# Diffraction and Interference

## Understanding Diffraction and Interference

### Section Key Terms

### Diffraction and Interference

We know that visible light is the type of electromagnetic wave to which our eyes responds. As we have seen previously, light obeys the equation
where
   m/s is the speed of light in vacuum, f is the frequency of the electromagnetic wave in Hz (or s–1), and
   is its wavelength in m. The range of visible wavelengths is approximately 380 to 750 nm. As is true for all waves, light travels in straight lines and acts like a ray when it interacts with objects several times as large as its wavelength. However, when it interacts with smaller objects, it displays its wave characteristics prominently. Interference is the identifying behavior of a wave.
In , both the ray and wave characteristics of light can be seen. The laser beam emitted by the observatory represents ray behavior, as it travels in a straight line. Passing a pure, one-wavelength beam through vertical slits with a width close to the wavelength of the beam reveals the wave character of light. Here we see the beam spreading out horizontally into a pattern of bright and dark regions that are caused by systematic constructive and destructive interference. As it is characteristic of wave behavior, interference is observed for water waves, sound waves, and light waves.
That interference is a characteristic of energy propagation by waves is demonstrated more convincingly by water waves.  shows water waves passing through gaps between some rocks. You can easily see that the gaps are similar in width to the wavelength of the waves and that this causes an interference pattern as the waves pass beyond the gaps. A cross-section across the waves in the foreground would show the crests and troughs characteristic of an interference pattern.
Light has wave characteristics in various media as well as in a vacuum. When light goes from a vacuum to some medium, such as water, its speed and wavelength change, but its frequency, f, remains the same. The speed of light in a medium is
 , where n is its index of refraction. If you divide both sides of the equation
   by n, you get
  . Therefore,
 , where
   is the wavelength in a medium, and
where
   is the wavelength in vacuum and n is the medium’s index of refraction. It follows that the wavelength of light is smaller in any medium than it is in vacuum. In water, for example, which has n = 1.333, the range of visible wavelengths is (380 nm)/1.333 to (760 nm)/1.333, or
  285–570 nm. Although wavelengths change while traveling from one medium to another, colors do not, since colors are associated with frequency.
The Dutch scientist Christiaan Huygens (1629–1695) developed a useful technique for determining in detail how and where waves propagate. He used wavefronts, which are the points on a wave’s surface that share the same, constant phase (such as all the points that make up the crest of a water wave). Huygens’s principle states, “Every point on a wavefront is a source of wavelets that spread out in the forward direction at the same speed as the wave itself. The new wavefront is a line tangent to all of the wavelets.”
 shows how Huygens’s principle is applied. A wavefront is the long edge that moves; for example, the crest or the trough. Each point on the wavefront emits a semicircular wave that moves at the propagation speed v. These are drawn later at a time, t, so that they have moved a distance
  . The new wavefront is a line tangent to the wavelets and is where the wave is located at time t. Huygens’s principle works for all types of waves, including water waves, sound waves, and light waves. It will be useful not only in describing how light waves propagate, but also in how they interfere.
What happens when a wave passes through an opening, such as light shining through an open door into a dark room? For light, you expect to see a sharp shadow of the doorway on the floor of the room, and you expect no light to bend around corners into other parts of the room. When sound passes through a door, you hear it everywhere in the room and, thus, you understand that sound spreads out when passing through such an opening. What is the difference between the behavior of sound waves and light waves in this case? The answer is that the wavelengths that make up the light are very short, so that the light acts like a ray. Sound has wavelengths on the order of the size of the door, and so it bends around corners.
If light passes through smaller openings, often called slits, you can use Huygens’s principle to show that light bends as sound does (see ). The bending of a wave around the edges of an opening or an obstacle is called diffraction. Diffraction is a wave characteristic that occurs for all types of waves. If diffraction is observed for a phenomenon, it is evidence that the phenomenon is produced by waves. Thus, the horizontal diffraction of the laser beam after it passes through slits in  is evidence that light has the properties of a wave.
Once again, water waves present a familiar example of a wave phenomenon that is easy to observe and understand, as shown in .
The fact that Huygens’s principle worked was not considered enough evidence to prove that light is a wave. People were also reluctant to accept light’s wave nature because it contradicted the ideas of Isaac Newton, who was still held in high esteem. The acceptance of the wave character of light came after 1801, when the English physicist and physician Thomas Young (1773–1829) did his now-classic double-slit experiment (see ).
When light passes through narrow slits, it is diffracted into semicircular waves, as shown in  (a). Pure constructive interference occurs where the waves line up crest to crest or trough to trough. Pure destructive interference occurs where they line up crest to trough. The light must fall on a screen and be scattered into our eyes for the pattern to be visible. An analogous pattern for water waves is shown in  (b). Note that regions of constructive and destructive interference move out from the slits at well-defined angles to the original beam. Those angles depend on wavelength and the distance between the slits, as you will see below.

### Calculations Involving Diffraction and Interference

The fact that the wavelength of light of one color, or monochromatic light, can be calculated from its two-slit diffraction pattern in Young’s experiments supports the conclusion that light has wave properties. To understand the basis of such calculations, consider how two waves travel from the slits to the screen. Each slit is a different distance from a given point on the screen. Thus different numbers of wavelengths fit into each path. Waves start out from the slits in phase (crest to crest), but they will end up out of phase (crest to trough) at the screen if the paths differ in length by half a wavelength, interfering destructively. If the paths differ by a whole wavelength, then the waves arrive in phase (crest to crest) at the screen, interfering constructively. More generally, if the paths taken by the two waves differ by any half-integral number of wavelengths
 , then destructive interference occurs. Similarly, if the paths taken by the two waves differ by any integral number of wavelengths
 , then constructive interference occurs.
 shows how to determine the path-length difference for waves traveling from two slits to a common point on a screen. If the screen is a large distance away compared with the distance between the slits, then the angle
   between the path and a line from the slits perpendicular to the screen (see the figure) is nearly the same for each path. That approximation and simple trigonometry show the length difference,
 , to be
 , where d is the distance between the slits,
To obtain constructive interference for a double slit, the path-length difference must be an integral multiple of the wavelength, or
Similarly, to obtain destructive interference for a double slit, the path-length difference must be a half-integral multiple of the wavelength, or
The number m is the order of the interference. For example, m = 4 is fourth-order interference.
 shows how the intensity of the bands of constructive interference decreases with increasing angle.
Light passing through a single slit forms a diffraction pattern somewhat different from that formed by double slits.  shows a single-slit diffraction pattern. Note that the central maximum is larger than those on either side, and that the intensity decreases rapidly on either side.
The analysis of single-slit diffraction is illustrated in . Assuming the screen is very far away compared with the size of the slit, rays heading toward a common destination are nearly parallel. That approximation allows a series of trigonometric operations that result in the equations for the minima produced by destructive interference.
or
When rays travel straight ahead, they remain in phase and a central maximum is obtained. However, when rays travel at an angle
   relative to the original direction of the beam, each ray travels a different distance to the screen, and they can arrive in or out of phase. Thus, a ray from the center travels a distance
   farther than the ray from the top edge of the slit, they arrive out of phase, and they interfere destructively. Similarly, for every ray between the top and the center of the slit, there is a ray between the center and the bottom of the slit that travels a distance
   farther to the common point on the screen, and so interferes destructively. Symmetrically, there will be another minimum at the same angle below the direct ray.
Below we summarize the equations needed for the calculations to follow.
The speed of light in a vacuum, c, the wavelength of the light,
 , and its frequency, f, are related as follows.
The wavelength of light in a medium,
 , compared to its wavelength in a vacuum,
 , is given by
To calculate the positions of constructive interference for a double slit, the path-length difference must be an integral multiple, m, of the wavelength.

where d is the distance between the slits and
   is the angle between a line from the slits to the maximum and a line perpendicular to the barrier in which the slits are located. To calculate the positions of destructive interference for a double slit, the path-length difference must be a half-integral multiple of the wavelength:
For a single-slit diffraction pattern, the width of the slit, D, the distance of the first (m = 1) destructive interference minimum, y, the distance from the slit to the screen, L, and the wavelength,
 , are given by
Also, for single-slit diffraction,
where
   is the angle between a line from the slit to the minimum and a line perpendicular to the screen, and m is the order of the minimum.

### Practice Problems

### Check Your Understanding

### Section Summary

1. The wavelength of light varies with the refractive index of the medium.
2. Slits produce a diffraction pattern if their width and separation are similar to the wavelength of light passing through them.
3. Interference bands of a single-slit diffraction pattern can be predicted.
4. Interference bands of a double-slit diffraction pattern can be predicted.

### Key Equations

### Concept Items

### Critical Thinking

### Problems

### Test Prep Multiple Choice

### Test Prep Short Answer

### Test Prep Extended Response


# Diffraction and Interference

## Applications of Diffraction, Interference, and Coherence

### Section Key Terms

### Wave-Based Applications of Light

In 1917, Albert Einstein was thinking about photons and excited atoms. He considered an atom excited by a certain amount of energy and what would happen if that atom were hit by a photon with the same amount of energy. He suggested that the atom would emit a photon with that amount of energy, and it would be accompanied by the original photon. The exciting part is that you would have two photons with the same energy and they would be in phase. Those photons could go on to hit other excited atoms, and soon you would have a stream of in-phase photons. Such a light stream is said to be coherent. Some four decades later, Einstein’s idea found application in a process called, light amplification by stimulated emission of radiation. Take the first letters of all the words (except by and “of”) and write them in order. You get the word  (see  (a)), which is the name of the device that produces such a beam of light.
Laser beams are directional, very intense, and narrow (only about 0.5 mm in diameter). These properties lead to a number of applications in industry and medicine. The following are just a few examples:
1. This chapter began with a picture of a compact disc (see ). Those audio and data-storage devices began replacing cassette tapes during the 1990s. CDs are read by interpreting variations in reflections of a laser beam from the surface.
2. Some barcode scanners use a laser beam.
3. Lasers are used in industry to cut steel and other metals.
4. Lasers are bounced off reflectors that astronauts left on the Moon. The time it takes for the light to make the round trip can be used to make precise calculations of the Earth-Moon distance.
5. Laser beams are used to produce holograms. The name hologram means entire picture (from the Greek holo-, as in holistic), because the image is three-dimensional. A viewer can move around the image and see it from different perspectives. Holograms take advantage of the wave properties of light, as opposed to traditional photography which is based on geometric optics. A holographic image is produced by constructive and destructive interference of a split laser beam.
6. One of the advantages of using a laser as a surgical tool is that it is accompanied by very little bleeding.
7. Laser eye surgery has improved the vision of many people, without the need for corrective lenses. A laser beam is used to change the shape of the lens of the eye, thus changing its focal length.
An interesting thing happens if you pass light through a large number of evenly-spaced parallel slits. Such an arrangement of slits is called a diffraction grating. An interference pattern is created that is very similar to the one formed by double-slit diffraction (see  and ). A diffraction grating can be manufactured by scratching glass with a sharp tool to form a number of precisely positioned parallel lines, which act like slits. Diffraction gratings work both for transmission of light, as in , and for reflection of light, as on the butterfly wings or the Australian opal shown in , or the CD pictured in the opening illustration of this chapter. In addition to their use as novelty items, diffraction gratings are commonly used for spectroscopic dispersion and analysis of light. What makes them particularly useful is the fact that they form a sharper pattern than do double slits. That is, their bright regions are narrower and brighter, while their dark regions are darker.  shows idealized graphs demonstrating the sharper pattern. Natural diffraction gratings occur in the feathers of certain birds. Tiny, fingerlike structures in regular patterns act as reflection gratings, producing constructive interference that gives the feathers colors not solely due to their pigmentation. The effect is called iridescence.
Where are diffraction gratings used? Diffraction gratings are key components of monochromators—devices that separate the various wavelengths of incoming light and allow a beam with only a specific wavelength to pass through. Monochromators are used, for example, in optical imaging of particular wavelengths from biological or medical samples. A diffraction grating can be chosen to specifically analyze a wavelength of light emitted by molecules in diseased cells in a biopsy sample, or to help excite strategic molecules in the sample with a selected frequency of light. Another important use is in optical fiber technologies where fibers are designed to provide optimum performance at specific wavelengths. A range of diffraction gratings is available for selecting specific wavelengths for such use.
Diffraction gratings are used in spectroscopes to separate a light source into its component wavelengths. When a material is heated to incandescence, it gives off wavelengths of light characteristic of the chemical makeup of the material. A pure substance will produce a spectrum that is unique, thus allowing identification of the substance. Spectroscopes are also used to measure wavelengths both shorter and longer than visible light. Such instruments have become especially useful to astronomers and chemists.  shows a diagram of a spectroscope.
Light diffracts as it moves through space, bending around obstacles and interfering constructively and destructively. While diffraction allows light to be used as a spectroscopic tool, it also limits the detail we can obtain in images.

 (a) shows the effect of passing light through a small circular aperture. Instead of a bright spot with sharp edges, a spot with a fuzzy edge surrounded by circles of light is obtained. This pattern is caused by diffraction similar to that produced by a single slit. Light from different parts of the circular aperture interferes constructively and destructively. The effect is most noticeable when the aperture is small, but the effect is there for large apertures, too.
How does diffraction affect the detail that can be observed when light passes through an aperture?  (b) shows the diffraction pattern produced by two point light sources that are close to one another. The pattern is similar to that for a single point source, and it is just barely possible to tell that there are two light sources rather than one. If they are closer together, as in  (c), you cannot distinguish them, thus limiting the detail, or resolution, you can obtain. That limit is an inescapable consequence of the wave nature of light.
There are many situations in which diffraction limits the resolution. The acuity of vision is limited because light passes through the pupil, the circular aperture of the eye. Be aware that the diffraction-like spreading of light is due to the limited diameter of a light beam, not the interaction with an aperture. Thus light passing through a lens with a diameter of D shows the diffraction effect and spreads, blurring the image, just as light passing through an aperture of diameter D does. Diffraction limits the resolution of any system having a lens or mirror. Telescopes are also limited by diffraction, because of the finite diameter, D, of their primary mirror.

### Calculations Involving Diffraction Gratings and Resolution

Early in the chapter, it was mentioned that when light passes from one medium to another, its speed and wavelength change, but its frequency remains constant. The equation
shows how to the wavelength in a given medium,
 , is related to the wavelength in a vacuum,
 , and the refractive index, n, of the medium. The equation is useful for calculating the change in wavelength of a monochromatic laser beam in various media. The analysis of a diffraction grating is very similar to that for a double slit. As you know from the discussion of double slits in Young’s double-slit experiment, light is diffracted by, and spreads out after passing through, each slit. Rays travel at an angle
   relative to the incident direction. Each ray travels a different distance to a common point on a screen far away. The rays start in phase, and they can be in or out of phase when they reach a screen, depending on the difference in the path lengths traveled. Each ray travels a distance that differs by
   from that of its neighbor, where d is the distance between slits. If
   equals an integral number of wavelengths, the rays all arrive in phase, and constructive interference (a maximum) is obtained. Thus, the condition necessary to obtain constructive interference for a diffraction grating is
where d is the distance between slits in the grating,
   is the wavelength of the light, and m is the order of the maximum. Note that this is exactly the same equation as for two slits separated by d. However, the slits are usually closer in diffraction gratings than in double slits, producing fewer maxima at larger angles.
Just what is the resolution limit of an aperture or lens? To answer that question, consider the diffraction pattern for a circular aperture, which, similar to the diffraction pattern of light passing through a slit, has a central maximum that is wider and brighter than the maxima surrounding it (see  (a)). It can be shown that, for a circular aperture of diameter D, the first minimum in the diffraction pattern occurs at
 , provided that the aperture is large compared with the wavelength of light, which is the case for most optical instruments. The accepted criterion for determining the diffraction limit to resolution based on diffraction was developed by Lord Rayleigh in the 19th century. The Rayleigh criterion for the diffraction limit to resolution states that two images are just resolvable when the center of the diffraction pattern of one is directly over the first minimum of the diffraction pattern of the other. See  (b). The first minimum is at an angle of
 , so that two point objects are just resolvable if they are separated by the angle
where
   is the wavelength of the light (or other electromagnetic radiation) and D is the diameter of the aperture, lens, mirror, etc., with which the two objects are observed. In the expression above,
   has units of radians.

### Practice Problems

### Check Your Understanding

### Section Summary

1. The focused, coherent radiation emitted by lasers has many uses in medicine and industry.
2. Characteristics of diffraction patterns produced with diffraction gratings can be determined.
3. Diffraction gratings have been incorporated in many instruments, including microscopes and spectrometers.
4. Resolution has a limit that can be predicted.

### Key Equations

### Concept Items

### Critical Thinking

### Problems

### Test Prep Multiple Choice

### Test Prep Short Answer

### Extended Response

### Performance Task


# Static Electricity

## Introduction

You may have been introduced to static electricity like the child sliding down the slide in the opening photograph (). The zap that he is likely to receive if he touches a playmate or parent tends to bring home the lesson. But static electricity is more than just fun and games—it is put to use in many industries. The forces between electrically charged particles are used in technologies such as printers, pollution filters, and spray guns used for painting cars and trucks. Static electricity is the study of phenomena that involve an imbalance of electrical charge. Although creating this imbalance typically requires moving charge around, once the imbalance is created, it often remains static for a long time. The study of charge in motion is called electromagnetism and will be covered in a later chapter. What is electrical charge, how is it associated with objects, and what forces does it create? These are just some of the questions that this chapter addresses.


# Static Electricity

## Electrical Charges, Conservation of Charge, and Transfer of Charge

### Section Key Terms

### Electric Charge

You may know someone who has an electric personality, which usually means that other people are attracted to this person. This saying is based on electric charge, which is a property of matter that causes objects to attract or repel each other. Electric charge comes in two varieties, which we call positive and negative. Like charges repel each other, and unlike charges attract each other. Thus, two positive charges repel each other, as do two negative charges. A positive charge and a negative charge attract each other.
How do we know there are two types of electric charge? When various materials are rubbed together in controlled ways, certain combinations of materials always result in a net charge of one type on one material and a net charge of the opposite type on the other material. By convention, we call one type of charge positive and the other type negative. For example, when glass is rubbed with silk, the glass becomes positively charged and the silk negatively charged. Because the glass and silk have opposite charges, they attract one another like clothes that have rubbed together in a dryer. Two glass rods rubbed with silk in this manner will repel one another, because each rod has positive charge on it. Similarly, two silk cloths rubbed in this manner will repel each other, because both cloths have negative charge.  shows how these simple materials can be used to explore the nature of the force between charges.
It took scientists a long time to discover what lay behind these two types of charges. The word electric itself comes from the Greek word elektron for amber, because the ancient Greeks noticed that amber, when rubbed by fur, attracts dry straw. Almost 2,000 years later, the English physicist William Gilbert proposed a model that explained the effect of electric charge as being due to a mysterious electrical fluid that would pass from one object to another. This model was debated for several hundred years, but it was finally put to rest in 1897 by the work of the English physicist J. J. Thomson and French physicist Jean Perrin. Along with many others, Thomson and Perrin were studying the mysterious cathode rays that were known at the time to consist of particles smaller than the smallest atom. Perrin showed that cathode rays actually carried negative electrical charge. Later, Thomson’s work led him to declare, “I can see no escape from the conclusion that [cathode rays] are charges of negative electricity carried by particles of matter.”
It took several years of further experiments to confirm Thomson’s interpretation of the experiments, but science had in fact discovered the particle that carries the fundamental unit of negative electrical charge. We now know this particle as the electron.
Atoms, however, were known to be electrically neutral, which means that they carry the same amount of positive and negative charge, so their net charge is zero. Because electrons are negative, some other part of the atom must contain positive charge. Thomson put forth what is called the plum pudding model, in which he described atoms as being made of thousands of electrons swimming around in a nebulous mass of positive charge, as shown by the left-side image of . His student, Ernest Rutherford, originally believed that this model was correct and used it (along with other models) to try to understand the results of his experiments bombarding gold foils with alpha particles (i.e., helium atoms stripped of their electrons). The results, however, did not confirm Thomson’s model but rather destroyed it! Rutherford found that most of the space occupied by the gold atoms was actually empty and that almost all of the matter of each atom was concentrated into a tiny, extremely dense nucleus, as shown by the right-side image of . The atomic nucleus was later found to contain particles called protons, each of which carries a unit of positive electric charge.Protons were later found to contain sub particles called
Protons and electrons are thus the fundamental particles that carry electric charge. Each proton carries one unit of positive charge, and each electron carries one unit of negative charge. To the best precision that modern technology can provide, the charge carried by a proton is exactly the opposite of that carried by an electron. The SI unit for electric charge is the coulomb (abbreviated as “C”), which is named after the French physicist Charles Augustin de Coulomb, who studied the force between charged objects. The proton carries
   and the electron carries
  . The number n of protons required to make +1.00 C is
The same number of electrons is required to make −1.00 C of electric charge. The fundamental unit of charge is often represented as e. Thus, the charge on a proton is e, and the charge on an electron is −e. Mathematically,

### Conservation of Charge

Because the fundamental positive and negative units of charge are carried on protons and electrons, we would expect that the total charge cannot change in any system that we define. In other words, although we might be able to move charge around, we cannot create or destroy it. This should be true provided that we do not create or destroy protons or electrons in our system. In the twentieth century, however, scientists learned how to create and destroy electrons and protons, but they found that charge is still conserved. Many experiments and solid theoretical arguments have elevated this idea to the status of a law. The law of conservation of charge says that electrical charge cannot be created or destroyed.
The law of conservation of charge is very useful. It tells us that the net charge in a system is the same before and after any interaction within the system. Of course, we must ensure that no external charge enters the system during the interaction and that no internal charge leaves the system. Mathematically, conservation of charge can be expressed as
where
   is the net charge of the system before the interaction, and
   is the net charge after the interaction.

### Practice Problems

### Conductors and Insulators

Materials can be classified depending on whether they allow charge to move. If charge can easily move through a material, such as metals, then these materials are called conductors. This means that charge can be conducted (i.e., move) through the material rather easily. If charge cannot move through a material, such as rubber, then this material is called an insulator.
Most materials are insulators. Their atoms and molecules hold on more tightly to their electrons, so it is difficult for electrons to move between atoms. However, it is not impossible. With enough energy, it is possible to force electrons to move through an insulator. However, the insulator is often physically destroyed in the process. In metals, the outer electrons are loosely bound to their atoms, so not much energy is required to make electrons move through metal. Such metals as copper, silver, and aluminum are good conductors. Insulating materials include plastics, glass, ceramics, and wood.
The conductivity of some materials is intermediate between conductors and insulators. These are called semiconductors. They can be made conductive under the right conditions, which can involve temperature, the purity of the material, and the force applied to push electrons through them. Because we can control whether semiconductors are conductors or insulators, these materials are used extensively in computer chips. The most commonly used semiconductor is silicon.  shows various materials arranged according to their ability to conduct electrons.
What happens if an excess negative charge is placed on a conducting object? Because like charges repel each other, they will push against each other until they are as far apart as they can get. Because the charge can move in a conductor, it moves to the outer surfaces of the object. (a) shows schematically how an excess negative charge spreads itself evenly over the outer surface of a metal sphere.
What happens if the same is done with an insulating object? The electrons still repel each other, but they are not able to move, because the material is an insulator. Thus, the excess charge stays put and does not distribute itself over the object. (b) shows this situation.

### Transfer and Separation of Charge

Most objects we deal with are electrically neutral, which means that they have the same amount of positive and negative charge. However, transferring negative charge from one object to another is fairly easy to do. When negative charge is transferred from one object to another, an excess of positive charge is left behind. How do we know that the negative charge is the mobile charge? The positive charge is carried by the proton, which is stuck firmly in the nucleus of atoms, and the atoms are stuck in place in solid materials. Electrons, which carry the negative charge, are much easier to remove from their atoms or molecules and can therefore be transferred more easily.
Electric charge can be transferred in several manners. One of the simplest ways to transfer charge is charging by contact, in which the surfaces of two objects made of different materials are placed in close contact. If one of the materials holds electrons more tightly than the other, then it takes some electrons with it when the materials are separated. Rubbing two surfaces together increases the transfer of electrons, because it creates a closer contact between the materials. It also serves to present fresh material with a full supply of electrons to the other material. Thus, when you walk across a carpet on a dry day, your shoes rub against the carpet, and some electrons are removed from the carpet by your shoes. The result is that you have an excess of negative charge on your shoes. When you then touch a doorknob, some of your excess of electrons transfer to the neutral doorknob, creating a small spark.
Touching the doorknob with your hand demonstrates a second way to transfer electric charge, which is charging by conduction. This transfer happens because like charges repel, and so the excess electrons that you picked up from the carpet want to be as far away from each other as possible. Some of them move to the doorknob, where they will distribute themselves over the outer surface of the metal. Another example of charging by conduction is shown in the top row of . A metal sphere with 100 excess electrons touches a metal sphere with 50 excess electrons, so 25 electrons from the first sphere transfer to the second sphere. Each sphere finishes with 75 excess electrons.
The same reasoning applies to the transfer of positive charge. However, because positive charge essentially cannot move in solids, it is transferred by moving negative charge in the opposite direction. For example, consider the bottom row of . The first metal sphere has 100 excess protons and touches a metal sphere with 50 excess protons, so the second sphere transfers 25 electrons to the first sphere. These 25 extra electrons will electrically cancel 25 protons so that the first metal sphere is left with 75 excess protons. This is shown in the bottom row of . The second metal sphere lost 25 electrons so it has 25 more excess protons, for a total of 75 excess protons. The end result is the same if we consider that the first ball transferred a net positive charge equal to that of 25 protons to the first ball.
In this discussion, you may wonder how the excess electrons originally got from your shoes to your hand to create the spark when you touched the doorknob. The answer is that no electrons actually traveled from your shoes to your hands. Instead, because like charges repel each other, the excess electrons on your shoe simply pushed away some of the electrons in your feet. The electrons thus dislodged from your feet moved up into your leg and in turn pushed away some electrons in your leg. This process continued through your whole body until a distribution of excess electrons covered the extremities of your body. Thus your head, your hands, the tip of your nose, and so forth all received their doses of excess electrons that had been pushed out of their normal positions. All this was the result of electrons being pushed out of your feet by the excess electrons on your shoes.
This type of charge separation is called polarization. As soon as the excess electrons leave your shoes (by rubbing off onto the floor or being carried away in humid air), the distribution of electrons in your body returns to normal. Every part of your body is again electrically neutral (i.e., zero excess charge).
The phenomenon of polarization is seen in . The child has accumulated excess positive charge by sliding on the slide. This excess charge repels itself and so becomes distributed over the extremities of the child’s body, notably in his hair. As a result, the hair stands on end, because the excess negative charge on each strand repels the excess negative charge on neighboring strands.
Polarization can be used to charge objects. Consider the two metallic spheres shown in . The spheres are electrically neutral, so they carry the same amounts of positive and negative charge. In the top picture ((a)), the two spheres are touching, and the positive and negative charge is evenly distributed over the two spheres. We then approach a glass rod that carries an excess positive charge, which can be done by rubbing the glass rod with silk, as shown in (b). Because opposite charges attract each other, the negative charge is attracted to the glass rod, leaving an excess positive charge on the opposite side of the right sphere. This is an example of charging by induction, whereby a charge is created by approaching a charged object with a second object to create an unbalanced charge in the second object. If we then separate the two spheres, as shown in (c), the excess charge is stuck on each sphere. The left sphere now has an excess negative charge, and the right sphere has an excess positive charge. Finally, in the bottom picture, the rod is removed, and the opposite charges attract each other, so they move as close together as they can get.

### Practice Problems

### Check Your Understanding

### Section Summary

1. Electric charge is a conserved quantity, which means it can be neither created nor destroyed.
2. Electric charge comes in two varieties, which are called positive and negative.
3. Charges with the same sign repel each other. Charges with opposite signs attract each other.
4. Charges can move easily in conducting material. Charges cannot move easily in an insulating material.
5. Objects can be charged in three ways: by contact, by conduction, and by induction.
6. Although a polarized object may be neutral, its electrical charge is unbalanced, so one side of the object has excess negative charge and the other side has an equal magnitude of excess positive charge.

### Concept Items

### Critical Thinking

### Problems

### Test Prep Multiple Choice

### Test Prep Short Answer

### Test Prep Extended Response


# Static Electricity

## Coulomb's law

### Section Key Terms

More than 100 years before Thomson and Rutherford discovered the fundamental particles that carry positive and negative electric charges, the French scientist Charles-Augustin de Coulomb mathematically described the force between charged objects. Doing so required careful measurements of forces between charged spheres, for which he built an ingenious device called a .
This device, shown in , contains an insulating rod that is hanging by a thread inside a glass-walled enclosure. At one end of the rod is the metallic sphere A. When no charge is on this sphere, it touches sphere B. Coulomb would touch the spheres with a third metallic ball (shown at the bottom of the diagram) that was charged. An unknown amount of charge would distribute evenly between spheres A and B, which would then repel each other, because like charges repel. This force would cause sphere A to rotate away from sphere B, thus twisting the wire until the torsion in the wire balanced the electrical force. Coulomb then turned the knob at the top, which allowed him to rotate the thread, thus bringing sphere A closer to sphere B. He found that bringing sphere A twice as close to sphere B required increasing the torsion by a factor of four. Bringing the sphere three times closer required a ninefold increase in the torsion. From this type of measurement, he deduced that the electrical force between the spheres was inversely proportional to the distance squared between the spheres. In other words,
where r is the distance between the spheres.
An electrical charge distributes itself equally between two conducting spheres of the same size. Knowing this allowed Coulomb to divide an unknown charge in half. Repeating this process would produce a sphere with one quarter of the initial charge, and so on. Using this technique, he measured the force between spheres A and B when they were charged with different amounts of charge. These measurements led him to deduce that the force was proportional to the charge on each sphere, or
where
   is the charge on sphere A, and
   is the charge on sphere B.
Combining these two proportionalities, he proposed the following expression to describe the force between the charged spheres.
This equation is known as Coulomb’s law, and it describes the electrostatic force between charged objects. The constant of proportionality k is called . In SI units, the constant k has the value

The direction of the force is along the line joining the centers of the two objects. If the two charges are of opposite signs, Coulomb’s law gives a negative result. This means that the force between the particles is attractive. If the two charges have the same signs, Coulomb’s law gives a positive result. This means that the force between the particles is repulsive. For example, if both
   and
   are negative or if both are positive, the force between them is repulsive. This is shown in (a). If
   is a negative charge and
   is a positive charge (or vice versa), then the charges are different, so the force between them is attractive. This is shown in (b).
Note that Coulomb’s law applies only to charged objects that are not moving with respect to each other. The law says that the force is proportional to the amount of charge on each object and inversely proportional to the square of the distance between the objects. If we double the charge
 , for instance, then the force is doubled. If we double the distance between the objects, then the force between them decreases by a factor of
  . Although Coulomb’s law is true in general, it is easiest to apply to spherical objects or to objects that are much smaller than the distance between the objects (in which case, the objects can be approximated as spheres).
Coulomb’s law is an example of an inverse-square law, which means the force depends on the square of the denominator. Another inverse-square law is Newton’s law of universal gravitation, which is
  . Although these laws are similar, they differ in two important respects: (i) The gravitational constant G is much, much smaller than k (
  ); and (ii) only one type of mass exists, whereas two types of electric charge exist. These two differences explain why gravity is so much weaker than the electrostatic force and why gravity is only attractive, whereas the electrostatic force can be attractive or repulsive.
Finally, note that Coulomb measured the distance between the spheres from the centers of each sphere. He did not explain this assumption in his original papers, but it turns out to be valid. From outside a uniform spherical distribution of charge, it can be treated as if all the charge were located at the center of the sphere.

### Practice Problems

### Check Your Understanding

### Section Summary

1. Coulomb’s law is an inverse square law and describes the electrostatic force between particles.
2. The electrostatic force between charged objects is proportional to the charge on each object and inversely proportional to the distance squared between the objects.
3. If Coulomb’s law gives a negative result, the force is attractive; if the result is positive, the force is repulsive.

### Key Equations

### Concept Items

### Critical Thinking

### Problems

### Test Prep Multiple Choice

### Test Prep Short Answer

### Test Prep Extended Response


# Static Electricity

## Electric Field

### Section Key Terms

You may have heard of a  in science fiction movies, where such fields apply forces at particular positions in space to keep a villain trapped or to protect a spaceship from enemy fire. The concept of a field is very useful in physics, although it differs somewhat from what you see in movies.
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 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. Michael Faraday, an English physicist of the nineteenth century, proposed the concept of an electric field. If you know the electric field, then you can easily calculate the force (magnitude and direction) applied to any electric charge that you place in the field.
An electric field is generated by electric charge and tells us the force per unit charge at all locations in space around a charge distribution. The charge distribution could be a single point charge; a distribution of charge over, say, a flat plate; or a more complex distribution of charge. The electric field extends into space around the charge distribution. Now consider placing a test charge in the field. A test charge is a positive electric charge whose charge is so small that it does not significantly disturb the charges that create the electric field. The electric field exerts a force on the test charge in a given direction. The force exerted is proportional to the charge of the test charge. For example, if we double the charge of the test charge, the force exerted on it doubles. Mathematically, saying that electric field is the force per unit charge is written as
where we are considering only electric forces. Note that the electric field is a vector field that points in the same direction as the force on the positive test charge. The units of electric field are N/C.
If the electric field is created by a point charge or a sphere of uniform charge, then the magnitude of the force between this point charge Q and the test charge is given by Coulomb’s law

where the absolute value is used, because we only consider the magnitude of the force. The magnitude of the electric field is then
This equation gives the magnitude of the electric field created by a point charge Q. The distance r in the denominator is the distance from the point charge, Q, or from the center of a spherical charge, to the point of interest.
If the test charge is removed from the electric field, the electric field still exists. To create a three-dimensional map of the electric field, imagine placing the test charge in various locations in the field. At each location, measure the force on the charge, and use the vector equation
   to calculate the electric field. Draw an arrow at each point where you place the test charge to represent the strength and the direction of the electric field. The length of the arrows should be proportional to the strength of the electric field. If you join together these arrows, you obtain lines.  shows an image of the three-dimensional electric field created by a positive charge.
Just drawing the electric field lines in a plane that slices through the charge gives the two-dimensional electric-field maps shown in . On the left is the electric field created by a positive charge, and on the right is the electric field created by a negative charge.
Notice that the electric field lines point away from the positive charge and toward the negative charge. Thus, a positive test charge placed in the electric field of the positive charge will be repelled. This is consistent with Coulomb’s law, which says that like charges repel each other. If we place the positive charge in the electric field of the negative charge, the positive charge is attracted to the negative charge. The opposite is true for negative test charges. Thus, the direction of the electric field lines is consistent with what we find by using Coulomb’s law.
The equation
   says that the electric field gets stronger as we approach the charge that generates it. For example, at 2 cm from the charge Q (r = 2 cm), the electric field is four times stronger than at 4 cm from the charge (r = 4 cm). Looking at  and  again, we see that the electric field lines become denser as we approach the charge that generates it. In fact, the density of the electric field lines is proportional to the strength of the electric field!
Electric-field maps can be made for several charges or for more complicated charge distributions. The electric field due to multiple charges may be found by adding together the electric field from each individual charge. Because this sum can only be a single number, we know that only a single electric-field line can go through any given point. In other words, electric-field lines cannot cross each other.
(a) shows a two-dimensional map of the electric field generated by a charge of +q and a nearby charge of −q. The three-dimensional version of this map is obtained by rotating this map about the axis that goes through both charges. A positive test charge placed in this field would experience a force in the direction of the field lines at its location. It would thus be repelled from the positive charge and attracted to the negative charge. (b) shows the electric field generated by two charges of −q. Note how the field lines tend to repel each other and do not overlap. A positive test charge placed in this field would be attracted to both charges. If you are far from these two charges, where far means much farther than the distance between the charges, the electric field looks like the electric field from a single charge of −2q.

### Practice Problems

### Check Your Understanding

### Section Summary

1. The electric field defines the force per unit charge in the space around a charge distribution.
2. For a point charge or a sphere of uniform charge, the electric field is inversely proportional to the distance from the point charge or from the center of the sphere.
3. Electric-field lines never cross each other.
4. More force is applied to a charge in a region with many electric field lines than in a region with few electric field lines.
5. Electric field lines start at positive charges and point away from positive charges. They end at negative charges and point toward negative charges.

### Key Equations

### Concept Items

### Critical Thinking

### Problems

### Test Prep Multiple Choice

### Test Prep Short Answer

### Test Prep Extended Response


# Static Electricity

## Electric Potential

### Section Key Terms

As you learned in studying gravity, a mass in a gravitational field has potential energy, which means it has the potential to accelerate and thereby increase its kinetic energy. This kinetic energy can be used to do work. For example, imagine you want to use a stone to pound a nail into a piece of wood. You first lift the stone high above the nail, which increases the potential energy of the stone-Earth system—because Earth is so large, it does not move, so we usually shorten this by saying simply that the potential energy of the stone increases. When you drop the stone, gravity converts the potential energy into kinetic energy. When the stone hits the nail, it does work by pounding the nail into the wood. The gravitational potential energy is the work that a mass can potentially do by virtue of its position in a gravitational field. Potential energy is a very useful concept, because it can be used with conservation of energy to calculate the motion of masses in a gravitational field.
Electric potential energy works much the same way, but it is based on the electric field instead of the gravitational field. By virtue of its position in an electric field, a charge has an electric potential energy. If the charge is free to move, the force due to the electric field causes it to accelerate, so its potential energy is converted to kinetic energy, just like a mass that falls in a gravitational field. This kinetic energy can be used to do work. The electric potential energy is the work that a charge can do by virtue of its position in an electric field.
The analogy between gravitational potential energy and electric potential energy is depicted in . On the left, the ball-Earth system gains gravitational potential energy when the ball is higher in Earth's gravitational field. On the right, the two-charge system gains electric potential energy when the positive charge is farther from the negative charge.
Let’s use the symbol
   to denote gravitational potential energy. When a mass falls in a gravitational field, its gravitational potential energy decreases. Conservation of energy tells us that the work done by the gravitational field to make the mass accelerate must equal the loss of potential energy of the mass. If we use the symbol
   to denote this work, then
where the minus sign reflects the fact that the potential energy of the ball decreases.
The work done by gravity on the mass is
where F is the force due to gravity, and
   and
   are the initial and final positions of the ball, respectively. The negative sign is because gravity points down, which we consider to be the negative direction. For the constant gravitational field near Earth’s surface,
  . The change in gravitational potential energy of the mass is
Note that
   is just the negative of the height h from which the mass falls, so we usually just write
  .
We now apply the same reasoning to a charge in an electric field to find the electric potential energy. The change
   in electric potential energy is the work done by the electric field to move a charge q from an initial position
   to a final position
   (
  ). The definition of work does not change, except that now the work is done by the electric field:
  . For a charge that falls through a constant electric field E, the force applied to the charge by the electric field is
  . The change in electric potential energy of the charge is thus
or
This equation gives the change in electric potential energy of a charge q when it moves from position
   to position
   in a constant electric field E.
 shows how this analogy would work if we were close to Earth’s surface, where gravity is constant. The top image shows a charge accelerating due to a constant electric field. Likewise, the round mass in the bottom image accelerates due to a constant gravitation field. In both cases, the potential energy of the particle decreases, and its kinetic energy increases.
If the electric field is not constant, then the equation
   is not valid, and deriving the electric potential energy becomes more involved. For example, consider the electric potential energy of an assembly of two point charges
   and
   of the same sign that are initially very far apart. We start by placing charge
   at the origin of our coordinate system. This takes no electrical energy, because there is no electric field at the origin (because charge
   is very far away). We then bring charge
   in from very far away to a distance r from the center of charge
  . This requires some effort, because the electric field of charge
   applies a repulsive force on charge
  . The energy it takes to assemble these two charges can be recuperated if we let them fly apart again. Thus, the charges have potential energy when they are a distance r apart. It turns out that the electric potential energy of a pair of point charges
   and
   a distance r apart is
To recap, if charges
   and
   are free to move, they can accumulate kinetic energy by flying apart, and this kinetic energy can be used to do work. The maximum amount of work the two charges can do (if they fly infinitely far from each other) is given by the equation above.
Notice that if the two charges have opposite signs, then the potential energy is negative. This means that the charges have more potential to do work when they are far apart than when they are at a distance r apart. This makes sense: Opposite charges attract, so the charges can gain more kinetic energy if they attract each other from far away than if they start at only a short distance apart. Thus, they have more potential to do work when they are far apart.  summarizes how the electric potential energy depends on charge and separation.

### Electric Potential

Recall that to find the force applied by a fixed charge Q on any arbitrary test charge q, it was convenient to define the electric field, which is the force per unit charge applied by Q on any test charge that we place in its electric field. The same Strategy is used here with electric potential energy: We now define the electric potential V, which is the electric potential energy per unit charge.
Normally, the electric potential is simply called the potential or . The units for the potential are J/C, which are given the name volt (V) after the Italian physicist Alessandro Volta (1745–1827). From the equation
 , the electric potential a distance r from a point charge
   is
This equation gives the energy required per unit charge to bring a charge
   from infinity to a distance r from a point charge
   Mathematically, this is written as
Note that this equation actually represents a difference in electric potential. However, because the second term is zero, it is normally not written, and we speak of the electric potential instead of the electric potential difference, or we just say the potential difference, or voltage). Below, when we consider the electric potential energy per unit charge between two points not infinitely far apart, we speak of electric potential difference explicitly. Just remember that electric potential and electric potential difference are really the same thing; the former is used just when the electric potential energy is zero in either the initial or final charge configuration.
Coming back now to the electric potential a distance r from a point charge
 , note that
   can be any arbitrary point charge, so we can drop the subscripts and simply write
Now consider the electric potential near a group of charges q1, q2, and q3, as drawn in . The electric potential is derived by considering the electric field. Electric fields follow the principle of superposition and can be simply added together, so the electric potential from different charges also add together. Thus, the electric potential of a point near a group of charges is
where
   are the distances from the center of charges
   to the point of interest, as shown in .
Now let’s consider the electric potential in a uniform electric field. From the equation
 , we see that the potential difference in going from
   to
   in a uniform electric field E is
Notice that a positive charge in a region with high potential will experience a force pushing it toward regions of lower potential. In this sense, potential is like pressure for fluids. Imagine a pipe containing fluid, with the fluid at one end of the pipe under high pressure and the fluid at the other end of the pipe under low pressure. If nothing prevents the fluid from flowing, it will flow from the high-pressure end to the low-pressure end. Likewise, a positive charge that is free to move will move from a region with high potential to a region with lower potential.

### Practice Problems

### Check Your Understanding

### Section Summary

1. Electric potential energy is a concept similar to gravitational potential energy: It is the potential that charges have to do work by virtue of their positions relative to each other.
2. Electric potential is the electric potential energy per unit charge.
3. The potential is always measured between two points, where one point may be at infinity.
4. Positive charges move from regions of high potential to regions of low potential.
5. Negative charges move from regions of low potential to regions of high potential.

### Key Equations

### Concept Items

### Critical Thinking

### Problems

### Test Prep Multiple Choice

### Test Prep Short Answer

### Test Prep Extended Response


# Static Electricity

## Capacitors and Dielectrics

### Section Key Terms

### Capacitors

Consider again the X-ray tube discussed in the previous sample problem. How can a uniform electric field be produced? A single positive charge produces an electric field that points away from it, as in . This field is not uniform, because the space between the lines increases as you move away from the charge. However, if we combine a positive and a negative charge, we obtain the electric field shown in (a). Notice that, between the charges, the electric field lines are more equally spaced.
What happens if we place, say, five positive charges in a line across from five negative charges, as in ? Now the region between the lines of charge contains a fairly uniform electric field.
We can extend this idea even further and into two dimensions by placing two metallic plates face to face and charging one with positive charge and the other with an equal magnitude of negative charge. This can be done by connecting one plate to the positive terminal of a battery and the other plate to the negative terminal, as shown in . The electric field between these charged plates will be extremely uniform.
Let’s think about the work required to charge these plates. Before the plates are connected to the battery, they are neutral—that is, they have zero net charge. Placing the first positive charge on the left plate and the first negative charge on the right plate requires very little work, because the plates are neutral, so no opposing charges are present. Now consider placing a second positive charge on the left plate and a second negative charge on the right plate. Because the first two charges repel the new arrivals, a force must be applied to the two new charges over a distance to put them on the plates. This is the definition of work, which means that, compared with the first pair, more work is required to put the second pair of charges on the plates. To place the third positive and negative charges on the plates requires yet more work, and so on. Where does this work come from? The battery! Its chemical potential energy is converted into the work required to separate the positive and negative charges.
Although the battery does work, this work remains within the battery-plate system. Therefore, conservation of energy tells us that, if the potential energy of the battery decreases to separate charges, the energy of another part of the system must increase by the same amount. In fact, the energy from the battery is stored in the electric field between the plates. This idea is analogous to considering that the potential energy of a raised hammer is stored in Earth’s gravitational field. If the gravitational field were to disappear, the hammer would have no potential energy. Likewise, if no electric field existed between the plates, no energy would be stored between them.
If we now disconnect the plates from the battery, they will hold the energy. We could connect the plates to a lightbulb, for example, and the lightbulb would light up until this energy was used up. These plates thus have the capacity to store energy. For this reason, an arrangement such as this is called a capacitor. A capacitor is an arrangement of objects that, by virtue of their geometry, can store energy an electric field.
Various real capacitors are shown in . They are usually made from conducting plates or sheets that are separated by an insulating material. They can be flat or rolled up or have other geometries.
The capacity of a capacitor is defined by its capacitance C, which is given by
where Q is the magnitude of the charge on each capacitor plate, and V is the potential difference in going from the negative plate to the positive plate. This means that both Q and V are always positive, so the capacitance is always positive. We can see from the equation for capacitance that the units of capacitance are C/V, which are called farads (F) after the nineteenth-century English physicist Michael Faraday.
The equation
   makes sense: A parallel-plate capacitor (like the one shown in ) the size of a football field could hold a lot of charge without requiring too much work per unit charge to push the charge into the capacitor. Thus, Q would be large, and V would be small, so the capacitance C would be very large. Squeezing the same charge into a capacitor the size of a fingernail would require much more work, so V would be very large, and the capacitance would be much smaller.
Although the equation
   makes it seem that capacitance depends on voltage, in fact it does not. For a given capacitor, the ratio of the charge stored in the capacitor to the voltage difference between the plates of the capacitor always remains the same. Capacitance is determined by the geometry of the capacitor and the materials that it is made from. For a parallel-plate capacitor with nothing between its plates, the capacitance is given by
where A is the area of the plates of the capacitor and d is their separation. We use
   instead of C, because the capacitor has nothing between its plates (in the next section, we’ll see what happens when this is not the case). The constant
   read epsilon zero is called the permittivity of free space, and its value is
Coming back to the energy stored in a capacitor, we can ask exactly how much energy a capacitor stores. If a capacitor is charged by putting a voltage V across it for example, by connecting it to a battery with voltage V—the electrical potential energy stored in the capacitor is
Notice that the form of this equation is similar to that for kinetic energy,
  .

### Practice Problems

### Dielectrics

Before working through some sample problems, let’s look at what happens if we put an insulating material between the plates of a capacitor that has been charged and then disconnected from the charging battery, as illustrated in . Because the material is insulating, the charge cannot move through it from one plate to the other, so the charge Q on the capacitor does not change. An electric field exists between the plates of a charged capacitor, so the insulating material becomes polarized, as shown in the lower part of the figure. An electrically insulating material that becomes polarized in an electric field is called a dielectric.
 shows that the negative charge in the molecules in the material shifts to the left, toward the positive charge of the capacitor. This shift is due to the electric field, which applies a force to the left on the electrons in the molecules of the dielectric. The right sides of the molecules are now missing a bit of negative charge, so their net charge is positive.
All electrically insulating materials are dielectrics, but some are better dielectrics than others. A good dielectric is one whose molecules allow their electrons to shift strongly in an electric field. In other words, an electric field pulls their electrons a fair bit away from their atom, but they do not escape completely from their atom (which is why they are insulators).
 shows a macroscopic view of a dielectric in a charged capacitor. Notice that the electric-field lines in the capacitor with the dielectric are spaced farther apart than the electric-field lines in the capacitor with no dielectric. This means that the electric field in the dielectric is weaker, so it stores less electrical potential energy than the electric field in the capacitor with no dielectric.
Where has this energy gone? In fact, the molecules in the dielectric act like tiny springs, and the energy in the electric field goes into stretching these springs. With the electric field thus weakened, the voltage difference between the two sides of the capacitor is smaller, so it becomes easier to put more charge on the capacitor. Placing a dielectric in a capacitor before charging it therefore allows more charge and potential energy to be stored in the capacitor. A parallel plate with a dielectric has a capacitance of
where
   (kappa) is a dimensionless constant called the dielectric constant. Because
   is greater than 1 for dielectrics, the capacitance increases when a dielectric is placed between the capacitor plates. The dielectric constant of several materials is shown in .

### Practice Problems

### Check Your Understanding

### Section Summary

1. The capacitance of a capacitor depends only on the geometry of the capacitor and the materials from which it is made. It does not depend on the voltage across the capacitor.
2. Capacitors store electrical energy in the electric field between their plates.
3. A dielectric material is an insulator that is polarized in an electric field.
4. Putting a dielectric between the plates of a capacitor increases the capacitance of the capacitor.

### Key Equations

### Concept Items

### Critical Thinking

### Problems

### Performance Task

### Test Prep Multiple Choice

### Test Prep Short Answer

### Test Prep Extended Response


# Electrical Circuits

## Introduction

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 into a station, a hydroelectric plant sending energy to metropolitan and rural users—these and many other examples of electricity involve electric current, which is the movement of charge. Humanity has harnessed electricity, the basis of this technology, to improve our quality of life. Whereas the previous chapter concentrated on static electricity and the fundamental force underlying its behavior, the next two chapters will be devoted to electric and magnetic phenomena involving current. In addition to exploring applications of electricity, we shall gain new insights into the workings of nature.


# Electrical Circuits

## Ohm's law

### Section Key Terms

### Direct and Alternating Current

Just as water flows from high to low elevation, electrons that are free to move will travel from a place with low potential to a place with high potential. A battery has two terminals that are at different potentials. If the terminals are connected by a conducting wire, an electric current (charges) will flow, as shown in . Electrons will then move from the low-potential terminal of the battery (the negative end) through the wire and enter the high-potential terminal of the battery (the positive end).
Electric current is the rate at which electric charge moves. A large current, such as that used to start a truck engine, moves a large amount very quickly, whereas a small current, such as that used to operate a hand-held calculator, moves a small amount of charge more slowly. In equation form, electric current I is defined as
where
   is the amount of charge that flows past a given area and
   is the time it takes for the charge to move past the area. The SI unit for electric current is the ampere (A), which is named in honor of the French physicist André-Marie Ampère (1775–1836). One ampere is one coulomb per second, or
Electric current moving through a wire is in many ways similar to water current moving through a pipe. To define the flow of water through a pipe, we can count the water molecules that flow past a given section of the pipe. As shown in , electric current is very similar. We count the number of electrical charges that flow past a section of a conductor; in this case, a wire.
Assume each particle q in  carries a charge
 , in which case the total charge shown would be
  . If these charges move past the area A in a time
 , then the current would be
Note that we assigned a positive charge to the charges in . Normally, negative charges—electrons—are the mobile charge in wires, as indicated in . Positive charges are normally stuck in place in solids and cannot move freely. However, because a positive current moving to the right is the same as a negative current of equal magnitude moving to the left, as shown in , we define conventional current to flow in the direction that a positive charge would flow if it could move. Thus, unless otherwise specified, an electric current is assumed to be composed of positive charges.
Also note that one Coulomb is a significant amount of electric charge, so 5 A is a very large current. Most often you will see current on the order of milliamperes (mA).
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, as we have seen, current is carried by electrons, so the negative charges move. In ionic solutions, such as salt water, both positively charged and negatively charged ions move. This is also true in nerve cells. Pure positive currents are relatively rare but do occur. History credits American politician and scientist Benjamin Franklin with describing current as the direction that positive charges flow through a wire. He named the type of charge associated with electrons negative long before they were known to carry current in so many situations.
As electrons move through a metal wire, they encounter obstacles such as other electrons, atoms, impurities, etc. The electrons scatter from these obstacles, as depicted in . Normally, the electrons lose energy with each interaction. This energy is transferred to the wire and becomes thermal energy, which is what makes wires hot when they carry a lot of current. To keep the electrons moving thus requires a force, which is supplied by an electric field. The electric field in a wire points from the end of the wire at the higher potential to the end of the wire at the lower potential. Electrons, carrying a negative charge, move on average (or drift) in the direction opposite the electric field, as shown in .
So far, we have discussed current that moves constantly in a single direction. This is called direct current, because the electric charge flows in only one direction. Direct current is often called DC current.
Many sources of electrical power, such as the hydroelectric dam shown at the beginning of this chapter, produce alternating current, in which the current direction alternates back and forth. Alternating current is often called AC current. Alternating current moves back and forth at regular time intervals, as shown in . The alternating current that comes from a normal wall socket does not suddenly switch directions. Rather, it increases smoothly up to a maximum current and then smoothly decreases back to zero. It then grows again, but in the opposite direction until it has reached the same maximum value. After that, it decreases smoothly back to zero, and the cycle starts over again.
Devices that use AC include vacuum cleaners, fans, power tools, hair dryers, and countless others. These devices obtain the power they require when you plug them into a wall socket. The wall socket is connected to the power grid that provides an alternating potential (AC potential). When your device is plugged in, the AC potential pushes charges back and forth in the circuit of the device, creating an alternating current.
Many devices, however, use DC, such as computers, cell phones, flashlights, and cars. One source of DC is a battery, which provides a constant potential (DC potential) between its terminals. With your device connected to a battery, the DC potential pushes charge in one direction through the circuit of your device, creating a DC current. Another way to produce DC current is by using a transformer, which converts AC potential to DC potential. Small transformers that you can plug into a wall socket are used to charge up your laptop, cell phone, or other electronic device. People generally call this a charger or a battery, but it is a transformer that transforms AC voltage into DC voltage. The next time someone asks to borrow your laptop charger, tell them that you don’t have a laptop charger, but that they may borrow your converter.

### Practice Problems

### Resistance and Ohm’s Law

As mentioned previously, electrical current in a wire is in many ways similar to water flowing through a pipe. The water current that can flow through a pipe is affected by obstacles in the pipe, such as clogs and narrow sections in the pipe. These obstacles slow down the flow of current through the pipe. Similarly, electrical current in a wire can be slowed down by many factors, including impurities in the metal of the wire or collisions between the charges in the material. These factors create a resistance to the electrical current. Resistance is a description of how much a wire or other electrical component opposes the flow of charge through it.
In the 19th century, the German physicist Georg Simon Ohm (1787–1854) found experimentally that current through a conductor is proportional to the voltage drop across a current-carrying conductor.
The constant of proportionality is the resistance R of the material, which leads to
This relationship is called Ohm’s law. It can be viewed as a cause-and-effect relationship, with voltage being the cause and the current being the effect. Ohm’s law is an empirical law like that for friction, which means that it is an experimentally observed phenomenon. The units of resistance are volts per ampere, or V/A. We call a V/A an ohm, which is represented by the uppercase Greek letter omega (
  ). Thus,
Ohm’s law holds for most materials and at common temperatures. At very low temperatures, resistance may drop to zero (superconductivity). At very high temperatures, the thermal motion of atoms in the material inhibits the flow of electrons, increasing the resistance. The many substances for which Ohm’s law holds are called ohmic. Ohmic materials include good conductors like copper, aluminum, and silver, and some poor conductors under certain circumstances. The resistance of ohmic materials remains essentially the same for a wide range of voltage and current.

### Practice Problems

### Check Your Understanding

### Section Summary

1. Direct current is constant over time; alternating current alternates smoothly back and forth over time.
2. Electrical resistance causes materials to extract work from the current that flows through them.
3. In ohmic materials, voltage drop along a path is proportional to the current that runs through the path.

### Key Equations

### Concept Items

### Critical Thinking

### Problems

### Test Prep Multiple Choice

### Test Prep Short Answer

### Test Prep Extended Response


# Electrical Circuits

## Series Circuits

### Section Key Terms

### Electric Circuits and Resistors

Now that we understand the concept of electric current, let’s see what we can do with it. As you are no doubt aware, the modern lifestyle relies heavily on electrical devices. These devices contain ingenious electric circuits, which are complete, closed pathways through which electric current flows. Returning to our water analogy, an electric circuit is to electric charge like a network of pipes is to water: The electric circuit guides electric charge from one point to the next, running the charge through various devices along the way to extract work or information.
Electric circuits are made from many materials and cover a huge range of sizes, as shown in . Computers and cell phones contain electric circuits whose features can be as small as roughly a billionth of a meter (a nanometer, or
  ). The pathways that guide the current in these devices are made by ultraprecise chemical treatments of silicon or other semiconductors. Large power systems, on the other hand, contain electric circuits whose features are on the scale of meters. These systems carry such large electric currents that their physical dimensions must be relatively large.
The pathways that form electric circuits are made from a conducting material, normally a metal in macroscopic circuits. For example, copper wires inside your school building form the electrical circuits that power lighting, projectors, screens, speakers, etc. To represent an electric circuit, we draw circuit diagrams. We use lines and symbols to represent the elements in the circuit. A simple electric circuit diagram is shown on the left side of . On the right side is an analogous water circuit, which we discuss below.
There are many different symbols that scientists and engineers use in circuit diagrams, but we will focus on four main symbols: the wire, the battery or voltage source, resistors, and the ground. The thin black lines in the electric circuit diagram represent the pathway that the electric charge must follow. These pathways are assumed to be perfect conductors, so electric charge can move along these pathways without losing any energy. In reality, the wires in circuits are not perfect, but they come close enough for our purposes.
The zigzag element labeled R is a resistor, which is a circuit element that provides a known resistance. Macroscopic resistors are often color coded to indicate their resistance, as shown in .
The red element in  is a battery, with its positive and negative terminals indicated; the longer line represents the positive terminal of the battery, and the shorter line represents the negative terminal. Note that the battery icon is not always colored red; this is done in  just to make it easy to identify.
Finally, the element labeled ground on the lower left of the circuit indicates that the circuit is connected to Earth, which is a large, essentially neutral object containing an infinite amount of charge. Among other things, the ground determines the potential of the negative terminal of the battery. Normally, the potential of the ground is defined to be zero:
  . This means that the entire lower wire in  is at a voltage of zero volts.
The electric current in  is indicted by the blue line labeled I. The arrow indicates the direction in which positive charge would flow in this circuit. Recall that, in metals, electrons are mobile charge carriers, so negative charges actually flow in the opposite direction around this circuit (i.e., counterclockwise). However, we draw the current to show the direction in which positive charge would move.
On the right side of  is an analogous water circuit. Water at a higher pressure leaves the top of the pump, which is like charges leaving the positive terminal of the battery. The water travels through the pipe, like the charges traveling through the wire. Next, the water goes through a sand filter, which heats up as the water squeezes through. This step is like the charges going through the resistor. When charges flow through a resistor, they do work to heat up the resistor. After flowing through the sand filter, the water has converted its potential energy into heat, so it is at a lower pressure. Likewise, the charges exiting the resistor have converted their potential energy into heat, so they are at a lower voltage. Recall that voltage is just potential energy per charge. Thus, water pressure is analogous to electric potential energy (i.e., voltage). Coming back to the water circuit again, we see that the water returns to the bottom of the pump, which is like the charge returning to the negative terminal of the battery. The water pump uses a source of energy to pump the water back up to a high pressure again, giving it the pressure required to go through the circuit once more. The water pump is like the battery, which uses chemical energy to increase the voltage of the charge up to the level of the positive terminal.
The potential energy per charge at the positive terminal of the battery is the voltage rating of the battery. This voltage is like water pressure in the upper pipe. Just like a higher pressure forces water to move toward a lower pressure, a higher voltage forces electric charge to flow toward a lower voltage. The pump takes water at low pressure and does work on it, ejecting water at a higher pressure. Likewise, a battery takes charge at a low voltage, does work on it, and ejects charge at a higher voltage.
Note that the current in the water circuit of  is the same throughout the circuit. In other words, if we measured the number of water molecules passing a cross-section of the pipe per unit time at any point in the circuit, we would get the same answer no matter where in the circuit we measured. The same is true of the electrical circuit in the same figure. The electric current is the same at all points in this circuit, including inside the battery and in the resistor. The electric current neither speeds up in the wires nor slows down in the resistor. This would create points where too much or too little charge would be bunched up. Thus, the current is the same at all points in the circuit shown in .
Although the current is the same everywhere in both the electric and water circuits, the voltage or water pressure changes as you move through the circuits. In the water circuit, the water pressure at the pump outlet stays the same until the water goes through the sand filter, assuming no energy loss in the pipe. Likewise, the voltage in the electrical circuit is the same at all points in a given wire, because we have assumed that the wires are perfect conductors. Thus, as indicated by the constant red color of the upper wire in , the voltage throughout this wire is constant at
  . The voltage then drops as you go through the resistor, but once you reach the blue wire, the voltage stays at its new level of
   all the way to the negative terminal of the battery (i.e., the blue terminal of the battery).
If we go from the blue wire through the battery to the red wire, the voltage increases from

 to
  . Likewise, if we go from the blue wire up through the resistor to the red wire, the voltage also goes from
   to
  . Thus, using Ohm’s law, we can write

Note that
   is measured from the bottom of the resistor to the top, meaning that the top of the resistor is at a higher voltage than the bottom of the resistor. Thus, current flows from the top of the resistor or higher voltage to the bottom of the resistor or lower voltage.
Other possible circuit elements include capacitors and switches. These are drawn as shown on the left side of . A switch is a device that opens and closes the circuit, like a light switch. It is analogous to a valve in a water circuit, as shown on the right side of . With the switch open, no current passes through the circuit. With the switch closed, it becomes part of the wire, so the current passes through it with no loss of voltage.
The capacitor is labeled C on the left of . A capacitor in an electrical circuit is analogous to a flexible membrane in a water circuit. When the switch is closed in the circuit of , the battery forces electrical current to flow toward the capacitor, charging the upper capacitor plate with positive charge. As this happens, the voltage across the capacitor plates increases. This is like the membrane in the water circuit: When the valve is opened, the pump forces water to flow toward the membrane, making it stretch to store the excess water. As this happens, the pressure behind the membrane increases.
Now if we open the switch, the capacitor holds the voltage between its plates because the charges have nowhere to go. Likewise, if we close the valve, the water has nowhere to go and the membrane maintains the water pressure in the pipe between itself and the valve.
If the switch is closed for a long time in the electric circuit or if the valve is open for a long time in the water circuit, the current will eventually stop flowing because the capacitor or the membrane will have become completely charged. Each circuit is now in the steady state, which means that its characteristics do not change over time. In this case, the steady state is characterized by zero current, and this does not change as long as the switch or valve remains in the same position. In the steady state, no electrical current passes through the capacitor, and no water current passes through the membrane. The voltage difference between the capacitor plates will be the same as the battery voltage. In the water circuit, the pressure behind the membrane will be the same as the pressure created by the pump.
Although the circuit in  may seem a bit pointless because all that happens when the switch is closed is that the capacitor charges up, it does show the capacitor’s ability to store charge. Thus, the capacitor serves as a reservoir for charge. This property of capacitors is used in circuits in many ways. For example, capacitors are used to power circuits while batteries are being charged. In addition, capacitors can serve as filters. To understand this, let’s go back to the water analogy. Suppose you have a water hose and are watering your garden. Your friend thinks he’s funny, and kinks the hose. While the hose is kinked, you experience no water flow. When he lets go, the water starts flowing again. If he does this really fast, you experience water-no water-water-no water, and that’s really no way to water your garden. Now imagine that the hose is filling up a big bucket, and you are watering from the bottom of the bucket. As long as you had water in your bucket to begin with and your friend doesn’t kink the water hose for too long, you would be able to water your garden without the interruptions. Your friend kinking the water hose is filtered by the big bucket’s supply of water, so it does not impact your ability to water the garden. We can think of the interruptions in the current (be it water or electrical current) as noise. Capacitors act in an analogous way as the water bucket to help filter out the noise. Capacitors have so many uses that it is very rare to find an electronic circuit that does not include some capacitors.

### Resistors in Series and Equivalent Resistance

Now that we have a basic idea of how electrical circuits work, let’s see what happens in circuits with more than one circuit element. In this section, we look at resistors in series. Components connected in series are connected one after the other in the same branch of a circuit, such as the resistors connected in series on the left side of .
We will now try to find a single resistance that is equivalent to the three resistors in series on the left side of . An equivalent resistor is a resistor that has the same resistance as the combined resistance of a set of other resistors. In other words, the same current will flow through the left and right circuits in  if we use the equivalent resistor in the right circuit.
According to Ohm’s law, the voltage drop V across a resistor when a current flows through it is
   where I is the current in amperes (A) and R is the resistance in ohms (
  ). Another way to think of this is that V is the voltage necessary to make a current I flow through a resistance R. Applying Ohm’s law to each resistor on the left circuit of , we find that the voltage drop across
   is
 , that across
   is
 , and that across
   is
  . The sum of these voltages equals the voltage output of the battery, that is
You may wonder why voltages must add up like this. One way to understand this is to go once around the circuit and add up the successive changes in voltage. If you do this around a loop and get back to the starting point, the total change in voltage should be zero, because you end up at the same place that you started. To better understand this, consider the analogy of going for a stroll through some hilly countryside. If you leave your car and walk around, then come back to your car, the total height you gained in your stroll must be the same as the total height you lost, because you end up at the same place as you started. Thus, the gravitational potential energy you gain must be the same as the gravitational potential energy you lose. The same reasoning holds for voltage in going around an electric circuit. Let’s apply this reasoning to the left circuit in . We start just below the battery and move up through the battery, which contributes a voltage gain of
  . Next, we got through the resistors. The voltage drops by
   in going through resistor
 , by
   in going through resistor
 , and by
   in going through resistor
  . After going through resistor
 , we arrive back at the starting point, so we add up these four changes in voltage and set the sum equal to zero. This gives
which is the same as the previous equation. Note that the minus signs in front of
   are because these are voltage drops, whereas
   is a voltage rise.
Ohm’s law tells us that
 ,
 , and
  . Inserting these values into equation
   gives
Applying this same logic to the right circuit in
 gives
Dividing the equation
   by
 , we get
This shows that the equivalent resistance for a series of resistors is simply the sum of the resistances of each resistor. In general, N resistors connected in series can be replaced by an equivalent resistor with a resistance of

### Check your Understanding

### Section Summary

1. Circuit diagrams are schematic representations of electric circuits.
2. Resistors in series are resistors that are connected head to tail.
3. The same current runs through all resistors in series; however, the voltage drop across each resistor can be different.
4. The voltage is the same at every point in a given wire.

### Key Equations

### Concept Items

### Critical Thinking

### Problems

### Test Prep Multiple Choice

### Test Prep Short Answer

### Test Prep Extended Response


# Electrical Circuits

## Parallel Circuits

### Section Key Terms

### Resistors in Parallel

In the previous section, we learned that resistors in series are resistors that are connected one after the other. If we instead combine resistors by connecting them next to each other, as shown in , then the resistors are said to be connected in parallel. Resistors are in parallel when both ends of each resistor are connected directly together.
Note that the tops of the resistors are all connected to the same wire, so the voltage at the top of the each resistor is the same. Likewise, the bottoms of the resistors are all connected to the same wire, so the voltage at the bottom of each resistor is the same. This means that the voltage drop across each resistor is the same. In this case, the voltage drop is the voltage rating V of the battery, because the top and bottom wires connect to the positive and negative terminals of the battery, respectively.
Although the voltage drop across each resistor is the same, we cannot say the same for the current running through each resistor. Thus,
   are not necessarily the same, because the resistors
   do not necessarily have the same resistance.
Note that the three resistors in  provide three different paths through which the current can flow. This means that the equivalent resistance for these three resistors must be less than the smallest of the three resistors. To understand this, imagine that the smallest resistor is the only path through which the current can flow. Now add on the alternate paths by connecting other resistors in parallel. Because the current has more paths to go through, the overall resistance (i.e., the equivalent resistance) will decrease. Therefore, the equivalent resistance must be less than the smallest resistance of the parallel resistors.
To find the equivalent resistance
   of the three resistors
 , we apply Ohm’s law to each resistor. Because the voltage drop across each resistor is V, we obtain
or
We also know from conservation of charge that the three currents
   must add up to give the current I that goes through the battery. If this were not true, current would have to be mysteriously created or destroyed somewhere in the circuit, which is physically impossible. Thus, we have
Inserting the expressions for
   into this equation gives
or
This formula is just Ohm’s law, with the factor in parentheses being the equivalent resistance.
Thus, the equivalent resistance for three resistors in parallel is
The same logic works for any number of resistors in parallel, so the general form of the equation that gives the equivalent resistance of N resistors connected in parallel is

### Practice Problems

### Resistors in Parallel and in Series

More complex connections of resistors are sometimes just combinations of series and parallel. Combinations of series and parallel resistors can be reduced to a single equivalent resistance by using the technique illustrated in . Various parts are identified as either series or parallel, reduced to their equivalents, and further reduced until a single resistance is left. The process is more time consuming than difficult.
Let’s work through the four steps in  to reduce the seven resistors to a single equivalent resistor. To avoid distracting algebra, we’ll assume each resistor is 10
  . In step 1, we reduce the two sets of parallel resistors circled by the blue dashed loop. The upper set has three resistors in parallel and will be reduced to a single equivalent resistor
  . The lower set has two resistors in parallel and will be reduced to a single equivalent resistor
  . Using the equation for the equivalent resistance of resistors in parallel, we obtain
These two equivalent resistances are encircled by the red dashed loop following step 1. They are in series, so we can use the equation for the equivalent resistance of resistors in series to reduce them to a single equivalent resistance
  . This is done in step 2, with the result being
The equivalent resistor
   appears in the green dashed loop following step 2. This resistor is in parallel with resistor
 , so the pair can be replaced by the equivalent resistor
 , which is given by
This is done in step 3. The resistor
   is in series with the resistor
 , as shown in the purple dashed loop following step 3. These two resistors are combined in the final step to form the final equivalent resistor
 , which is
Thus, the entire combination of seven resistors may be replaced by a single resistor with a resistance of about 14.5
  .
That was a lot of work, and you might be asking why we do it. It’s important for us to know the equivalent resistance of the entire circuit so that we can calculate the current flowing through the circuit. Ohm’s law tells us that the current flowing through a circuit depends on the resistance of the circuit and the voltage across the circuit. But to know the current, we must first know the equivalent resistance.
Here is a general approach to find the equivalent resistor for any arbitrary combination of resistors:
1. Identify a group of resistors that are only in parallel or only in series.
2. For resistors in series, use the equation for the equivalent resistance of resistors in series to reduce them to a single equivalent resistance. For resistors in parallel, use the equation for the equivalent resistance of resistors in parallel to reduce them to a single equivalent resistance.
3. Draw a new circuit diagram with the resistors from step 1 replaced by their equivalent resistor.
4. If more than one resistor remains in the circuit, return to step 1 and repeat. Otherwise, you are finished.

### Practice Problems

### Check Your Understanding

### Section Summary

1. The equivalent resistance of a group of N identical resistors R connected in parallel is R/N.
2. Connecting resistors in parallel provides more paths for the current to go through, so the equivalent resistance is always less than the smallest resistance of the parallel resistors.
3. The same voltage drop occurs across all resistors in parallel; however, the current through each resistor can differ.

### Key Equations

### Concept Items

### Critical Thinking

### Problems

### Test Prep Multiple Choice

### Test Prep Short Answer

### Test Prep Extended Response