated for a uniform field and extended for the more general case using the concept of the “average value” of the electric field (Essential Knowledge 2.E.3).
The concept that an electric field is caused by charged objects (Enduring Understanding 2.C) supports Big Idea 2, that fields exist in space and can be used to explain interactions. The relationship between the electric field, electric charge, and electric force (Essential Knowledge 2.C.1) is used to describe the behavior of charged particles. The uniformity of the electric field between two oppositely charged parallel plates with uniformly distributed electric charge (Essential Knowledge 2.C.5), as well as the properties of materials and their geometry, are used to develop understanding of the capacitance of a capacitor (Essential Knowledge 4.E.4).
This chapter also supports Big Idea 4, that interactions between systems result in changes in those systems. This idea is applied to electric properties of various systems of charged objects, demonstrating the effect of electric interactions on electric properties of systems (Enduring Understanding 4.E). This fact in turn supports Big Idea 5, that changes due to interactions are governed by conservation laws. In particular, the energy of a system is conserved (Enduring Understanding 5.B). Any system that has internal structure can have internal energy. For a system of charged objects, internal energy can change as a result of changes in the arrangement of charges and their geometric configuration as long as work is done on, or by, the system (Essential Knowledge 5.B.2). When objects within the system interact with conservative forces, such as electric forces, the internal energy is defined by the potential energy of that interaction (Essential Knowledge 5.B.3). In general, the internal energy of a system is the sum of the kinetic energies of all its objects and the potential energy of interaction between the objects within the system (Essential Knowledge 5.B.4).
The concepts in this chapter support:
Big Idea 1 Objects and systems have properties such as mass and charge. Systems may have internal structure.
Enduring Understanding 1.E Materials have many macroscopic properties that result from the arrangement and interactions of the atoms and molecules that make up the material.
Essential Knowledge 1.E.4 Matter has a property called electric permittivity.
Big Idea 2 Fields existing in space can be used to explain interactions.
Enduring Understanding 2.C An electric field is caused by an object with electric charge.
Essential Knowledge 2.C.1 The magnitude of the electric force F exerted on an object with electric charge q by an electric field F= qE. The direction of the force is determined by the direction of the field and the sign of the charge, with positively charged objects accelerating in the direction of the field and negatively charged objects accelerating in the direction opposite the field. This should include a vector field map for positive point charges, negative point charges, spherically symmetric charge distribution, and uniformly charged parallel plates.
Essential Knowledge 2.C.5 Between two oppositely charged parallel plates with uniformly distributed electric charge, at points far from the edges of the plates, the electric field is perpendicular to the plates and is constant in both magnitude and direction.
Enduring Understanding 2.E Physicists often construct a map of isolines connecting points of equal value for some quantity related to a field and use these maps to help visualize the field.
Essential Knowledge 2.E.1 Isolines on a topographic (elevation) map describe lines of approximately equal gravitational potential energy per unit mass (gravitational equipotential). As the distance between two different isolines decreases, the steepness of the surface increases. [Contour lines on topographic maps are useful teaching tools for introducing the concept of equipotential lines. Students are encouraged to use the analogy in their answers when explaining gravitational and electrical potential and potential differences.]
Essential Knowledge 2.E.2 Isolines in a region where an electric field exists represent lines of equal electric potential, referred to as equipotential lines.
Essential Knowledge 2.E.3 The average value of the electric field in a region equals the change in electric potential across that region divided by the change in position (displacement) in the relevant direction.
Big Idea 4 Interactions between systems can result in changes in those systems.
Enduring Understanding 4.E The electric and magnetic properties of a system can change in response to the presence of, or changes in, other objects or systems.
Essential Knowledge 4.E.4 The resistance of a resistor, and the capacitance of a capacitor, can be understood from the basic properties of electric fields and forces, as well as the properties of materials and their geometry.
Big Idea 5 Changes that occur as a result of interactions are constrained by conservation laws.
Enduring Understanding 5.B The energy of a system is conserved.
Essential Knowledge 5.B.2 A system with internal structure can have internal energy, and changes in a system’s internal structure can result in changes in internal energy. [Physics 1: includes mass–spring oscillators and simple pendulums. Physics 2: charged object in electric ﬁelds and examining changes in internal energy with changes in conﬁguration.]
Essential Knowledge 5.B.3 A system with internal structure can have potential energy. Potential energy exists within a system if the objects within that system interact with conservative forces.
Essential Knowledge 5.B.4 The internal energy of a system includes the kinetic energy of the objects that make up the system and the potential energy of the conﬁguration of the objects that make up the system.


# Electric Potential and Electric Field

## Electric Potential Energy: Potential Difference

### Learning Objectives

By the end of this section, you will be able to:
1. Define electric potential and electric potential energy.
2. Describe the relationship between potential difference and electrical potential energy.
3. Explain electron volt and its usage in submicroscopic process.
4. Determine electric potential energy given potential difference and amount of charge.
When a free positive charge  is accelerated by an electric field, such as shown in , it is given kinetic energy. The process is analogous to an object being accelerated by a gravitational field. It is as if the charge is going down an electrical hill where its electric potential energy is converted to kinetic energy. Let us explore the work done on a charge  by the electric field in this process, so that we may develop a definition of electric potential energy.
The electrostatic or Coulomb force is conservative, which means that the work done on  is independent of the path taken. This is exactly analogous to the gravitational force in the absence of dissipative forces such as friction. When a force is conservative, it is possible to define a potential energy associated with the force, and it is usually easier to deal with the potential energy (because it depends only on position) than to calculate the work directly.
We use the letters PE to denote electric potential energy, which has units of joules (J). The change in potential energy, , is crucial, since the work done by a conservative force is the negative of the change in potential energy; that is, . For example, work

   done to accelerate a positive charge from rest is positive and results from a loss in PE, or a negative . There must be a minus sign in front of  to make

   positive. PE can be found at any point by taking one point as a reference and calculating the work needed to move a charge to the other point.
Gravitational potential energy and electric potential energy are quite analogous. Potential energy accounts for work done by a conservative force and gives added insight regarding energy and energy transformation without the necessity of dealing with the force directly. It is much more common, for example, to use the concept of voltage (related to electric potential energy) than to deal with the Coulomb force directly.
Calculating the work directly is generally difficult, since  and the direction and magnitude of  can be complex for multiple charges, for odd-shaped objects, and along arbitrary paths. But we do know that, since , the work, and hence , is proportional to the test charge  To have a physical quantity that is independent of test charge, we define electric potential  (or simply potential, since electric is understood) to be the potential energy per unit charge:
Since PE is proportional to  , the dependence on  cancels. Thus  does not depend on . The change in potential energy  is crucial, and so we are concerned with the difference in potential or potential difference  between two points, where
The potential difference between points A and B, , is thus defined to be the change in potential energy of a charge  moved from A to B, divided by the charge. Units of potential difference are joules per coulomb, given the name volt (V) after Alessandro Volta.
The familiar term voltage is the common name for potential difference. Keep in mind that whenever a voltage is quoted, it is understood to be the potential difference between two points. For example, every battery has two terminals, and its voltage is the potential difference between them. More fundamentally, the point you choose to be zero volts is arbitrary. This is analogous to the fact that gravitational potential energy has an arbitrary zero, such as sea level or perhaps a lecture hall floor.
In summary, the relationship between potential difference (or voltage) and electrical potential energy is given by
Voltage is not the same as energy. Voltage is the energy per unit charge. Thus a motorcycle battery and a car battery can both have the same voltage (more precisely, the same potential difference between battery terminals), yet one stores much more energy than the other since . The car battery can move more charge than the motorcycle battery, although both are 12 V batteries.
Note that the energies calculated in the previous example are absolute values. The change in potential energy for the battery is negative, since it loses energy. These batteries, like many electrical systems, actually move negative charge—electrons in particular. The batteries repel electrons from their negative terminals (A) through whatever circuitry is involved and attract them to their positive terminals (B) as shown in . The change in potential is  and the charge  is negative, so that  is negative, meaning the potential energy of the battery has decreased when  has moved from A to B.

### The Electron Volt

The energy per electron is very small in macroscopic situations like that in the previous example—a tiny fraction of a joule. But on a submicroscopic scale, such energy per particle (electron, proton, or ion) can be of great importance. For example, even a tiny fraction of a joule can be great enough for these particles to destroy organic molecules and harm living tissue. The particle may do its damage by direct collision, or it may create harmful x rays, which can also inflict damage. It is useful to have an energy unit related to submicroscopic effects.  shows a situation related to the definition of such an energy unit. An electron is accelerated between two charged metal plates as it might be in an old-model television tube or oscilloscope. The electron is given kinetic energy that is later converted to another form—light in the television tube, for example. (Note that downhill for the electron is uphill for a positive charge.) Since energy is related to voltage by  we can think of the joule as a coulomb-volt.
On the submicroscopic scale, it is more convenient to define an energy unit called the electron volt (eV), which is the energy given to a fundamental charge accelerated through a potential difference of 1 V.  In equation form,
An electron accelerated through a potential difference of 1 V is given an energy of 1 eV. It follows that an electron accelerated through 50 V is given 50 eV. A potential difference of 100,000 V (100 kV) will give an electron an energy of 100,000 eV (100 keV), and so on. Similarly, an ion with a double positive charge accelerated through 100 V will be given 200 eV of energy. These simple relationships between accelerating voltage and particle charges make the electron volt a simple and convenient energy unit in such circumstances.
The electron volt is commonly employed in submicroscopic processes—chemical valence energies and molecular and nuclear binding energies are among the quantities often expressed in electron volts. For example, about 5 eV of energy is required to break up certain organic molecules. If a proton is accelerated from rest through a potential difference of 30 kV, it is given an energy of 30 keV (30,000 eV) and it can break up as many as 6000 of these molecules (). Nuclear decay energies are on the order of 1 MeV (1,000,000 eV) per event and can, thus, produce significant biological damage.

### Conservation of Energy

The total energy of a system is conserved if there is no net addition (or subtraction) of work or heat transfer. For conservative forces, such as the electrostatic force, conservation of energy states that mechanical energy is a constant.
Mechanical energy is the sum of the kinetic energy and potential energy of a system; that is, . A loss of PE of a charged particle becomes an increase in its KE. Here PE is the electric potential energy. Conservation of energy is stated in equation form as
or
where i and f stand for initial and final conditions. As we have found many times before, considering energy can give us insights and facilitate problem solving.

### Test Prep for AP Courses

### Section Summary

1. Electric potential is potential energy per unit charge.
2. The potential difference between points A and B, , defined to be the change in potential energy of a charge  moved from A to B, is equal to the change in potential energy divided by the charge, Potential difference is commonly called voltage, represented by the symbol
.
3. An electron volt is the energy given to a fundamental charge accelerated through a potential difference of 1 V. In equation form,

4. Mechanical energy is the sum of the kinetic energy and potential energy of a system, that is,  This sum is a constant.

### Conceptual Questions

### Problems & Exercises


# Electric Potential and Electric Field

## Electric Potential in a Uniform Electric Field

### Learning Objectives

By the end of this section, you will be able to:
1. Describe the relationship between voltage and electric field.
2. Derive an expression for the electric potential and electric field.
3. Calculate electric field strength given distance and voltage.
In the previous section, we explored the relationship between voltage and energy. In this section, we will explore the relationship between voltage and electric field. For example, a uniform electric field  is produced by placing a potential difference (or voltage)  across two parallel metal plates, labeled A and B. (See .) Examining this will tell us what voltage is needed to produce a certain electric field strength; it will also reveal a more fundamental relationship between electric potential and electric field. From a physicist’s point of view, either  or  can be used to describe any charge distribution.  is most closely tied to energy, whereas  is most closely related to force.  is a scalar quantity and has no direction, while  is a vector quantity, having both magnitude and direction. (Note that the magnitude of the electric field strength, a scalar quantity, is represented by  below.) The relationship between  and  is revealed by calculating the work done by the force in moving a charge from point A to point B. But, as noted in Electric Potential Energy: Potential Difference, this is complex for arbitrary charge distributions, requiring calculus. We therefore look at a uniform electric field as an interesting special case.
The work done by the electric field in  to move a positive charge  from A, the positive plate, higher potential, to B, the negative plate, lower potential, is
 The potential difference between points A and B is
Entering this into the expression for work yields
Work is ; here , since the path is parallel to the field, and so . Since
, we see that . Substituting this expression for work into the previous equation gives
The charge cancels, and so the voltage between points A and B is seen to be
where  is the distance from A to B, or the distance between the plates in . Note that the above equation implies the units for electric field are volts per meter. We already know the units for electric field are newtons per coulomb; thus the following relation among units is valid:
In more general situations, regardless of whether the electric field is uniform, it points in the direction of decreasing potential, because the force on a positive charge is in the direction of  and also in the direction of lower potential . Furthermore, the magnitude of  equals the rate of decrease of  with distance. The faster  decreases over distance, the greater the electric field. In equation form, the general relationship between voltage and electric field is
where  is the distance over which the change in potential, , takes place. The minus sign tells us that  points in the direction of decreasing potential. The electric field is said to be the gradient (as in grade or slope) of the electric potential.
For continually changing potentials,  and  become infinitesimals and differential calculus must be employed to determine the electric field.

### Test Prep for AP Courses

### Section Summary

1. The voltage between points A and B is

where  is the distance from A to B, or the distance between the plates.
2. In equation form, the general relationship between voltage and electric field is

where  is the distance over which the change in potential, , takes place. The minus sign tells us that  points in the direction of decreasing potential.) The electric field is said to be the gradient (as in grade or slope) of the electric potential.

### Conceptual Questions

### Problems & Exercises


# Electric Potential and Electric Field

## Electrical Potential Due to a Point Charge

### Learning Objectives

By the end of this section, you will be able to:
1. Explain point charges and express the equation for electric potential of a point charge.
2. Distinguish between electric potential and electric field.
3. Determine the electric potential of a point charge given charge and distance.
Point charges, such as electrons, are among the fundamental building blocks of matter. Furthermore, spherical charge distributions (like on a metal sphere) create external electric fields exactly like a point charge. The electric potential due to a point charge is, thus, a case we need to consider. Using calculus to find the work needed to move a test charge  from a large distance away to a distance of  from a point charge , and noting the connection between work and potential , it can be shown that the electric potential  is
where k is a constant equal to

  .
The potential at infinity is chosen to be zero. Thus  for a point charge decreases with distance, whereas  for a point charge decreases with distance squared:
Recall that the electric potential  is a scalar and has no direction, whereas the electric field  is a vector. To find the voltage due to a combination of point charges, you add the individual voltages as numbers. To find the total electric field, you must add the individual fields as vectors, taking magnitude and direction into account. This is consistent with the fact that  is closely associated with energy, a scalar, whereas  is closely associated with force, a vector.
The voltages in both of these examples could be measured with a meter that compares the measured potential with ground potential. Ground potential is often taken to be zero (instead of taking the potential at infinity to be zero). It is the potential difference between two points that is of importance, and very often there is a tacit assumption that some reference point, such as Earth or a very distant point, is at zero potential. As noted in Electric Potential Energy: Potential Difference, this is analogous to taking sea level as  when considering gravitational potential energy, .

### Section Summary

1.  Electric potential of a point charge is .
2. Electric potential is a scalar, and electric field is a vector. Addition of voltages as numbers gives the voltage due to a combination of point charges, whereas addition of individual fields as vectors gives the total electric field.

### Conceptual Questions

### Problems & Exercises


# 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

## Connection for AP® Courses

In our daily lives, we see and experience many examples of electricity which involve electric current, the movement of charge. These include 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, and a hydroelectric plant sending energy to metropolitan and rural users.
Humankind has indeed harnessed electricity, the basis of technology, to improve the quality of life. While 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 electric current. In addition to exploring applications of electricity, we shall gain new insights into its nature – in particular, the fact that all magnetism results from electric current.
This chapter supports learning objectives covered under Big Ideas 1, 4, and 5 of the AP Physics Curriculum Framework. Electric charge is a property of a system (Big Idea 1) that affects its interaction with other charged systems (Enduring Understanding 1.B), whereas electric current is fundamentally the movement of charge through a conductor and is based on the fact that electric charge is conserved within a system (Essential Knowledge 1.B.1). The conservation of charge also leads to the concept of an electric circuit as a closed loop of electrical current. In addition, this chapter discusses examples showing that the current in a circuit is resisted by the elements of the circuit and the strength of the resistance depends on the material of the elements. The macroscopic properties of materials, including resistivity, depend on their molecular and atomic structure (Enduring Understanding 1.E). In addition, resistivity depends on the temperature of the material (Essential Knowledge 1.E.2).
The chapter also describes how the interaction of systems of objects can result in changes in those systems (Big Idea 4). For example, electric properties of a system of charged objects can change in response to the presence of, or changes in, other charged objects or systems (Enduring Understanding 4.E). A simple circuit with a resistor and an energy source is an example of such a system. The current through the resistor in the circuit is equal to the difference of potentials across the resistor divided by its resistance (Essential Knowledge 4.E.4).
The unifying theme of the physics curriculum is that any changes in the systems due to interactions are governed by laws of conservation (Big Idea 5). This chapter applies the idea of energy conservation (Enduring Understanding 5.B) to electric circuits and connects concepts of electric energy and electric power as rates of energy use (Essential Knowledge 5.B.5). While the laws of conservation of energy in electric circuits are fully described by Kirchoff's rules, which are introduced in the next chapter (Essential Knowledge 5.B.9), the specific definition of power (based on Essential Knowledge 5.B.9) is that it is the rate at which energy is transferred from a resistor as the product of the electric potential difference across the resistor and the current through the resistor.
Big Idea 1 Objects and systems have properties such as mass and charge. Systems may have internal structure.
Enduring Understanding 1.B Electric charge is a property of an object or system that affects its interactions with other objects or systems containing charge.
Essential Knowledge 1.B.1 Electric charge is conserved. The net charge of a system is equal to the sum of the charges of all the objects in the system.
Enduring Understanding 1.E Materials have many macroscopic properties that result from the arrangement and interactions of the atoms and molecules that make up the material.
Essential Knowledge 1.E.2 Matter has a property called resistivity.
Big Idea 4 Interactions between systems can result in changes in those systems.
Enduring Understanding 4.E The electric and magnetic properties of a system can change in response to the presence of, or changes in, other objects or systems.
Essential Knowledge 4.E.4 The resistance of a resistor, and the capacitance of a capacitor, can be understood from the basic properties of electric fields and forces, as well as the properties of materials and their geometry.
Big Idea 5: Changes that occur as a result of interactions are constrained by conservation laws.
Enduring Understanding 5.B The energy of a system is conserved.
Essential Knowledge 5.B.5 Energy can be transferred by an external force exerted on an object or system that moves the object or system through a distance; this energy transfer is called work. Energy transfer in mechanical or electrical systems may occur at different rates. Power is deﬁned as the rate of energy transfer into, out of, or within a system. [A piston ﬁlled with gas getting compressed or expanded is treated in Physics 2 as a part of thermodynamics.]
Essential Knowledge 5.B.9 Kirchhoff's loop rule describes conservation of energy in electrical circuits. [The application of Kirchhoff's laws to circuits is introduced in Physics 1 and further developed in Physics 2 in the context of more complex circuits, including those with capacitors.]


# 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