Unit 25 · Physics: Energy, Waves and Electricity

Hold a heavy backpack at arm’s length for one minute. Your arm burns, but by the physics definition you have done no work on the backpack. Work needs two things: a force, and movement along the direction of that force. The backpack did not move, so the work is zero. Your muscles get tired because they keep twitching and burning fuel to hold the force, but none of that energy went into the backpack.

Now lift the backpack one meter straight up. You push up with a force about equal to its weight, and it moves up one meter. Work equals force times distance, written W = F d. A 10-newton backpack lifted 1 meter takes 10 newton-meters of work. Physicists call one newton-meter a joule, after the man in the story. A joule is small: lifting an apple from the table to your mouth takes roughly one joule.

Direction matters. If you carry the backpack across a flat floor at steady speed, the upward force from your arm is at right angles to the sideways motion, so it does no work at all. When a force pushes at an angle, only the part along the motion counts. A force that points against the motion, like friction on a sliding box, does negative work: it takes energy out of the box’s motion and turns it into heat.

This narrow definition is what makes energy a useful idea. Because work is defined so precisely, we can add it up, track where it goes, and always find the total. A joule spent lifting is a joule banked in height; a joule spent against friction is a joule of heat in the floor. That bookkeeping, following every joule from where it starts to where it ends, is the whole story of this chapter.

A bowling ball rolling down a lane can knock over pins. It carries kinetic energy, the energy of motion. The formula is KE = 1/2 m v squared, where m is mass in kilograms and v is speed in meters per second. The square is the important part. Double a car’s speed and its kinetic energy is four times larger. That is why highway crashes are so much worse than parking-lot bumps: stopping the car means removing all of that energy, and it has to go somewhere, usually into crumpled metal.

Now imagine the bowling ball resting on a shelf two meters up. It is not moving, but it could. Push it off and gravity will speed it up. We say it holds gravitational potential energy, energy stored by position. The formula is PE = m g h, where g is the strength of gravity, about 9.8 newtons per kilogram near Earth’s surface, and h is the height. Lifting the ball to the shelf took work, and that work is now banked in the ball’s height.

The two forms trade back and forth. As the ball falls, its height shrinks and its speed grows. Potential energy becomes kinetic energy, joule for joule. A 7-kilogram ball on a 2-meter shelf holds about 137 joules of potential energy. Just before it hits the floor, if air resistance is small, it has about 137 joules of kinetic energy, which works out to a speed of roughly 6.3 meters per second.

Height is always measured from somewhere, and you get to choose where. Potential energy on a shelf is 137 joules relative to the floor, but zero relative to the shelf itself. Only changes in potential energy matter, so pick the reference that makes the problem easiest.

Two students carry identical boxes up the same stairs. One takes ten seconds, the other takes a full minute. They do the same work, because the force and the height are the same. But the first student is more powerful. Power is the rate of doing work, energy divided by time. Its unit is the watt, one joule per second, named for James Watt, the Scottish engineer who improved the steam engine in the 1770s.

Watts show up on almost everything you plug in. A phone charger might draw 20 watts, a hair dryer 1500 watts, a microwave oven about 1000 watts. That means the microwave turns about 1000 joules of electrical energy into heat and microwaves every second. A resting human body runs at roughly 100 watts, most of it leaving as heat, which is why a crowded room warms up.

Your electric bill is not in watts but in kilowatt-hours. A kilowatt-hour is one thousand watts running for one hour, which equals 3.6 million joules. It is an energy unit, not a power unit, even though it has the word watt inside it. Running a 1000-watt microwave for six minutes uses one tenth of a kilowatt-hour. Illinois households pay a price per kilowatt-hour, so the cost of a device depends on both its wattage and how long it runs.

Power also explains why a car needs a big engine to climb a hill fast but a small one to climb it slowly. The work of lifting the car is fixed by its weight and the hill’s height. Doing that work in less time takes more power, and an engine’s power is what limits its speed uphill.

Drop a rubber ball and it bounces, but each bounce is lower than the last. Where did the energy go? Not nowhere. Each time the ball squashes against the floor, some of its energy turns into a bit of heat in the rubber and a bit of sound in the air. The ball’s mechanical energy shrinks, but the total energy of ball, floor and air does not change. This is the law of conservation of energy: energy cannot be created or destroyed, only changed from one form to another or moved from one object to another.

The cleanest way to see it is an energy bar chart. Draw a bar for each form of energy at the start: kinetic, gravitational potential, elastic potential, thermal. Draw the same bars at the end. If the system is closed, the total height of all the bars is the same in both pictures. A roller coaster at the top of its first hill has a tall potential bar and a tiny kinetic bar. At the bottom, the bars have swapped, and a small thermal bar has grown from friction on the rails.

The word system is the key. If you decide the system is just the ball, then the floor and air are outside it, and energy leaves the system as heat and sound. If you decide the system is ball, floor and air together, nothing leaves. Both are correct. The law says only that whatever crosses the boundary must be counted as work or heat flowing in or out.

Engineers use this bookkeeping every day. Energy that leaks out as heat is usually wasted, so tracking where each joule goes tells you where to make a machine better. A car designer who finds that a third of the fuel’s energy leaves through the exhaust knows exactly where to look, and a bar chart of the losses is often the first drawing on the whiteboard.

Pull a spring a little and it pulls back a little. Pull it twice as far and it pulls back twice as hard. Robert Hooke described this in the 1670s, and it is now called Hooke's law: the restoring force equals a constant times the stretch, F = k x. The constant k is the spring constant, measured in newtons per meter. A stiff car suspension spring might have k in the tens of thousands of newtons per meter; a slinky’s is far smaller.

Because the force grows as you stretch, the work to stretch a spring is not simply force times distance. The average force during the stretch is half the final force, so the stored elastic potential energy is PE = 1/2 k x squared. Like kinetic energy, it has a square in it. Stretch a spring twice as far and you store four times the energy. That is why a fully drawn bow launches an arrow so much faster than a half-drawn one.

Hooke’s law only holds up to a point. Stretch a spring too far and it stops springing back; it has passed its elastic limit and is permanently bent. Many materials obey the law for small stretches, which is why a diving board, a guitar string, a trampoline and even a steel bridge girder all behave a bit like springs. Engineers measure k for each and design so that loads stay well within the elastic range.

Rubber bands look like springs but are not perfect ones. Their force does not grow in a straight line with stretch, and they warm up noticeably when stretched and released quickly, a sign that some of the work becomes heat instead of stored energy. Hold a thick band against your lip, stretch it fast, and you can feel the warmth.

Pull a mass hanging from a spring down a few centimeters and let go. It rises, overshoots, falls back, overshoots again, and keeps going. This repeating motion is oscillation. Energy sloshes between elastic potential energy at the ends of the swing, where the mass momentarily stops, and kinetic energy in the middle, where it moves fastest. Nothing pushes it after the first tug; the spring’s own restoring force does all the work.

The time for one full back-and-forth is the period. For a mass on a spring, a heavier mass makes the period longer and a stiffer spring makes it shorter. For a pendulum, something surprising happens: the period depends on the length of the string and the strength of gravity, but not on the mass of the bob, and for small swings not on how wide the swing is. Galileo is said to have noticed this watching a lamp swing in a cathedral. A pendulum about one meter long takes almost exactly two seconds for a full swing on Earth.

That steady period is why pendulums ran clocks for three hundred years. A grandfather clock’s pendulum ticks once per second in each direction. Friction and air drag slowly steal energy, so the swings shrink; a weight or spring inside the clock gives a tiny push each swing to make up the loss. Without that push the oscillation is damped and eventually stops.

The site’s drum machine and oscilloscope are oscillators too. A drumhead struck in the middle springs up and down hundreds of times a second, and that rate sets the pitch. The same physics that swings a pendulum also makes a bridge sway in wind and an atom vibrate in a crystal.

A bathtub of warm water and a single spark from a sparkler: which is hotter? The spark, by far, at well over a thousand degrees. Which holds more thermal energy? The bathtub, by an enormous margin. That is the difference between temperature and thermal energy. Temperature measures how hard, on average, each molecule is jiggling. Thermal energy is the total for all the molecules put together. The spark has few molecules moving very fast; the tub has trillions of trillions moving gently.

Heat is a third idea, and the word is used carelessly in everyday speech. In physics, heat is energy in transit, moving from a hotter object to a colder one because of the temperature difference. A hot pan does not contain heat; it contains thermal energy. When you touch it, heat flows into your hand. Once the energy arrives it is thermal energy again. Joule’s paddle wheel showed that this flowing energy is the same stuff as work, measured in the same joules.

Heat moves three ways. Conduction is molecule bumping molecule, the way a metal spoon in soup grows hot at the handle. Convection is hot fluid physically moving, the way warm air rises off a radiator. Radiation is light, including infrared light you cannot see, the way the Sun warms your face across empty space. On a Chicago January morning all three are pulling heat out of you: the cold air by conduction, the wind by convection, and the dark sky by radiation.

Temperature has a floor. Cool a gas and its molecules slow down; at about minus 273 degrees Celsius they would have the least energy physically possible. That is absolute zero, the zero of the Kelvin scale. Nothing has ever reached it, though laboratories have come within billionths of a degree.

On a hot August afternoon the sand at a Lake Michigan beach can burn your feet while the water is still cool. Sand and water sit under the same sun, yet the sand heats up fast and the water hardly changes. The property behind this is specific heat: the energy needed to raise one kilogram of a material by one degree Celsius. Water’s is about 4200 joules per kilogram per degree, among the highest of any common substance. Dry sand’s is roughly 800, so the same sunlight warms sand about five times as much.

The formula is Q = m c delta T: heat added equals mass times specific heat times the temperature change. Heat 2 kilograms of water from 20 to 100 degrees and you need 2 times 4200 times 80, about 670,000 joules. That is why a kettle takes minutes even at 1500 watts. Heat the same mass of iron by the same amount and you need only about a ninth as much energy, because iron’s specific heat is around 450.

Water’s high specific heat shapes Chicago’s weather. Lake Michigan warms slowly in spring and cools slowly in fall, so lakefront neighborhoods are cooler than the western suburbs in May and milder in November. Weather forecasters say cooler near the lake for exactly this reason. The oceans do the same on a planetary scale, soaking up enormous amounts of heat and smoothing the seasons for coastal regions.

Specific heat also explains Joule’s choice of water. Because water takes so much energy to warm, his temperature rises were tiny and hard to read. But water is easy to measure and stir, and its specific heat was already the standard definition of the calorie, so his result could be compared with everyone else’s.

Thermodynamics began as the science of steam engines and grew into rules that govern everything from stars to cells. The first law of thermodynamics is conservation of energy written for heat. The change in a system’s internal energy equals the heat added to it minus the work it does on its surroundings. Pump air into a bicycle tire and the pump gets warm: you did work on the air, and its internal energy and temperature rose. Let compressed air rush out and it feels cold: the air did work pushing outward, and its internal energy fell.

The first law allows many things that never happen. Nothing in it forbids a cold cup of coffee from spontaneously pulling heat out of the room and becoming hot again. Energy would still be conserved. Yet it never occurs. The second law of thermodynamics explains why: heat flows on its own only from hotter to colder, never the reverse. Rudolf Clausius stated it this way in the 1850s. You can move heat from cold to hot, as a refrigerator does, but only by doing work, and the work always ends up as extra heat somewhere.

There is a zeroth law, named later because it turned out to be more basic. If two objects are each in thermal balance with a third, they are in balance with each other. That is what makes a thermometer work: it settles at the temperature of whatever it touches. And a third law says absolute zero can be approached but never reached in a finite number of steps.

These laws are not derived from anything deeper. They are summaries of what has been observed, without exception, in every experiment ever done. Patent offices refuse machines that would violate them, and no proposed perpetual-motion machine has ever run.

A car engine burns gasoline and turns perhaps a quarter of that chemical energy into motion. The rest leaves as hot exhaust and heat through the radiator. This is not bad engineering; it is a law of nature. Any heat engine takes heat from a hot source, turns some of it into work, and must dump the rest into a cold sink. If it dumped nothing, heat would be fully converted into work with no other change, and the second law forbids that.

In 1824 a young French engineer, Sadi Carnot, worked out the best any engine could possibly do. The ideal efficiency depends only on the temperatures of the hot source and cold sink, measured in kelvins: efficiency equals 1 minus cold temperature over hot temperature. An engine running between steam at 600 kelvin and a river at 300 kelvin could at most convert half its heat to work. Real engines, with friction and leaks, do worse. The only ways to improve are to make the hot side hotter or the cold side colder.

This is why power-plant engineers push for higher steam temperatures and why plants sit beside rivers and lakes for cooling. It is also why a car’s engine needs a radiator: the cold sink has to stay cold, or the efficiency collapses. Combined-cycle gas plants reach around 60 percent by running a gas turbine and then using its hot exhaust to boil water for a second steam turbine.

Carnot was working with the old caloric theory and still got the right answer for the limit. His book sold poorly and he died of cholera at 36. Twenty-five years later, Kelvin and Clausius rebuilt his argument on Joule’s finding that heat is energy, and thermodynamics became a science.

Film an egg falling and breaking, then play the film backward. Everyone can tell which direction is real. Yet the laws of motion work equally well forward and backward; nothing in Newton’s equations says the shards cannot leap up and reassemble. The one-way character of the world comes from the second law, and the quantity that captures it is entropy.

Ludwig Boltzmann, in the 1870s, gave entropy a concrete meaning. Think of the energy in a room as spread among trillions of molecules. There are only a few ways to arrange it so that all the fast molecules are in one corner and the slow ones in another. There are unimaginably many ways to arrange it so that fast and slow are mixed everywhere. Entropy counts those ways. Left alone, a system drifts toward the arrangements that are most numerous, which means the ones where energy is spread out and useless for doing work. The egg has far more ways to be broken than whole.

This is why heat flows from hot to cold, why a drop of ink spreads through water and never gathers back up, and why every engine wastes energy. Entropy can fall in one place, as when your freezer makes ice, but only by raising it more somewhere else, in the warm coils behind the freezer. Add it all up and the total always climbs. Boltzmann’s equation linking entropy to the number of arrangements is carved on his gravestone in Vienna.

Living things are not exceptions. A seed grows into an ordered tree, but it does so by taking in concentrated sunlight and giving off spread-out heat. The tree’s order is paid for by a larger disorder in the Sun and sky. Life rides the flow of energy downhill; it does not reverse it.

Nearly every watt in an Illinois wall socket began as something spinning. In a coal or natural gas plant, burning fuel boils water; in a nuclear plant, splitting uranium atoms does the boiling. The steam drives a turbine, the turbine turns a generator, and a magnet spinning past coils of wire pushes electrons into motion. Wind turbines skip the steam and let the wind turn the generator directly. Solar panels are the odd ones out: they convert light straight to electricity with no moving parts.

Illinois is unusual. Its six nuclear plants hold eleven reactors, more than any other state. They supply about half of the electricity generated here. The chain began in Chicago. On December 2, 1942, Enrico Fermi’s team ran the first controlled nuclear chain reaction under the stands of the University of Chicago’s football field. Dresden Unit 1, near Morris, opened in 1960 as the first privately financed commercial nuclear plant in the country. Every one of these plants is still a heat engine bound by Carnot’s limit. Each converts roughly a third of its heat into electricity and sends the rest into cooling towers and lakes.

The grid is the network of high-voltage lines, transformers and substations that links plants to homes. Electricity cannot easily be stored in large amounts. So at every instant the power generated must match the power being used. Grid operators forecast demand hour by hour, ramp plants up and down, and watch the frequency, which drifts if supply and demand slip apart. Transformers step voltage up to hundreds of thousands of volts for long-distance lines. Higher voltage means lower current and less energy lost as heat in the wires. Near your house, other transformers step it back down.

Every source has trade-offs the evidence makes clear. Fossil fuels are cheap and controllable but release carbon dioxide. Nuclear plants emit almost none but produce long-lived waste. Wind and solar are clean and now inexpensive but depend on weather, which is pushing utilities toward batteries and other storage. No option is free of costs, and deciding among them is a public choice informed by, but not settled by, physics.

Drop two pebbles into a still pond a hand’s width apart. Two sets of ripples spread out and cross each other. Where they cross, the water does something remarkably simple: it adds. If a crest from one meets a crest from the other, the water rises twice as high. If a crest meets a trough, they cancel and the surface is briefly flat. This is superposition, and it holds for water, sound, light and every other wave. The waves pass through each other unharmed and continue on their way.

The pattern that results is called interference. Along some lines from the two pebbles the crests always arrive together, and the water heaves strongly. These are lines of constructive interference. Between them run lines where crest always meets trough and the water is nearly calm, destructive interference. Whether a spot is calm or heaving depends on the difference in distance to the two sources. If one wave has traveled exactly half a wavelength farther, the two arrive out of step and cancel.

Young’s card in a sunbeam was the same experiment with light. The bright stripes were constructive interference and the dark stripes destructive. Because the spacing of the stripes depends on the wavelength, measuring the stripes measures the wavelength. This is the standard way to measure very small lengths even today, and it works because the stripes are much larger and easier to see than the wavelength itself.

Noise-canceling headphones are interference at work. A tiny microphone hears the outside noise, and the headphone plays the same wave upside down. Crest meets trough at your eardrum, and the roar of an airplane cabin fades. The trick works best on steady, low rumbles, because the electronics must produce the canceling wave before the original arrives.

Shake one end of a rope tied to a wall. The wave runs down, reflects, and runs back. At most shaking rates the returning wave and the new one jumble together. But at certain rates, something clicks: the rope settles into a fixed pattern with points that never move and loops that swing wildly between them. This is a standing wave. The still points are nodes; the widest swings are antinodes. It is really two waves, one going each way, interfering to make a pattern that seems to stand still.

A string fixed at both ends can only hold standing waves that have a node at each end. The simplest fits half a wavelength between the ends and is the fundamental, the lowest note the string can make. The next fits a whole wavelength and sounds an octave higher; the next fits one and a half. These are the harmonics. On the site’s oscilloscope you can see them directly: change the driving frequency slowly and the string snaps from one loop to two to three. Shorten a guitar string with your finger and the fundamental rises, because a shorter string fits a shorter wavelength.

A drumhead is a two-dimensional string. On the drum machine, sprinkling sand on a vibrating plate collects it along the node lines, drawing the pattern of the standing wave. Unlike a string, a drumhead’s higher modes are not simple multiples of the fundamental, which is why a drum has a thud rather than a clear pitch.

Every object that can vibrate has natural frequencies, and pushing it at one of those frequencies with even small nudges builds up a large motion. This is resonance. A child on a swing, a singer shattering a glass, a car that buzzes at one particular engine speed: all are resonance. Engineers test bridges and buildings for it. The famous Tacoma Narrows bridge collapse in 1940 is often called resonance, though the wind actually fed energy through a more complex self-driven twisting; the lesson, that small periodic forces can build huge motions, still stands.

An ambulance races past with its siren on. As it approaches, the siren sounds high; the instant it passes, the note drops. The siren itself never changes. What changes is how the sound waves reach you. While the ambulance drives toward you, each new crest is sent from a little closer than the last, so the crests arrive bunched up: shorter wavelength, higher frequency, higher pitch. As it drives away, each crest is sent from farther back, the crests are stretched apart, and the pitch falls. Christian Doppler described this in 1842, and it is the Doppler effect.

For sound, the size of the shift depends on the source’s speed compared with the speed of sound. That speed is about 343 meters per second in air at room temperature. A car at 34 meters per second, about 76 miles per hour, moves at one tenth the speed of sound. Its horn shifts up by roughly ten percent as it approaches. Police radar guns work backward. They bounce radio waves off a car and measure the shift in the returning wave to find the speed.

Light shifts too. A star or galaxy moving away stretches its light toward longer, redder wavelengths, a redshift. One moving toward us shifts blue. Astronomers read the shift from the dark lines in a spectrum, which sit at known positions for each element. In the 1920s Edwin Hubble found that almost every galaxy is redshifted, and the farther it is, the larger the shift. The universe is expanding, and the Doppler idea, born from a passing siren, was the key that unlocked it.

Doctors use ultrasound Doppler to hear blood flow: sound bounces off moving blood cells and comes back shifted. Weather radar uses the same trick to see wind inside a storm, which is how forecasters in Illinois spot rotation in a thunderstorm before a tornado forms.

Look into a flat mirror and a twin looks back from exactly as far behind the glass as you stand in front of it. The rule behind this is simple. Light bounces off a smooth surface with the angle of reflection equal to the angle of incidence. Both are measured from a line perpendicular to the surface called the normal. Trace a few rays from your nose to the mirror and back to your eye. Extend them straight behind the glass, and they meet at the image, a point where light only seems to come from. No light is actually there, so it is a virtual image.

A rough surface, like paper, obeys the same rule at every tiny spot, but the spots face every direction, so the light scatters. That is why you can see the page from anywhere but cannot see your face in it. A puddle at night looks like a mirror because its surface is smooth.

Curve the mirror and things get interesting. A concave mirror, hollowed like the inside of a spoon, bounces parallel rays toward a single point, the focus, which sits halfway between the mirror and the center of its curve. A reflecting telescope is a concave mirror pointed at the sky, gathering faint starlight to that point. Put an object closer than the focus and the mirror acts like a magnifying shaving mirror, showing an enlarged, upright image. Put it farther and the image flips upside down and can be caught on a screen: a real image, made of actual light.

A convex mirror, bulging like the back of the spoon, spreads rays apart. It shows a small, upright image of a wide field of view, which is why it is used for passenger-side car mirrors and at blind corners in parking garages. The warning etched on it, that objects are closer than they appear, is a statement about how a curved surface makes images.

Put a pencil in a glass of water and it looks broken at the surface. A swimming pool looks shallower than it is. Both happen because light slows down when it enters water, and a slowing wave that hits a boundary at an angle changes direction. This bending is refraction. Light in a vacuum travels at about 300,000 kilometers per second; in water it travels about three quarters as fast, and in glass about two thirds. The ratio of the vacuum speed to the speed in a material is that material’s index of refraction, n. Water’s is about 1.33, ordinary glass about 1.5, diamond about 2.4.

The rule for the bend was worked out in the 1600s and is called Snell's law. It says n1 times the sine of the angle in the first material equals n2 times the sine of the angle in the second. Both angles are measured from the normal. Light entering a slower material bends toward the normal; leaving it, light bends away. A ray hitting water at 45 degrees from the normal continues at about 32 degrees. The same geometry explains why the pool bottom looks raised.

Different colors travel at slightly different speeds in glass, so they bend by slightly different amounts. Violet bends most, red least. That is dispersion, and it is why a prism spreads white light into a rainbow and why raindrops make a real one, with the Sun behind you and each drop bending and reflecting the light back at about 42 degrees.

Go the other way, from glass toward air, and at a steep enough angle Snell’s law demands a sine greater than one, which is impossible. The light cannot leave and reflects entirely back inside. This total internal reflection is what keeps light trapped inside a glass fiber for kilometers, carrying internet traffic under Chicago streets and across oceans.

A converging lens is thicker in the middle than at the edges. Parallel rays entering it bend toward the axis and meet at the focal point. The distance from lens to that point is the focal length. A magnifying glass is a converging lens used with the object closer than the focal length. Rays leave spread out, your eye traces them back, and you see a large upright virtual image. Move the object farther than the focal length and the lens throws a real, inverted image on the other side. That is how a camera, or your own eye, forms a picture on its sensor or retina.

A ray diagram predicts all this with three rays. One comes in parallel to the axis and leaves through the focal point. One passes straight through the center undeflected. One passes through the near focal point and leaves parallel. Where the rays meet, or where they seem to come from, is the image. The same three rays, with the rules reversed, handle a diverging lens, which is thinner in the middle and always makes a small upright virtual image, like the eyeglasses of a nearsighted person.

The refracting telescope, built by Galileo in 1609 from lenses he ground himself, uses two converging lenses. The large front lens, the objective, forms a real image of the Moon or a planet near its focal point. The small eyepiece then acts as a magnifying glass on that image. The magnification is the objective’s focal length divided by the eyepiece’s. The site’s telescope works this way, and the largest refractor ever put to serious use, the 40-inch at Yerkes Observatory in Wisconsin, was built for the University of Chicago in 1897.

The microscope flips the arrangement. Its objective has a very short focal length and sits just above the specimen, forming an enlarged real image inside the tube. The eyepiece magnifies that image again. Robert Hooke used one to see the cells in cork in 1665; the site’s microscope shows the same cells four centuries later. In both instruments, the limit is not the glass but the wave nature of light itself, as the next section explains.

Stand outside an open doorway and you can hear people talking inside even though you cannot see them. Sound bends around the door frame. This spreading of waves past edges and through gaps is diffraction. It happens to every wave, but it is only obvious when the gap is about the size of the wavelength. Sound wavelengths are centimeters to meters, the size of doors, so sound diffracts strongly. Light wavelengths are about half a micrometer, so light passing a doorway barely spreads at all, and shadows look sharp.

Squeeze light through a very narrow slit, though, and it spreads into a fan of bright and dark bands. Shine a laser through a single hair and the same thing happens. The bands are interference between light from different parts of the same slit. Fresnel’s mathematics of diffraction, and the bright spot found in the middle of a disk’s shadow in 1818, are what finally convinced physicists that Young was right.

Diffraction sets a limit on every optical instrument. A microscope cannot show detail smaller than roughly half the wavelength of the light it uses, about 200 nanometers for visible light, because the light waves spread too much to be focused finer. This is why viruses are invisible in a classroom microscope and why electron microscopes, which use electron waves thousands of times shorter, can see them. A telescope’s ability to separate two close stars is limited the same way, which is one reason astronomers build mirrors as wide as they can.

A diffraction grating, a surface with thousands of fine lines per millimeter, turns diffraction into a tool. Each line acts as a slit, and their interference sends each wavelength off at its own angle, spreading light into a spectrum far sharper than a prism can. The rainbow shimmer on the underside of a CD is a grating made of its data tracks.

Just when the wave theory of light seemed complete, a small experiment refused to fit. Shine ultraviolet light on a clean metal surface and electrons pop out. Physicists around 1900 expected that brighter light, carrying more energy, would knock electrons out faster and harder. It did knock more of them out, but no harder. And dim light of a high enough frequency kicked out electrons instantly, while intense red light, below a threshold frequency, kicked out none at all, no matter how long it shone. Waves should not behave this way; a wave delivers energy steadily and should eventually shake any electron loose.

In 1905 Albert Einstein proposed that light arrives in packets, later called photons, each carrying an energy proportional to the light’s frequency. A single photon gives all its energy to a single electron. If the packet is too small, below the threshold, the electron cannot escape, and more packets do not help. If it is large enough, the electron flies off with whatever energy is left over. Brighter light means more photons and therefore more electrons, but each still gets the same energy. Robert Millikan, at the University of Chicago, spent a decade testing the idea, expecting to disprove it, and in 1916 confirmed it in detail. Einstein received the Nobel Prize for this work, not for relativity.

So which is light, wave or particle? The honest answer is that it is neither of the everyday things those words name. It travels and interferes like a wave, in Young’s stripes and every diffraction grating. It arrives and is absorbed in lumps, one photon at a time, in every solar cell and camera sensor. Send photons through two slits one at a time and each lands as a single dot, yet the dots build up into Young’s interference pattern. This wave-particle duality is not a paradox to be resolved but a description of how light actually behaves, and it turned out to apply to electrons and atoms as well.

The debate that began with Newton and Young did not end with one side winning. It ended with a new picture, quantum physics, that contains both. Solar panels on Illinois farms, the sensor in your phone camera and the smoke detector in the hallway all run on the photoelectric effect and its cousins.

Pull a wool sweater over your head in a dry Illinois winter and hear it crackle. Rub a balloon on your hair and it sticks to the wall. Something has moved between the materials. That something is electric charge, carried by electrons that rub off one surface onto another. There are two kinds, called positive and negative; like charges repel, unlike charges attract. Rubbing does not create charge, it separates it. The sweater and your hair end up with equal and opposite amounts, and the total charge, like energy, is conserved.

Charles-Augustin de Coulomb measured the force between charges in 1785 with a delicate twisting balance. His law looks just like Newton’s law of gravity: the force is proportional to the product of the two charges and falls off with the square of the distance between them. Double the distance and the force drops to a quarter. But electric force is enormously stronger than gravity. It is electric attraction between nuclei and electrons that holds atoms together, and electric repulsion between electron clouds that keeps your hand from passing through a table.

Rather than think of charges acting on each other across empty space, Michael Faraday pictured each charge surrounded by an electric field, a condition of space that would push on any charge placed there. Draw arrows showing the direction a positive test charge would be pushed, and you have a field map. Arrows point away from positive charges and toward negative ones. Near a charge the lines crowd together, where the force is strong; far away they spread thin.

A thundercloud is a giant charge separator. The field beneath it grows until air itself breaks down and conducts, and that is lightning. The same field idea explains why charge on a metal object gathers at sharp points. That is why lightning rods work, and why you feel a shock at the tip of your finger rather than your palm.

A battery does not store electricity the way a tank stores water. It stores chemical energy and uses it to push charge. The push is measured as voltage, which is energy per unit of charge: a 9-volt battery gives each coulomb of charge 9 joules of energy to spend. A coulomb is a large amount of charge, about six billion billion electrons. Voltage is like the height of a water tower; it tells you how much energy each unit of charge will release on its way down.

Current is how much charge actually flows, measured in amperes, one coulomb per second. Whether much flows depends on the resistance of the path, measured in ohms. Georg Ohm found in 1827 that for a metal wire at steady temperature, current equals voltage divided by resistance, I = V over R. This is Ohm's law. Double the voltage across a wire and the current doubles; double the resistance and the current halves. A long thin wire has more resistance than a short thick one, the way a narrow pipe restricts water.

Resistance turns electrical energy into heat. Power dissipated is voltage times current, P = V I, which with Ohm’s law becomes I squared times R. That is the glow of a toaster wire and the danger of a frayed cord. It is also why the grid uses high voltage: for a given power, higher voltage means lower current, and losses in the lines fall with the square of the current.

Not everything obeys Ohm’s law. An LED barely conducts until the voltage reaches a threshold and then conducts strongly; a light bulb’s filament resistance rises as it heats. Ohm’s law is a good description of metals under ordinary conditions, not a universal law of nature, and knowing the difference is part of using it well.

Old strings of holiday lights had a famous flaw: one dead bulb and the whole string went dark. Those bulbs were wired in series, one after another along a single path. Current has only one road, so it is the same through every bulb, and if any bulb breaks, the road is cut. Resistances in series simply add. Three 2-ohm bulbs in series make 6 ohms, and the battery’s voltage is shared among them: each bulb gets a third.

Your house is wired in parallel. Every outlet and light is its own branch connected across the same two wires, so each gets the full 120 volts whether or not anything else is on. Unplug the toaster and the lamp does not care. In parallel, current divides among the branches according to their resistance, and adding a branch always lowers the total resistance because it opens another road. Two 2-ohm bulbs in parallel have a combined resistance of 1 ohm.

Parallel wiring has a cost. Every device you plug in adds current on the main wires. Too many heaters on one circuit and the current exceeds what the wire can carry safely; the wire heats, and a circuit breaker trips to cut the flow. The breaker is a safety device that responds to current, not voltage, and that is exactly why: current is what makes heat.

Most real circuits mix the two. A car’s headlights are parallel with each other but in series with the switch; the switch must carry the total current. Reading a circuit means tracing the paths: where the current has one road, think series; where it splits, think parallel.

In 1820 the Danish physicist Hans Christian Ørsted noticed a compass needle twitch when he switched on a current in a nearby wire. Electricity and magnetism, studied separately for centuries, were connected. Every electric current makes a magnetic field circling around it. Wind the wire into a coil and the fields add up into a strong magnet you can switch on and off, an electromagnet. Permanent magnets work the same way at a smaller scale: electrons in iron atoms act as tiny current loops, and in a magnet vast numbers of them line up.

Michael Faraday asked the reverse question: if current makes magnetism, can magnetism make current? In 1831 he found the answer, and it is subtle. A magnet sitting still next to a coil does nothing. But move the magnet, or move the coil, or change the field in any way, and a voltage appears across the coil for as long as the change lasts. This is electromagnetic induction. The faster the change, the larger the voltage. Every generator in the world, from Dresden’s reactors to a bicycle dynamo, is a coil and a magnet moving past each other.

The site’s turntables make the rule audible. In the groove of a record, a wavy wall wiggles the diamond stylus back and forth thousands of times per second. In a typical cartridge the stylus carries a tiny magnet that moves between fixed coils, or a tiny coil that moves inside a magnet. Either way the field through the coil changes, and induction produces a voltage of a few thousandths of a volt that copies the shape of the groove. The amplifier makes it larger, the speaker turns it back into motion, and you hear the music. The cartridge is a generator so small it is driven by a scratch in plastic.

Run induction backward and you have a motor: a current in a magnetic field feels a force, which spins a coil. Speakers are motors too, a coil in a magnet pushing a paper cone. In the 1860s James Clerk Maxwell combined the electric and magnetic rules into one set of equations and found that a changing field of either kind creates the other, and that the pair could travel through empty space as a wave moving at, remarkably, the measured speed of light. Light, he concluded, was an electromagnetic wave. Radio, discovered by Hertz in 1887, is the same wave at a longer wavelength.