Electrostatics and Coulomb's Law

Pull a wool jumper over your head in dry weather and you hear it: a faint crackle, and if the room is dark, tiny sparks. Shuffle across a carpet and touch a doorknob and you get a sharp little zap. Rub a balloon on your hair and it clings to the wall as if glued there. All of these are the same phenomenon — electric charge — sitting still and making itself felt. That is what electrostatics studies: charges at rest, and the forces they push and pull on one another with.

Those forces are not small. The same electric attraction that makes a balloon stick to a wall is the force that holds every atom together, binds atoms into molecules, and is the reason a chair is solid enough to sit on rather than letting you fall straight through. Master a single equation — Coulomb's law — and you hold the key to almost all of chemistry, materials, and everyday matter. This page builds that one law carefully, from what charge is to how to add up the forces from many charges at once.

Two kinds of charge

Rub a plastic rod with fur and it picks up a charge; rub a glass rod with silk and it picks up a charge too — but of the opposite kind. Bring the two charged rods together and they attract. Bring two plastic rods together and they repel. Benjamin Franklin gave the two kinds their names: positive and negative. The rule that governs all of electrostatics is beautifully simple:

Charge is carried by the particles inside atoms: the proton carries one unit of positive charge, the electron exactly one unit of negative charge, and the neutron none. Charge is measured in coulombs (C). It comes in three iron rules worth memorising:

A coulomb is an enormous amount of charge — it takes about 6.24\times 10^{18} electrons to make up just one. That is why the charges you meet in the lab are usually measured in microcoulombs (\mu\text{C}, 10^{-6}\ \text{C}) or nanocoulombs.

Conductors, insulators, and how things get charged

Why does charge flow freely through a copper wire but sit stubbornly on a balloon? It comes down to how tightly a material grips its electrons.

There are two everyday ways to charge an object. Charging by friction is the balloon trick: rubbing two different insulators together transfers electrons from one to the other. Charging by induction is sneakier and needs no contact at all: bring a charged rod near a conductor and the conductor's free electrons shift — repelled or attracted by the rod — so one side of the conductor becomes negative and the other positive, even though its total charge is still zero. This separation of charge without touching is why a charged comb can pick up small neutral scraps of paper.

Coulomb's law: the force between two charges

Knowing that like charges repel and unlike attract is qualitative. In the 1780s Charles-Augustin de Coulomb made it quantitative, measuring exactly how strong the force is. His result is one of the cleanest laws in all of physics:

Read the formula slowly, because every piece is telling you something:

Coulomb's law also has a vector form that carries the direction with it. If \hat{\mathbf{r}} is the unit vector pointing from charge 1 to charge 2, the force on charge 2 is

\mathbf{F}_{12} = k\,\frac{q_1\,q_2}{r^2}\,\hat{\mathbf{r}}.

Here the sign does the work automatically: if q_1 q_2 > 0 (like charges) the force points along \hat{\mathbf{r}}, away from charge 1 — repulsion. If q_1 q_2 < 0 (unlike charges) it points back the other way — attraction. No need to remember which; the algebra tells you.

Seeing the inverse-square law

The 1/r^2 falloff is worth feeling, not just reading. The graph below plots the force magnitude F = k q^2 / r^2 between two equal charges of size q as you slide them apart. Drag the slider to change the charge and watch how steeply the curve dives as r grows.

Two features stand out. Near r = 0 the curve rockets upward — bring point charges very close and the force becomes ferociously strong. And as r increases the curve flattens towards (but never touches) zero: the force has infinite range, yet becomes negligible at large distances. Turning up the charge q lifts the whole curve, because the force grows with the product of the charges.

Worked example 1 — a straight force calculation

Two point charges, q_1 = +3.0\ \mu\text{C} and q_2 = -5.0\ \mu\text{C}, sit r = 0.20\ \text{m} apart. What force do they feel? Put the magnitudes into Coulomb's law (a micro is 10^{-6}):

F = k\,\frac{|q_1 q_2|}{r^2} = (8.99\times 10^{9})\,\frac{(3.0\times 10^{-6})(5.0\times 10^{-6})}{(0.20)^2}. F = (8.99\times 10^{9})\,\frac{1.5\times 10^{-11}}{0.040} \approx 3.4\ \text{N}.

Because the charges have opposite signs, the force is attractive — each charge is pulled toward the other with about 3.4\ \text{N}, roughly the weight of a large apple. Not bad for two specks of charge you can't even see.

Worked example 2 — Coulomb versus gravity

Just how strong is the electric force? Compare it with gravity for the two most fundamental particles we can: two protons. Each proton has charge e = 1.6\times 10^{-19}\ \text{C} and mass m_p = 1.67\times 10^{-27}\ \text{kg}. Both forces obey an inverse square law, so when we take their ratio the distance r cancels completely — the answer is the same at any separation:

\frac{F_{\text{elec}}}{F_{\text{grav}}} = \frac{k\,e^2/r^2}{G\,m_p^2/r^2} = \frac{k\,e^2}{G\,m_p^2}.

Plugging in k = 8.99\times 10^{9} and the gravitational constant G = 6.67\times 10^{-11}:

\frac{F_{\text{elec}}}{F_{\text{grav}}} = \frac{(8.99\times 10^{9})(1.6\times 10^{-19})^2}{(6.67\times 10^{-11})(1.67\times 10^{-27})^2} \approx 1.2\times 10^{36}.

The electric repulsion between two protons is about a trillion trillion trillion times stronger than their gravitational attraction. Gravity only dominates the universe at large scales because big objects are electrically neutral — their vast positive and negative charges cancel. Charge doesn't cancel gravity; it simply overwhelms it whenever it isn't balanced.

Superposition: adding up many charges

Coulomb's law is a rule for one pair of charges. But the real world has charges everywhere. What happens when three or four or a million charges all act on one? The answer is the superposition principle, and it is wonderfully simple:

The word vector is doing heavy lifting there. You cannot just add the magnitudes — you must add the forces as arrows, taking their directions into account. Two forces of 3\ \text{N} pointing the same way give 6\ \text{N}; pointing opposite ways they cancel to zero; at right angles they combine to \sqrt{3^2 + 3^2} \approx 4.2\ \text{N}. The figure below shows a test charge Q feeling a pull from one source and a push from another, and how the two arrows combine into a single net force by completing the parallelogram.

Worked example 3 — three charges in a line

Put three charges on a straight line. At x = 0 a charge q_1 = +2.0\ \mu\text{C}; at x = 0.10\ \text{m} a charge q_2 = +4.0\ \mu\text{C}; and at x = 0.30\ \text{m} the charge we care about, q_3 = +1.0\ \mu\text{C}. What is the net force on q_3?

Compute each Coulomb force separately. From q_1, a distance 0.30\ \text{m} away:

F_{13} = (8.99\times 10^{9})\,\frac{(2.0\times 10^{-6})(1.0\times 10^{-6})}{(0.30)^2} \approx 0.20\ \text{N}.

From q_2, a distance 0.20\ \text{m} away:

F_{23} = (8.99\times 10^{9})\,\frac{(4.0\times 10^{-6})(1.0\times 10^{-6})}{(0.20)^2} \approx 0.90\ \text{N}.

Both source charges are positive, and they both sit to the left of q_3, so both forces push q_3 to the right — they point the same way. Here the vector sum is just the ordinary sum:

F_{\text{net}} = F_{13} + F_{23} \approx 0.20 + 0.90 = 1.1\ \text{N, to the right.}

If one of the source charges had been negative, its force would have pointed the other way and we would have subtracted. Always draw the arrows first — the signs and directions are the whole game.

The single most common slip with superposition is to compute each Coulomb force, add the magnitudes, and call it done. That is only correct when all the forces happen to point the same direction, as they did in Example 3. In general they don't.

Coulomb's law gives you the force from one pair of charges at a time — a single arrow with a size and a direction. To get the net force you must add those arrows as vectors: break each into x and y components, add the components separately, then recombine. Two 5\ \text{N} forces at right angles do not make 10\ \text{N} — they make \sqrt{5^2+5^2} \approx 7.1\ \text{N} at 45^\circ. Forget the directions and your answer can be off by a huge factor, or point the wrong way entirely. When in doubt, draw the arrows.

With astonishing cleverness. Coulomb built a torsion balance: a light insulating rod hung horizontally from a thin silk (later metal) fibre, with a small charged ball on one end. He brought a second charged ball nearby, and the electric force twisted the rod, winding up the fibre. Because a fine fibre resists twisting only very slightly, even a minuscule force produced a measurable rotation — and the twist angle told him the force. By varying the distance he confirmed the 1/r^2 law directly. (The same torsion-balance trick was used a few years later by Henry Cavendish to weigh the Earth via gravity — it is one of the great instruments of physics.)

There is a lovely footnote here: the 1/r^2 form is now known to be exact to extraordinary precision. If the exponent were even slightly different from 2 — say 2.000001 — it would betray a tiny mass for the photon and break deep symmetries of electromagnetism. Experiments confirm the "2" to better than one part in 10^{16}. Coulomb's eighteenth-century twist of a fibre pointed straight at a law of nature that still holds today.

This is a genuinely deep puzzle. An atomic nucleus crams many positively charged protons into a space about 10^{-15}\ \text{m} across. At that tiny separation the Coulomb repulsion between them is colossal — by the inverse-square law, squeezing them a hundred thousand times closer than atomic distances makes the repulsion ten billion times fiercer. Left to electrostatics alone, every nucleus would blow itself apart instantly.

Something stronger must be holding it together, and there is: the aptly named strong nuclear force. It is far more powerful than the electric force but has an extremely short range, acting only over distances comparable to the nucleus itself. Within that range it easily overpowers the Coulomb repulsion and glues the protons and neutrons together. Push a nucleus too big, though — like uranium — and the long-range Coulomb repulsion starts to win against the short-range glue, which is exactly why heavy nuclei are unstable and undergo radioactive decay and fission. Coulomb's law is still there in the heart of the atom; it just meets its match.