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# Chapter 5. Electricity

Where the telescope ends, the microscope begins. Which of the two has the grander view? -- Victor Hugo

His father died during his mother's pregnancy. Rejected by her as a boy, he was packed off to boarding school when she remarried. He himself never married, but in middle age he formed an intense relationship with a much younger man, a relationship that he terminated when he underwent a psychotic break. Following his early scientific successes, he spent the rest of his professional life mostly in frustration over his inability to unlock the secrets of alchemy.

The man being described is Isaac Newton, but not the triumphant Newton of the standard textbook hagiography. Why dwell on the sad side of his life? To the modern science educator, Newton's lifelong obsession with alchemy may seem an embarrassment, a distraction from his main achievement, the creation the modern science of mechanics. To Newton, however, his alchemical researches were naturally related to his investigations of force and motion. What was radical about Newton's analysis of motion was its universality: it succeeded in describing both the heavens and the earth with the same equations, whereas previously it had been assumed that the sun, moon, stars, and planets were fundamentally different from earthly objects. But Newton realized that if science was to describe all of nature in a unified way, it was not enough to unite the human scale with the scale of the universe: he would not be satisfied until he fit the microscopic universe into the picture as well.

It should not surprise us that Newton failed. Although he was a firm believer in the existence of atoms, there was no more experimental evidence for their existence than there had been when the ancient Greeks first posited them on purely philosophical grounds. Alchemy labored under a tradition of secrecy and mysticism. Newton had already almost single-handedly transformed the fuzzyheaded field of “natural philosophy” into something we would recognize as the modern science of physics, and it would be unjust to criticize him for failing to change alchemy into modern chemistry as well. The time was not ripe. The microscope was a new invention, and it was cutting-edge science when Newton's contemporary Hooke discovered that living things were made out of cells.

## 5.1 The Quest for the Atomic Force

a / Four pieces of tape are prepared, 1, as described in the text. Depending on which combination is tested, the interaction can be either repulsive, 2, or attractive, 3.

Newton was not the first of the age of reason. He was the last of the magicians. -- John Maynard Keynes

Nevertheless it will be instructive to pick up Newton's train of thought and see where it leads us with the benefit of modern hindsight. In uniting the human and cosmic scales of existence, he had reimagined both as stages on which the actors were objects (trees and houses, planets and stars) that interacted through attractions and repulsions. He was already convinced that the objects inhabiting the microworld were atoms, so it remained only to determine what kinds of forces they exerted on each other.

His next insight was no less brilliant for his inability to bring it to fruition. He realized that the many human-scale forces --- friction, sticky forces, the normal forces that keep objects from occupying the same space, and so on --- must all simply be expressions of a more fundamental force acting between atoms. Tape sticks to paper because the atoms in the tape attract the atoms in the paper. My house doesn't fall to the center of the earth because its atoms repel the atoms of the dirt under it.

Here he got stuck. It was tempting to think that the atomic force was a form of gravity, which he knew to be universal, fundamental, and mathematically simple. Gravity, however, is always attractive, so how could he use it to explain the existence of both attractive and repulsive atomic forces? The gravitational force between objects of ordinary size is also extremely small, which is why we never notice cars and houses attracting us gravitationally. It would be hard to understand how gravity could be responsible for anything as vigorous as the beating of a heart or the explosion of gunpowder. Newton went on to write a million words of alchemical notes filled with speculation about some other force, perhaps a “divine force” or “vegetative force” that would for example be carried by the sperm to the egg.

Luckily, we now know enough to investigate a different suspect as a candidate for the atomic force: electricity. Electric forces are often observed between objects that have been prepared by rubbing (or other surface interactions), for instance when clothes rub against each other in the dryer. A useful example is shown in figure 5.1/1: stick two pieces of tape on a tabletop, and then put two more pieces on top of them. Lift each pair from the table, and then separate them. The two top pieces will then repel each other, 5.1/2, as will the two bottom pieces. A bottom piece will attract a top piece, however, 5.1/3. Electrical forces like these are similar in certain ways to gravity, the other force that we already know to be fundamental:

• Electrical forces are universal. Although some substances, such as fur, rubber, and plastic, respond more strongly to electrical preparation than others, all matter participates in electrical forces to some degree. There is no such thing as a “nonelectric” substance. Matter is both inherently gravitational and inherently electrical.
• Experiments show that the electrical force, like the gravitational force, is an inverse square force. That is, the electrical force between two spheres is proportional to $$1/r^2$$, where $$r$$ is the center-to-center distance between them.

Furthermore, electrical forces make more sense than gravity as candidates for the fundamental force between atoms, because we have observed that they can be either attractive or repulsive.

## 5.2 Charge, Electricity and Magnetism

b / A charged piece of tape attracts uncharged pieces of paper from a distance, and they leap up to it.

c / The paper has zero total charge, but it does have charged particles in it that can move.

d / Examples of the construction of atoms: hydrogen (top) and helium (bottom). On this scale, the electrons' orbits would be the size of a college campus.

e / André Marie Ampère (1775-1836).

### Charge

“Charge” is the technical term used to indicate that an object participates in electrical forces. This is to be distinguished from the common usage, in which the term is used indiscriminately for anything electrical. For example, although we speak colloquially of “charging” a battery, you may easily verify that a battery has no charge in the technical sense, e.g., it does not exert any electrical force on a piece of tape that has been prepared as described in section 5.1.

#### Two types of charge

We can easily collect reams of data on electrical forces between different substances that have been charged in different ways. We find for example that cat fur prepared by rubbing against rabbit fur will attract glass that has been rubbed on silk. How can we make any sense of all this information? A vast simplification is achieved by noting that there are really only two types of charge. Suppose we pick cat fur rubbed on rabbit fur as a representative of type A, and glass rubbed on silk for type B. We will now find that there is no “type C.” Any object electrified by any method is either A-like, attracting things A attracts and repelling those it repels, or B-like, displaying the same attractions and repulsions as B. The two types, A and B, always display opposite interactions. If A displays an attraction with some charged object, then B is guaranteed to undergo repulsion with it, and vice-versa.

#### The coulomb

Although there are only two types of charge, each type can come in different amounts. The metric unit of charge is the coulomb (rhymes with “drool on”), defined as follows:

One Coulomb (C) is the amount of charge such that a force of $$9.0\times10^9$$ N occurs between two pointlike objects with charges of 1 C separated by a distance of 1 m.

The notation for an amount of charge is $$q$$. The numerical factor in the definition is historical in origin, and is not worth memorizing. The definition is stated for pointlike, i.e., very small, objects, because otherwise different parts of them would be at different distances from each other.

#### A model of two types of charged particles

Experiments show that all the methods of rubbing or otherwise charging objects involve two objects, and both of them end up getting charged. If one object acquires a certain amount of one type of charge, then the other ends up with an equal amount of the other type. Various interpretations of this are possible, but the simplest is that the basic building blocks of matter come in two flavors, one with each type of charge. Rubbing objects together results in the transfer of some of these particles from one object to the other. In this model, an object that has not been electrically prepared may actually possesses a great deal of both types of charge, but the amounts are equal and they are distributed in the same way throughout it. Since type A repels anything that type B attracts, and vice versa, the object will make a total force of zero on any other object. The rest of this chapter fleshes out this model and discusses how these mysterious particles can be understood as being internal parts of atoms.

#### Use of positive and negative signs for charge

Because the two types of charge tend to cancel out each other's forces, it makes sense to label them using positive and negative signs, and to discuss the total charge of an object. It is entirely arbitrary which type of charge to call negative and which to call positive. Benjamin Franklin decided to describe the one we've been calling “A” as negative, but it really doesn't matter as long as everyone is consistent with everyone else. An object with a total charge of zero (equal amounts of both types) is referred to as electrically $$neutral$$.

self-check:

Criticize the following statement: “There are two types of charge, attractive and repulsive.”

#### Coulomb's law

A large body of experimental observations can be summarized as follows:

Coulomb's law: The magnitude of the force acting between pointlike charged objects at a center-to-center distance $$r$$ is given by the equation

$\begin{equation*} |\mathbf{F}| = k\frac{|q_1||q_2|}{r^2} , \end{equation*}$

where the constant $$k$$ equals $$9.0\times10^9\ \text{N}\!\cdot\!\text{m}^2/\text{C}^2$$. The force is attractive if the charges are of different signs, and repulsive if they have the same sign.

### Conservation of charge

An even more fundamental reason for using positive and negative signs for electrical charge is that experiments show that with the signs defined this way, the total amount of charge is a conserved quantity. This is why we observe that rubbing initially uncharged substances together always has the result that one gains a certain amount of one type of charge, while the other acquires an equal amount of the other type. Conservation of charge seems natural in our model in which matter is made of positive and negative particles. If the charge on each particle is a fixed property of that type of particle, and if the particles themselves can be neither created nor destroyed, then conservation of charge is inevitable.

### Electrical forces involving neutral objects

As shown in figure 5.2.3, an electrically charged object can attract objects that are uncharged. How is this possible? The key is that even though each piece of paper has a total charge of zero, it has at least some charged particles in it that have some freedom to move. Suppose that the tape is positively charged, 5.2.3. Mobile particles in the paper will respond to the tape's forces, causing one end of the paper to become negatively charged and the other to become positive. The attraction between the paper and the tape is now stronger than the repulsion, because the negatively charged end is closer to the tape.

self-check:

What would have happened if the tape was negatively charged?

### The atom, and subatomic particles

I once had a student whose father had been an electrician. He told me that his father had never really believed that an electrical current in a wire could be carried by moving electrons, because the wire was solid, and it seemed to him that physical particles moving through it would eventually have drilled so many holes through it that it would have crumbled. It may sound as though I'm trying to make fun of the father, but actually he was behaving very much like the model of the skeptical scientist: he didn't want to make hypotheses that seemed more complicated than would be necessary in order to explain his observations. Physicists before about 1905 were in exactly the same situation. They knew all about electrical circuits, and had even invented radio, but knew absolutely nothing about subatomic particles. In other words, it hardly ever matters that electricity really is made of charged particles, and it hardly ever matters what those particles are. Nevertheless, it may avoid some confusion to give a brief review of how an atom is put together:

 charge mass in units of the proton’s mass location in atom proton + e 1 in nucleus neutron 0 1.001 in nucleus electron − e 1/1836 orbiting nucleus

The symbol $$e$$ in this table is an abbreviation for $$1.60\times10^{-19}$$ C. The physicist Robert Millikan discovered in 1911 that any material object (he used oil droplets) would have a charge that was a multiple of this number, and today we interpret that as being a consequence of the fact that matter is made of atoms, and atoms are made of particles whose charges are plus and minus this amount.

### Electric current

If the fundamental phenomenon is the motion of charged particles, then how can we define a useful numerical measurement of it? We might describe the flow of a river simply by the velocity of the water, but velocity will not be appropriate for electrical purposes because we need to take into account how much charge the moving particles have, and in any case there are no practical devices sold at Radio Shack that can tell us the velocity of charged particles. Experiments show that the intensity of various electrical effects is related to a different quantity: the number of coulombs of charge that pass by a certain point per second. By analogy with the flow of water, this quantity is called the electric current, $$I$$. Its units of coulombs/second are more conveniently abbreviated as amperes,
[3] 1 A=1 C/s. (In informal speech, one usually says “amps.”)

The main subtlety involved in this definition is how to account for the two types of charge. The stream of water coming from a hose is made of atoms containing charged particles, but it produces none of the effects we associate with electric currents. For example, you do not get an electrical shock when you are sprayed by a hose. This type of experiment shows that the effect created by the motion of one type of charged particle can be canceled out by the motion of the opposite type of charge in the same direction. In water, every oxygen atom with a charge of $$+8e$$ is surrounded by eight electrons with charges of $$-e$$, and likewise for the hydrogen atoms.

We therefore refine our definition of current as follows:

##### definition of electric current

When charged particles are exchanged between regions of space A and B, the electric current flowing from A to B is

$\begin{equation*} I = \frac{\text{change in B's charge}}{t} , \end{equation*}$

where the transfer occurs over a period of time $$t$$.

In the garden hose example, your body picks up equal amounts of positive and negative charge, resulting in no change in your total charge, so the electrical current flowing into you is zero.

##### Example 1: Ions moving across a cell membrane
$$\triangleright$$ Figure f shows ions, labeled with their charges, moving in or out through the membranes of four cells. If the ions all cross the membranes during the same interval of time, how would the currents into the cells compare with each other?

$$\triangleright$$ Cell A has positive current going into it because its charge is increased, i.e., has a positive change in its charge.

Cell B has the same current as cell A, because by losing one unit of negative charge it also ends up increasing its own total charge by one unit.

Cell C's total charge is reduced by three units, so it has a large negative current going into it.

Cell D loses one unit of charge, so it has a small negative current into it.

f / Example 1

It may seem strange to say that a negatively charged particle going one way creates a current going the other way, but this is quite ordinary. As we will see, currents flow through metal wires via the motion of electrons, which are negatively charged, so the direction of motion of the electrons in a circuit is always opposite to the direction of the current. Of course it would have been convenient if Benjamin Franklin had defined the positive and negative signs of charge the opposite way, since so many electrical devices are based on metal wires.

##### Example 2: Number of electrons flowing through a lightbulb

$$\triangleright$$ If a lightbulb has 1.0 A flowing through it, how many electrons will pass through the filament in 1.0 s?

$$\triangleright$$ We are only calculating the number of electrons that flow, so we can ignore the positive and negative signs. Solving for $$(\text{charge})= I t$$ gives a charge of 1.0 C flowing in this time interval. The number of electrons is

\begin{align*} \text{number of electrons} &= \text{coulombs}\times\frac{\text{electrons}}{\text{coulomb}} \\ &= \text{coulombs}/\frac{\text{coulombs}}{\text{electron}} \\ &= 1.0\ \text{C} / e \\ &= 6.2\times10^{18} \end{align*}

That's a lot of electrons!

## 5.3 Circuits

g / 1. Static electricity runs out quickly. 2. A practical circuit. 3. An open circuit. 4. How an ammeter works. 5. Measuring the current with an ammeter.

How can we put electric currents to work? The only method of controlling electric charge we have studied so far is to charge different substances, e.g., rubber and fur, by rubbing them against each other. Figure g/1 shows an attempt to use this technique to light a lightbulb. This method is unsatisfactory. True, current will flow through the bulb, since electrons can move through metal wires, and the excess electrons on the rubber rod will therefore come through the wires and bulb due to the attraction of the positively charged fur and the repulsion of the other electrons. The problem is that after a zillionth of a second of current, the rod and fur will both have run out of charge. No more current will flow, and the lightbulb will go out.

Figure g/2 shows a setup that works. The battery pushes charge through the circuit, and recycles it over and over again. (We will have more to say later in this chapter about how batteries work.) This is called a complete circuit. Today, the electrical use of the word “circuit” is the only one that springs to mind for most people, but the original meaning was to travel around and make a round trip, as when a circuit court judge would ride around the boondocks, dispensing justice in each town on a certain date.

Note that an example like g/3 does not work. The wire will quickly begin acquiring a net charge, because it has no way to get rid of the charge flowing into it. The repulsion of this charge will make it more and more difficult to send any more charge in, and soon the electrical forces exerted by the battery will be canceled out completely. The whole process would be over so quickly that the filament would not even have enough time to get hot and glow. This is known as an open circuit. Exactly the same thing would happen if the complete circuit of figure g/2 was cut somewhere with a pair of scissors, and in fact that is essentially how an ordinary light switch works: by opening up a gap in the circuit.

The definition of electric current we have developed has the great virtue that it is easy to measure. In practical electrical work, one almost always measures current, not charge. The instrument used to measure current is called an ammeter. A simplified ammeter, g/4, simply consists of a coiled-wire magnet whose force twists an iron needle against the resistance of a spring. The greater the current, the greater the force. Although the construction of ammeters may differ, their use is always the same. We break into the path of the electric current and interpose the meter like a tollbooth on a road, g/5. There is still a complete circuit, and as far as the battery and bulb are concerned, the ammeter is just another segment of wire.

Does it matter where in the circuit we place the ammeter? Could we, for instance, have put it in the left side of the circuit instead of the right? Conservation of charge tells us that this can make no difference. Charge is not destroyed or “used up” by the lightbulb, so we will get the same current reading on either side of it. What is “used up” is energy stored in the battery, which is being converted into heat and light energy.

## 5.4 Voltage

h / Alessandro Volta (1745-1827).

i / Example 3.

### The volt unit

Electrical circuits can be used for sending signals, storing information, or doing calculations, but their most common purpose by far is to manipulate energy, as in the battery-and-bulb example of the previous section. We know that lightbulbs are rated in units of watts, i.e., how many joules per second of energy they can convert into heat and light, but how would this relate to the flow of charge as measured in amperes? By way of analogy, suppose your friend, who didn't take physics, can't find any job better than pitching bales of hay. The number of calories he burns per hour will certainly depend on how many bales he pitches per minute, but it will also be proportional to how much mechanical work he has to do on each bale. If his job is to toss them up into a hayloft, he will got tired a lot more quickly than someone who merely tips bales off a loading dock into trucks. In metric units,

$\begin{equation*} \frac{\text{joules}}{\text{second}} = \frac{\text{haybales}}{\text{second}} \times \frac{\text{joules}}{\text{haybale}} . \end{equation*}$

Similarly, the rate of energy transformation by a battery will not just depend on how many coulombs per second it pushes through a circuit but also on how much mechanical work it has to do on each coulomb of charge:

$\begin{equation*} \frac{\text{joules}}{\text{second}} = \frac{\text{coulombs}}{\text{second}} \times \frac{\text{joules}}{\text{coulomb}} \end{equation*}$

or

$\begin{equation*} \text{power} = \text{current} \times \text{work per unit charge} . \end{equation*}$

Units of joules per coulomb are abbreviated as volts, 1 V=1 J/C, named after the Italian physicist Alessandro Volta. Everyone knows that batteries are rated in units of volts, but the voltage concept is more general than that; it turns out that voltage is a property of every point in space.

To gain more insight, let's think again about the analogy with the haybales. It took a certain number of joules of gravitational energy to lift a haybale from one level to another. Since we're talking about gravitational energy, it really makes more sense to talk about units of mass, rather than using the haybale as our measure of the quantity of matter. The gravitational version of voltage would then be joules per kilogram. Gravitational energy equals $$mgh$$, but if we calculate how much of that we have per kilogram, we're canceling out the $$m$$, giving simply $$gh$$. For any point in the Earth's gravitational field, we can assign a number, $$gh$$, which tells us how hard it is to get a given amount of mass to that point. For instance, the top of Mount Everest would have a big value of $$gh$$, because of the big height. That tells us that it's expensive in terms of energy to lift a given amount of mass from some reference level (sea level, say) to the top of Mount Everest.

Voltage does the same thing, but using electrical energy. We can visualize an electrical circuit as being like a roller-coaster. The battery is like the part of the roller-coaster where they lift you up to the top. The height of this initial hill is analogous to the voltage of the battery. When you roll downhill later, that's like a lightbulb. In the roller-coaster, the initial gravitational energy is turned into heat and sound as the cars go down the hill. In our circuit, the initial electrical energy is turned into heat by the lightbulb (and the hot filament of the lightbulb then glows, turning the heat into light).

##### Example 3: Energy stored in a battery
$$\triangleright$$ The 1.2 V rechargeable battery in figure i is labeled 1800 milliamp-hours. What is the maximum amount of energy the battery can store?

$$\triangleright$$ An ampere-hour is a unit of current multiplied by a unit of time. Current is charge per unit time, so an ampere-hour is in fact a funny unit of charge:

\begin{align*} \text{(1 A)(1 hour)} &= \text{(1 C/s)(3600 s)} \\ &= \text{3600 C} \end{align*}

1800 milliamp-hours is therefore $$1800\times10^{-3}\times 3600\ \text{C}=6.5\times10^3\ \text{C}$$. That's a huge number of charged particles, but the total loss of electrical energy will just be their total charge multiplied by the voltage difference across which they move:

\begin{align*} \text{energy} &= (6.5\times10^3\ \text{C})(1.2\ \text{V}) \\ &= 7.8\ \text{kJ} \end{align*}

Using the definition of voltage, $$V$$, we can rewrite the equation $$\text{power} = \text{current} \times \text{work per unit charge}$$ more concisely as $$P=IV$$.

##### Example 4: Units of volt-amps

$$\triangleright$$ Doorbells are often rated in volt-amps. What does this combination of units mean?

$$\triangleright$$ Current times voltage gives units of power, $$P= I V$$, so volt-amps are really just a nonstandard way of writing watts. They are telling you how much power the doorbell requires.

##### Example 5: Power dissipated by a battery and bulb

$$\triangleright$$ If a 9.0-volt battery causes 1.0 A to flow through a lightbulb, how much power is dissipated?

$$\triangleright$$ The voltage rating of a battery tells us what voltage difference $$\Delta V$$ it is designed to maintain between its terminals.

\begin{align*} P &= I \ \Delta \text{V} \\ &= 9.0 \ \text{A}\cdot\text{V} \\ &= 9.0 \ \frac{\text{C}}{\text{s}}\cdot\frac{\text{J}}{\text{C}} \\ &= 9.0 \ \text{J/s} \\ &= 9.0 \ \text{W} \end{align*}

The only nontrivial thing in this problem was dealing with the units. One quickly gets used to translating common combinations like $$\text{A}\cdot\text{V}$$ into simpler terms.

##### Discussion Questions

In the roller-coaster metaphor, what would a high-voltage roller coaster be like? What would a high-current roller coaster be like?

Criticize the following statements:

• “He touched the wire, and 10000 volts went through him.”
• “That battery has a charge of 9 volts.”
• “You used up the charge of the battery.”

When you touch a 9-volt battery to your tongue, both positive and negative ions move through your saliva. Which ions go which way?

I once touched a piece of physics apparatus that had been wired incorrectly, and got a several-thousand-volt voltage difference across my hand. I was not injured. For what possible reason would the shock have had insufficient power to hurt me?

## 5.5 Resistance

k / Georg Simon Ohm (1787-1854).

l / 1. A simplified diagram of how a voltmeter works. 2. Measuring the voltage difference across a lightbulb. 3. The same setup drawn in schematic form. 4. The setup for measuring current is different.

What's the physical difference between a 100-watt lightbulb and a 200-watt one? They both plug into a 110-volt outlet, so according to the equation $$P=IV$$, the only way to explain the double power of the 200-watt bulb is that it must pull in, or “draw,” twice as much current. By analogy, a fire hose and a garden hose might be served by pumps that give the same pressure (voltage), but more water will flow through the fire hose, because there's simply more water in the hose that can flow. Likewise, a wide, deep river could flow down the same slope as a tiny creek, but the number of liters of water flowing through the big river is greater. If you look at the filaments of a 100-watt bulb and a 200-watt bulb, you'll see that the 200-watt bulb's filament is thicker. In the charged-particle model of electricity, we expect that the thicker filament will contain more charged particles that are available to flow. We say that the thicker filament has a lower electrical resistance than the thinner one.

j / A fat pipe has less resistance than a skinny pipe.

Although it's harder to pump water rapidly through a garden hose than through a fire hose, we could always compensate by using a higher-pressure pump. Similarly, the amount of current that will flow through a lightbulb depends not just on its resistance but also on how much of a voltage difference is applied across it. For many substances, including the tungsten metal that lightbulb filaments are made of, we find that the amount of current that flows is proportional to the voltage difference applied to it, so that the ratio of voltage to current stays the same. We then use this ratio as a numerical definition of resistance,

$\begin{equation*} R = \frac{V}{I} , \end{equation*}$

which is known as Ohm's law. The units of resistance are ohms, symbolized with an uppercase Greek letter Omega, $$\Omega$$. Physically, when a current flows through a resistance, the result is to transform electrical energy into heat. In a lightbulb filament, for example, the heat is what causes the bulb to glow.

Ohm's law states that many substances, including many solids and some liquids, display this kind of behavior, at least for voltages that are not too large. The fact that Ohm's law is called a “law” should not be taken to mean that all materials obey it, or that it has the same fundamental importance as the conservation laws, for example. Materials are called ohmic or nonohmic, depending on whether they obey Ohm's law.

On an intuitive level, we can understand the idea of resistance by making the sounds “hhhhhh” and “ffffff.” To make air flow out of your mouth, you use your diaphragm to compress the air in your chest. The pressure difference between your chest and the air outside your mouth is analogous to a voltage difference. When you make the “h” sound, you form your mouth and throat in a way that allows air to flow easily. The large flow of air is like a large current. Dividing by a large current in the definition of resistance means that we get a small resistance. We say that the small resistance of your mouth and throat allows a large current to flow. When you make the “f” sound, you increase the resistance and cause a smaller current to flow. In this mechanical analogy, resistance is like friction: the air rubs against your lips. Mechanical friction converts mechanical forms of energy to heat, as when you rub your hands together. Electrical friction --- resistance --- converts electrical energy to heat.

If objects of the same size and shape made from two different ohmic materials have different resistances, we can say that one material is more resistive than the other, or equivalently that it is less conductive. Materials, such as metals, that are very conductive are said to be good conductors. Those that are extremely poor conductors, for example wood or rubber, are classified as insulators. There is no sharp distinction between the two classes of materials. Some, such as silicon, lie midway between the two extremes, and are called semiconductors.

### Applications

#### Superconductors

All materials display some variation in resistance according to temperature (a fact that is used in thermostats to make a thermometer that can be easily interfaced to an electric circuit). More spectacularly, most metals have been found to exhibit a sudden change to zero resistance when cooled to a certain critical temperature. They are then said to be superconductors. A current flowing through a superconductor doesn't create any heat at all.

Theoretically, superconductors should make a great many exciting devices possible, for example coiled-wire magnets that could be used to levitate trains. In practice, the critical temperatures of all metals are very low, and the resulting need for extreme refrigeration has made their use uneconomical except for such specialized applications as particle accelerators for physics research.

But scientists have recently made the surprising discovery that certain ceramics are superconductors at less extreme temperatures. The technological barrier is now in finding practical methods for making wire out of these brittle materials. Wall Street is currently investing billions of dollars in developing superconducting devices for cellular phone relay stations based on these materials. In 2001, the city of Copenhagen replaced a short section of its electrical power trunks with superconducing cables, and they are now in operation and supplying power to customers.

There is currently no satisfactory theory of superconductivity in general, although superconductivity in metals is understood fairly well. Unfortunately I have yet to find a fundamental explanation of superconductivity in metals that works at the introductory level.

#### Constant voltage throughout a conductor

The idea of a superconductor leads us to the question of how we should expect an object to behave if it is made of a very good conductor. Superconductors are an extreme case, but often a metal wire can be thought of as a perfect conductor, for example if the parts of the circuit other than the wire are made of much less conductive materials. What happens if the resistance equals zero in the equation

$\begin{equation*} R = \frac{V}{I} ? \end{equation*}$

The result of dividing two numbers can only be zero if the number on top equals zero. This tells us that if we pick any two points in a perfect conductor, the voltage difference between them must be zero. In other words, the entire conductor must be at the same voltage. Using the water metaphor, a perfect conductor is like a perfectly calm lake or canal, whose surface is flat. If you take an eyedropper and deposit a drop of water anywhere on the surface, it doesn't flow away, because the water is still. In electrical terms, a charge located anywhere in the interior of a perfect conductor will always feel a total electrical force of zero.

Suppose, for example, that you build up a static charge by scuffing your feet on a carpet, and then you deposit some of that charge onto a doorknob, which is a good conductor. How can all that charge be in the doorknob without creating any electrical force at any point inside it? The only possible answer is that the charge moves around until it has spread itself into just the right configuration. In this configuration, the forces exerted by all the charge on any charged particle within the doorknob exactly cancel out.

We can explain this behavior if we assume that the charge placed on the doorknob eventually settles down into a stable equilibrium. Since the doorknob is a conductor, the charge is free to move through it. If it was free to move and any part of it did experience a nonzero total force from the rest of the charge, then it would move, and we would not have an equilibrium.

It also turns out that charge placed on a conductor, once it reaches its equilibrium configuration, is entirely on the surface, not on the interior. We will not prove this fact formally, but it is intuitively reasonable (see discussion question B).

#### Short circuits

So far we have been assuming a perfect conductor. What if it's a good conductor, but not a perfect one? Then we can solve for

$\begin{equation*} V=IR . \end{equation*}$

An ordinary-sized current will make a very small result when we multiply it by the resistance of a good conductor such as a metal wire. The voltage throughout the wire will then be nearly constant. If, on the other hand, the current is extremely large, we can have a significant voltage difference. This is what happens in a short-circuit: a circuit in which a low-resistance pathway connects the two sides of a voltage source. Note that this is much more specific than the popular use of the term to indicate any electrical malfunction at all. If, for example, you short-circuit a 9-volt battery as shown in the figure, you will produce perhaps a thousand amperes of current, leading to a very large value of $$P=IV$$. The wire gets hot!

#### The voltmeter

A voltmeter is nothing more than an ammeter with an additional high-value resistor through which the current is also forced to flow, l/1. Ohm's law relates the current through the resistor directly to the voltage difference across it, so the meter can be calibrated in units of volts based on the known value of the resistor. The voltmeter's two probes are touched to the two locations in a circuit between which we wish to measure the voltage difference, l/2. Note how cumbersome this type of drawing is, and how difficult it can be to tell what is connected to what. This is why electrical drawing are usually shown in schematic form. Figure l/3 is a schematic representation of figure l/2.

The setups for measuring current and voltage are different. When we're measuring current, we're finding “how much stuff goes through,” so we place the ammeter where all the current is forced to go through it. Voltage, however, is not “stuff that goes through,” it is a measure of electrical energy. If an ammeter is like the meter that measures your water use, a voltmeter is like a measuring stick that tells you how high a waterfall is, so that you can determine how much energy will be released by each kilogram of falling water. We don't want to force the water to go through the measuring stick! The arrangement in figure l/3 is a parallel circuit: one which in there are “forks in the road” where some of the current will flow one way and some will flow the other. Figure l/4 is said to be wired in series: all the current will visit all the circuit elements one after the other.

If you inserted a voltmeter incorrectly, in series with the bulb and battery, its large internal resistance would cut the current down so low that the bulb would go out. You would have severely disturbed the behavior of the circuit by trying to measure something about it.

Incorrectly placing an ammeter in parallel is likely to be even more disconcerting. The ammeter has nothing but wire inside it to provide resistance, so given the choice, most of the current will flow through it rather than through the bulb. So much current will flow through the ammeter, in fact, that there is a danger of burning out the battery or the meter or both! For this reason, most ammeters have fuses or circuit breakers inside. Some models will trip their circuit breakers and make an audible alarm in this situation, while others will simply blow a fuse and stop working until you replace it.

##### Discussion Questions

In figure g/4 on page 96, what would happen if you had the ammeter on the left rather than on the right?

Imagine a charged doorknob, as described on page 103. Why is it intuitively reasonable to believe that all the charge will end up on the surface of the doorknob, rather than on the interior?

## Homework Problems

m / Problems 2 and 3.

1. A hydrogen atom consists of an electron and a proton. For our present purposes, we'll think of the electron as orbiting in a circle around the proton.

The subatomic particles called muons behave exactly like electrons, except that a muon's mass is greater by a factor of 206.77. Muons are continually bombarding the Earth as part of the stream of particles from space known as cosmic rays. When a muon strikes an atom, it can displace one of its electrons. If the atom happens to be a hydrogen atom, then the muon takes up an orbit that is on the average 206.77 times closer to the proton than the orbit of the ejected electron. How many times greater is the electric force experienced by the muon than that previously felt by the electron?

2. The figure shows a circuit containing five lightbulbs connected to a battery. Suppose you're going to connect one probe of a voltmeter to the circuit at the point marked with a dot. How many unique, nonzero voltage differences could you measure by connecting the other probe to other wires in the circuit? Visualize the circuit using the same waterfall metaphor.

3. The lightbulbs in the figure are all identical. If you were inserting an ammeter at various places in the circuit, how many unique currents could you measure? If you know that the current measurement will give the same number in more than one place, only count that as one unique current.

(c) 1998-2013 Benjamin Crowell, licensed under the Creative Commons Attribution-ShareAlike license. Photo credits are given at the end of the Adobe Acrobat version.