The results of their experiments were summarized in laws, and these laws were brought together by the Scottish physicist James Clerk Maxwell in 1861. Maxwell’s four equations, which in their abbreviated mathematical form have become a popular adornment of physics department sweatshirts, play the same role in electromagnetism that Newton’s laws do for motion and gravity: they summarize everything there is to know on the subject.
Electrical Charge and Coulomb’s Law
When you run a stack of papers through a photocopy machine, individual sheets may stick together. The force that holds the sheets together is said to be the electrical force, and objects that respond to the electrical force are said to have an electrical charge. Some simple experiments show that there are in fact two kinds of electrical charge. If you run two plastic combs through your hair, the force between them will be repulsive: they will be pushed apart. If you take one of those combs and bring it near a piece of glass that has been rubbed with fur, however, the objects experience an attractive force: they will be pulled together. There are two kinds of electrical forces, so it is reasonable to suppose they are generated by two kinds of electrical charge, which for historical reasons are called positive and negative.
Electrical charge is carried by subatomic particles—the building blocks of atoms (Chapter 4). The atom’s massive central nucleus is positively charged, while lighter, negatively charged electrons orbit the nucleus. An object is electrically charged if its atoms possess either an excess of electrons (in which case it has a negative charge) or a deficit of electrons (in which case it has a positive charge). In most situations the ponderous nuclei of atoms move very slowly, while electrons move easily. Thus, large objects usually acquire an electric charge by having electrons removed or added to their bulk. When you comb your hair, for example, electrons are stripped from your hair and pulled into the comb. As a result, the comb acquires a negative charge. This is why it will pick up bits of dust and paper (try this experiment yourself next time you comb your hair on a dry day). It also explains why the comb will then attract your hair and why the individual strands of hair repel each other—why they can “stand up.” Vigorous combing can even make your hair stand on end as the deficit of electrons (and consequent positive charge) increases.
The French physicist Charles Coulomb (1736–1806) first wrote down the law that describes forces between electric charges:
Like charges repel each other; unlike charges attract.
and
Between any two charged objects is a force proportional to the size of the two charges, divided by the square of the distance between them.
This law says that if two objects have an excess of electrons (and therefore have negative charges), they will repel each other, but if one has an excess and one a deficit, they will attract. It also says that the form of the equation that describes the electrical force is strikingly similar to the one that describes gravity.
Coulomb’s law describes the force between electrical charges that do not move—what is called static electricity. Electrostatic forces dominate the world as we know it. Plus attracts minus in chemical bonds, and thus holds materials together. Every object you see is made from atoms, themselves collections of negative electrons attracted to positive nuclei. Just as the gravitational force keeps the Earth and the other planets in orbit around the sun, electrostatic attraction keeps negative electrons in orbit around the positive nucleus of an atom.
The repulsion of electrons by electrons, on the other hand, keeps one object from passing through another. You can’t put your hand through this book, for example, because electrons in atoms in your hand are repelled by electrons in atoms in the book. You don’t fall through the floor, because electrons in your shoes repel electrons in the floor. Every time you touch or feel something, you are making use of the electrostatic force.
Photocopies are products of electrostatic forces at work. A polished plate of selenium metal can hold an electrical charge for extended periods of time. When exposed to light, however, the charge leaks off. The key to xerography is to project a pattern of light and dark (such as a printed page) onto the charged plate. A similar pattern, consisting of charged and uncharged regions, is then created on the plate as charge leaves the lighted areas. Electrostatic forces cause a special black plastic powder to cling only to the charged areas of the selenium plate. The powder is transferred to paper, then melted in place, producing a copy of the original document.
Magnetism
Human beings have known about magnets and magnetism for thousands of years. Naturally occurring magnets, called lode-stones, were scientific curiosities in the ancient world, and slivers of lodestone that lined up in a north-south direction were the first compasses. That a magnetic force exists can be verified by anyone who uses magnets to hang notes and miscellany on the refrigerator.
All known magnets share one feature: each magnet, whether it be the size of an atom, a compass needle, or the planet Earth itself, has two poles. Each pole is usually labeled north or south, depending on which end of the Earth they would point to if they were allowed to act as a compass. Magnetic poles have properties reminiscent of electric charge. Poles with the same character always repel, while opposite poles always attract (north attracts south, but repels north).
There is, however, an important difference between electrical charges and magnetic poles—a difference enshrined in Max well’s second equation. No matter how hard you try, the law says, you can never create an isolated magnetic pole. Unlike electrical charges (which can exist as independent positive or negative particles), magnetic poles always come in pairs. If you cut a 2-inch-long bar magnet in half, you don’t get one north end and one south end. You get two 1-inch-long bar magnets, each with its own north and south pole. Cut those pieces in half and you just get more magnets. Even the individual atoms are tiny “dipole” (two-pole) magnets. Thus, Maxwell’s second equation states:
There are no isolated magnetic poles.
Maxwell’s second equation says nothing about how magnetic fields come to be. Static electricity and magnetism seem to be very different things, and there is no obvious connection between a photocopy machine and refrigerator decorations. The nature of magnetism, and the connection between it and electricity, is the subject addressed in Maxwell’s third and fourth equations.
Two Sides of the Same Coin
The relationship between electricity and magnetism can be stated succinctly: every time an electric charge moves, a magnetic field is created; and every time a magnetic field varies, an electric field is created. Electricity and magnetism are two inseparable aspects of one phenomenon: you cannot have one without the other.
If we could see or feel electric and magnetic fields, their close ties would be obvious because we’d always see them together. But in day-to-day life we are not usually aware of electrical effects when we use magnets, nor do we sense magnetic fields when we use electricity. We have to use instruments to tell us about the connection between the two.
The story of the discovery of this connection is a curious one. The Danish physicist Hans Oersted (1777–1851) was giving a physics lecture when he noticed that flipping a switch to start the flow of an electric current caused a nearby compass needle to twitch. Further experiments convinced him that a magnetic field is present whenever electrical charge flows through a wire. This finding (usually expressed in a suitable mathematical form) is Maxwell’s third equation:
Moving electric charges create magnetic fields.
One common application of this law of nature is a device called the electromagnet. The simplest electromagnet is a loop of wire carrying an electric current. Because the current produces a magnetic field, the loop acts as a magnet. Unlike the permanent magnets that you use to hold things on your refrigerator, however, an electromagnet can be turned off and on by opening and closing the switch that controls the current.
A single loop produces a magnetic field with a north and a south pole. In fact, you can think of a
current-carrying loop as equivalent to a small bar magnet with its north pole coinciding with the north end of the magnetic field created by the current. The only difference is that the polarity of the loop’s field can be reversed by reversing the direction of the current. Electromagnets are found in many devices and machines, from ordinary doorbells to the large magnets that lift cars around in auto junkyards.
The other side of the electric/magnetic coin concerns the ability of magnetic fields to produce electrical forces. If the magnetic field in the region of a loop of wire is changed (by moving a magnet near the wire, for example), electrons will flow in the wire, even though there seems to be nothing in the wire to make them accelerate. This phenomenon, called electromagnetic induction, is described in the last of the Maxwell equations:
Electric current passing through a loop of wire creates a simple electromagnet, an essential component of every electric motor.
Magnetic effects can accelerate electrical charges.
Physicists Oersted, Henry, Faraday, and Maxwell did not know that their work would someday lead to large-scale generation and use of electricity. They could not have foreseen our twenty-first-century technological society, which nevertheless is almost entirely based on their discoveries. Electric motors and generators are simply practical applications of Maxwell’s third and fourth equations.
Electric Motors and Generators
Your home contains dozens of electric motors. Fans, hair dryers, razors, mixers, can openers, and virtually all major appliances incorporate at least one. All of these motors convert electricity into magnetic fields, which in turn cause useful rotary motion. What happens when you flip the switch?
The simplest electric motors combine a permanent magnet with an electromagnet. Stationary electrical contacts pass current through rotating loops of wire, thus turning each loop into the equivalent of a small bar magnet. The north and south poles of the electromagnet are oriented so that each is attracted to the appropriate pole of the permanent magnet. The result: the loop starts to rotate as like poles repel and opposite poles attract. As soon as the rotating loop completes half a turn the current switches direction, causing the poles of the electromagnet to flip. Each pole of the rotating electromagnet now finds itself attracted to the next pole of the permanent magnet, so the loop continues to rotate.
Most motors are more complex than the simple one described above. Typically, a motor incorporates multiple sets of permanent magnets or several synchronized electromagnets. Different arrangements of the basic components lead to motors that turn with a constant speed or a high torque or by small steps. In every case, however, the basic principle is the same: electricity is converted into magnetic fields.
The simplest electric motor incorporates a permanent magnet and an electromagnet. Magnetic forces drive the rotary motion.
Electrical generators are the exact opposite of electric motors: they convert rotary motion into electrical energy. The basic generator, first put to practical use by Thomas Edison, is little more than a loop of wire spinning in a magnetic field. Because of the rotation, the field seen by the loop is constantly changing, so a current flows in the wire, first one way and then the other. This alternating (AC) current comes out of the generator on wires and can be used to run electrical circuits. Almost all electricity used in the United States is produced in this way.
Anything that can turn an axle can power a generator. Flowing water, pressurized steam, wind, or a gasoline engine can drive a rotating turbine that houses coils of copper wire. In a large generating plant, powerful electromagnets surround the wire loops. As a rotating loop of copper wire cuts through the magnetic field lines, electrons are pushed back and forth—60 times per second in the United States—to produce the 60-cycle alternating current that lights cities and runs air conditioners.
ELECTRICAL CIRCUITS
The aspect of electricity and magnetism that we encounter most often in everyday life is the electrical circuit. A circuit is a continuous path of material through which electrical charge can flow. The most common circuits are made from copper wires.
Any flow of electrical charges is called an electric current. Although both positive and negative charges can constitute a current, in everyday situations we give this name to the movement of electrons. A toaster and a lightbulb, for example, both get their energy from the movement of electrons through the copper wires of your home.
Electric current is usually discussed in terms of a unit called the ampere, or amp. The amp measures how many charges go by a particular point in the wire (you can imagine a microscope traffic counter sitting in the wire and pushing a button every time an electron goes by, then adding things up every second). Typical household currents run from one amp (in a 100-watt lightbulb) to 50 amps (in an electric stove with all burners going and the oven on full blast).
Every circuit must also have a source—a device that supplies the energy to push electrons through the wires—which can be either a battery or a generator. In a battery, stored chemical energy is expended to provide the kinetic energy needed for electrons to move through the circuit. Current from a battery always flows in one direction, and is called DC (direct current). Current from a generator, on the other hand, alternates in direction and is called AC (alternating current).
The “pressure” with which electrons are pushed through a wire—the electrical potential—is measured in a unit called the volt. The higher the potential—the more volts—the more electrons can be pushed through a given wire. Some typical voltages encountered in everyday situations: flashlight batteries—1.5 V; car batteries—12 V; household current—115 V; high-voltage transmission lines—500,000 V.
ELECTROMAGNETIC RADIATION
The Nature of Light
Since all of the four experimental laws we have discussed so far were discovered by other people, you may be wondering why they are called Maxwell’s equations. There are three reasons: (1) he was the first to see that the equations formed a coherent system; (2) he added a small piece to the third law (he proved that there was a kind of electrical current that no one had thought about up to that point); and (3) most important, he realized that the four equations predicted the existence of a new kind of energy wave—one that we now call electromagnetic radiation.
The third and fourth equations show that every field, magnetic or electric, induces a corresponding electric or magnetic field. Back and forth, ad infinitum, the fields create and modify each other. This sort of eternal oscillation, Maxwell realized, creates a wave that moves through space. Like ripples from a pebble thrown in a pond, these energy waves radiate out from their source.
All waves can be described by three closely linked characteristics: speed, wavelength, and frequency. Each wave consists of a series of crests and troughs. Wavelength is the distance between adjacent crests, speed is measured by the movement of the crests, and frequency is a measure of how many crests pass a given point in a second. The most common unit for measuring frequency is the hertz (named after the German radio pioneer Heinrich Hertz [1857–94]). One hertz (1Hz) corresponds to one crest going by a point each second. Look at the plate on any appliance in your home—it will say 60Hz, another reminder that household electrical current changes direction 120 times each second.
When Maxwell saw that his equations predicted the existence of waves, the first thing he did was calculate how fast those waves would move. He found that the speed of the mysterious waves depended on things like the force that one electrical charge or magnet exerts on another. These numbers had been measured in laboratories, so Maxwell was able to predict the velocity with high accuracy. His result: the waves move at 186,000 miles per second. This, of course, is what he knew, and we know, as the speed of light. Light itself is the mysterious electromagnetic wave.
The speed of light is so important that physicists denote it by a special letter—c. It is the only speed that is actually built in to the laws of nature. It figures prominently in many fundamental theories, like the theory o
f relativity and its famous equation, E = mc2. It also denotes the speed of other types of radiation like X-rays and radio waves.
According to Maxwell’s calculation, his waves were actually composed of electrical and magnetic fields alternately creating each other as they move through space. The frequency of an electromagnetic wave is simply the frequency of the oscillating field that caused it. If you wave a charged comb in the air once a second, for example, you create an electromagnetic wave with a frequency of 1 hertz and a wavelength of 186,000 miles. Atoms can vibrate a trillion times a second, giving waves about a hundredth of an inch in length. It takes a lot more energy to wiggle an electron trillions of times per second than just once, so higher-frequency waves are also higher-energy waves. The important point, however, is that Maxwell’s equations predict that electromagnetic waves should exist at all frequencies and all wavelengths, not just for the narrow band we call visible light.
The Electromagnetic Spectrum
Think about waves on the ocean. They range from mini-ripples to swells a few yards long to tidal effects that span the oceans. These ocean waves are all intrinsically the same, differing only in the size, frequency, and energy contained in the moving water. If you travel in an ocean liner you notice only a narrow range of these waves—the ones that make the ship rise and fall. Other waves are all around, but you don’t sense them.
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