Charge & the electric field

Charge is a property some particles carry, the way they carry mass. It comes in two flavors, which we call positive and negative. Protons carry one, electrons the other, and neutrons carry none. Like flavors push each other away; opposite flavors pull together. That's the whole rule. Nearly all of chemistry, biology and electronics is that rule playing out in detail.

The names are an accident. In the 1740s Benjamin Franklin guessed that rubbing glass gave it an excess of a single "electric fluid," and he called that state positive. When J. J. Thomson discovered the electron in 1897, it turned out to be the thing that usually moves, and by Franklin's convention it's negative. That's why conventional current is drawn flowing one way while the electrons in the wire drift the other. Nobody was wrong; the label just got there first.

Charge is quantized. Every electron carries exactly −e and every proton exactly +e, where e = 1.602 × 10⁻¹⁹ coulombs. Robert Millikan measured it in 1909 by watching tiny charged oil droplets hover in an electric field: their charges always came in whole multiples of one tiny lump. Quarks carry thirds (+⅔e and −⅓e), but they never turn up alone. Every particle you can isolate carries a whole number of e.

Charge is conserved. Rubbing a balloon on your hair doesn't create charge; it moves electrons from hair to rubber. The balloon ends up negative by exactly as much as your hair ends up positive. Even when matter is made from pure energy, as when a gamma ray turns into an electron and a positron, the new particles come as a pair whose charges add to zero.

The unit, the coulomb, is enormous: 6.24 × 10¹⁸ electrons' worth. The spark that jumps from your finger to a doorknob on a dry winter day moves about a millionth of a coulomb.

In 1785 Charles-Augustin de Coulomb hung a light rod from a fine wire, put a small charged ball on one end, and measured how far the wire twisted when he brought another charged ball near. The force fell with the square of the distance: double the gap, a quarter the force.

The square is geometry. Picture a charge's influence spreading out evenly over a sphere around it. A sphere twice as wide has four times the surface area, so the influence on each square meter is a quarter as strong. Gravity spreads the same way and obeys the same inverse square.

But look at the size of k. Two charges of one coulomb each, one meter apart, push on each other with 8.99 × 10⁹ newtons. That's the weight of about 916,000 metric tons, roughly nine aircraft carriers, from two specks of charge.

Between an electron and a proton, the pair inside every hydrogen atom, the electric pull is 2.27 × 10³⁹ times the gravitational pull. Since both forces fall as 1/r², that ratio is the same at every distance. If the electric force were a bar as long as the observable universe is wide (about 8.8 × 10²⁶ m), gravity's bar would be 0.39 picometers: less than a hundredth of the width of a hydrogen atom.

So why does gravity run the solar system? Because electric charges cancel. Matter is neutral so precisely that its titanic internal pushes and pulls balance out. Gravity has only one sign. It never cancels; it just piles up, and a planet is about 10⁵⁰ atoms all pulling the same way.

Richard Feynman liked this one. Stand one meter from a friend, and give each of you 1% more electrons than protons. The repulsion would be about 1.2 × 10²⁵ newtons, the weight, at Earth's surface, of an object a fifth as massive as the Earth itself.

Coulomb's law tells you how hard two charges push. It doesn't say how one charge knows the other is there. Newton was uneasy about the same puzzle for gravity. Michael Faraday's answer, worked out in the 1830s and 1840s, was to stop thinking about pairs at all. A charge sets up a condition in the space around it, at every point, whether or not anything is there to feel it. Drop a second charge somewhere, and it responds to the condition right where it sits.

That condition is the electric field, E. You define it by what it does: the force per coulomb on a small positive test charge placed at that spot. Its units, newtons per coulomb, turn out to be the same thing as volts per meter, which is how engineers usually quote it.

Field lines are a way to draw it. They start on positive charges and end on negative ones (or run off to infinity). At any point the field points along the line, and where lines crowd together the field is strong. They never cross, because the field has only one direction at each point. The lines are a drawing tool, not physical strings, but they are an honest one: the number of lines crossing a surface measures the field through it.

Superposition means fields simply add, arrow by arrow. To find the field near two charges, work out each one's arrow at that point and add them head to tail. That one rule explains the swooping lines of a dipole, and the dead spot halfway between two like charges, where their arrows are equal and opposite.

Real numbers help. Earth's surface sits in a fair-weather field of about 100 volts per meter, pointing down. You don't feel it, partly because your conducting body reshapes the field around you. Air sparks at about 3,000,000 V/m. And inside a hydrogen atom, where the electron sits, the proton's field is 5.1 × 10¹¹ V/m.

Is the field real, or just bookkeeping? The answer comes from motion. Jiggle a charge here, and a charge 300 meters away doesn't notice for about a microsecond: the news travels through the field at the speed of light. In between, the field itself carries energy and momentum. That traveling ripple is light, radio and Wi-Fi; Chapter 4 shows one being born.

In a metal, each atom gives one or two electrons to a shared sea that belongs to the whole lump. Copper has about 8.5 × 10²⁸ free electrons per cubic meter. Push them with a field and they move: that's a current. In an insulator such as rubber, glass, plastic or dry air, every electron is tied to its own atom or chemical bond. Charge you put there stays where you put it.

That difference has a striking consequence. Put extra charge on a piece of metal and the free electrons shuffle until the field inside the metal is exactly zero, with all the excess charge sitting on the outer surface. A metal box shields its inside from outside fields: a Faraday cage. A car struck by lightning protects you because of its metal shell, not its tires; a few centimeters of rubber mean nothing to a spark that just crossed a kilometer of air.

Static electricity comes from contact. Different materials grip their outer electrons with different strength; touch two together and pull them apart, and some electrons stay on the greedier one. Rubber takes electrons from hair; wool gives them up to plastic. Moist air coats surfaces with a thin film of water that lets charge leak away, which is why shocks are a winter thing. Shuffling across carpet in dry air can charge your body to 10,000 volts or more. Your body's capacitance is about 100 picofarads, so that is about 1 microcoulomb and 5 millijoules: enough to sting (most people feel a spark above about 3,000 volts), nowhere near enough to hurt.

The balloon on the wall is the clever part. A wall is an insulator, so it can't send charge to cancel the balloon's. But its molecules can stretch. The negative balloon nudges each molecule's electrons slightly away and pulls its positive nuclei slightly closer. This is polarization: each molecule becomes a tiny dipole with its positive end toward the balloon. The positive ends are nearer, so their attraction beats the repulsion from the farther negative ends. The net pull is small, but a balloon weighs only a few grams. A charged comb picks up scraps of paper the same way.

The field tells you the push. Voltage tells you the energy. Moving a positive charge against a field takes work, just as carrying a ball uphill does. The electric potential at a point is the work per coulomb needed to bring a positive charge there. The unit is the volt: one joule per coulomb.

Picture a landscape. Height is the potential; the slope is the field. A positive charge rolls from high potential to low, and the steeper the slope, the harder it's pushed. The field points straight downhill, and its strength is how many volts the potential drops per meter.

An AA cell gives each coulomb that passes through it 1.5 joules. A US wall outlet swings around 120 volts. In atoms the natural unit is the electron-volt, the energy one electron gains falling through one volt: 1.602 × 10⁻¹⁹ joules. It's the same unit used for photon energies in Chapter 6.

Only differences in voltage matter. A bird on a power line is at thousands of volts, but both feet are at the same voltage, so no current runs through it. The trouble starts if it touches something at a different voltage.

High voltage and strong field are not the same thing, either. A nerve cell holds only about 70 millivolts across its membrane, but the membrane is about 7 nanometers thick, so the field inside it is 10 million volts per meter, more than three times what makes air spark.

A capacitor is two conductors separated by a gap. Charge one positive and the other negative, and a nearly uniform field fills the gap. The energy you spent separating the charges doesn't live in the plates. It lives in the field between them, at ½ ε₀ E² joules per cubic meter.

Two plates of 1 square meter, 1 millimeter apart in air, make just 8.85 nanofarads. Air is a poor place to store energy: even at the breakdown field the energy density is only about 40 joules per cubic meter. So real capacitors cheat. They fill the gap with a material whose molecules polarize and multiply the capacitance (some ceramics by thousands), roll thin films into tight cylinders, or, in supercapacitors, use a gap only a few molecules wide.

A camera flash charges about 220 microfarads to 300 volts, roughly 10 joules, and dumps it into the bulb in about a millisecond. A large 3,000-farad supercapacitor at 2.7 volts holds about 11,000 joules, around what's in a single AA battery, but can deliver it in seconds and recharge it hundreds of thousands of times. Your phone's touchscreen is a grid of tiny capacitors too: your conducting fingertip changes the field, and the controller notices.

Inside a thundercloud, updrafts fling ice crystals and soft hail (graupel) into each other. Each collision moves a little charge. The light crystals get carried up; the heavy graupel sinks. In the simplest picture the top of the cloud ends up positive and the main charge near the bottom negative. That negative base induces positive charge on the ground beneath it, and the field between them grows.

Air is a fine insulator until it isn't. There are always a few free electrons around, knocked loose by cosmic rays and natural radioactivity. In a strong enough field, one gains enough energy between collisions to knock an electron off a nitrogen or oxygen molecule. Now there are two, then four, then an avalanche. For dry air at sea level, that happens at about 3 million volts per meter: 3 kilovolts per millimeter.

A lightning flash starts with a faint stepped leader that works its way down in jumps of tens of meters. When it nears the ground, streamers rise to meet it, and the circuit closes. The return stroke carries a typical peak current of around 30,000 amps and heats the channel to about 30,000 kelvin, roughly five times hotter than the surface of the Sun. The air expands explosively, and you hear it as thunder. Sound covers a kilometer in about three seconds, so count the gap between flash and bang and divide by three for the distance in kilometers.

The voltage between cloud and ground runs to something like a hundred million volts. Worldwide, there are about 44 flashes every second.