Charge and Coulomb's law

Rub a plastic rod on wool and it will pick up scraps of paper against the pull of the whole Earth, which is the first hint that the electric force is not a small correction to anything.

This course assumes the mechanics in Classical Mechanics (Newton's laws, work and energy, conservative forces, circular motion and torque) and the calculus in Calculus (derivatives, definite integrals, and separation of variables). It assumes no vector calculus at all: every field equation in it, including the four that Maxwell assembled, is written and used in integral form. What it does assume is a willingness to take a force law and follow it, because almost everything in the next twelve lessons is a consequence of the one law derived here.

Two kinds of charge

The oldest recorded fact in the subject is that amber, rubbed, attracts light objects, which is where the word electricity comes from: the Greek for amber is elektron. For two thousand years that was the whole of it. The useful step was Stephen Gray's, in 1729, when he showed the attracting property could be conducted along a moist thread for hundreds of feet, and so was something transferable rather than a property of a rubbed surface. Materials divided at once into conductors, which pass it, and insulators, which do not.

Charles du Fay found the crucial complication in 1733. Two glass rods, each rubbed with silk, repel each other. Two amber rods, each rubbed with fur, also repel each other. But a rubbed glass rod and a rubbed amber rod attract. There is no way to fit that into a single quantity that objects have more or less of, so du Fay proposed two electricities, vitreous and resinous, with like repelling like and unlike attracting.

Benjamin Franklin recast it around 1750 as a single fluid present in every body in a natural amount. Rubbing transfers some from one body to the other, so one ends with a surplus, which he called positive, and the other with a deficit, which he called negative. Franklin's picture is closer to the truth than du Fay's, since what actually moves is electrons, and it has one famous cost. Franklin guessed that the fluid flowed from the glass into the silk, so he called the glass negative and the silk positive, and the guess was backwards. The carrier of ordinary electric current in a metal is the electron, which by Franklin's convention carries a negative charge, so conventional current is drawn flowing the way positive charge would flow, which is opposite to the actual drift of the electrons. Every circuit diagram ever drawn still carries that error, harmlessly, because nothing physical depends on which sign is called positive.

What survives from all of this is one experimental rule: like charges repel, unlike charges attract, and nothing else in the subject has ever been observed. There is no third sign.

Conserved, and quantised

Two further facts, neither of them obvious, make charge a useful quantity rather than a description of rubbed rods.

The first is conservation. The total charge of an isolated system never changes. Rubbing does not create charge, it separates it: the wool ends up as positive as the rod is negative, and if you weigh the pair together the net charge is what it was. This survives into contexts Franklin never imagined. A neutron decays into a proton, an electron and an antineutrino, and the charges before and after are both zero. A gamma ray of sufficient energy converts into an electron and a positron, never into one alone. Charge conservation is tested harder than almost any law in physics: if the electron could decay into neutral particles, charge would vanish, and the Borexino detector's search for that decay puts the electron's lifetime above about 10²⁸ years, which is eighteen orders of magnitude longer than the age of the universe.

The second is quantisation. Charge comes in integer multiples of a fixed amount. Robert Millikan's oil drop experiment, run from 1909 and refined to 1913, sprayed a fine mist of oil between two horizontal plates and balanced individual droplets against gravity by adjusting the field. The charge on each droplet came out as a multiple of a single unit, and no droplet ever showed a fraction of it. His published value was $1.592 \times 10^{- 19}$ coulombs, low by 0.64 per cent against the modern figure, and the reason is instructive: he used a value for the viscosity of air that was slightly wrong, and the error propagated straight into $e$ . Since the 2019 redefinition of SI units the elementary charge is not measured at all but fixed by definition, at exactly

e = 1.602176634 \times 10^{- 19} C

and the coulomb is now defined as the charge of $1 / e$ elementary charges. Quarks carry $\pm e / 3$ and $\pm 2 e / 3$ , but no experiment has ever isolated one, so every free object carries an integer multiple of $e$ .

The quantum is small enough that it is invisible in ordinary electrostatics. A charge of one microcoulomb, easily produced by rubbing, is about $6 \times 10^{12}$ elementary charges, so treating charge as a smooth continuous fluid costs nothing at laboratory scales, exactly as treating water as continuous costs nothing when the vessel holds $10^{25}$ molecules.

Coulomb's measurement

Knowing that charges push and pull says nothing about how hard. Several people guessed an inverse square by analogy with gravity, and Joseph Priestley argued for it in 1767 from an experiment of Franklin's that this course reaches in the third lesson. The measurement that settled it was Charles Augustin de Coulomb's, published in 1785.

His instrument was a torsion balance, and the reason it worked is that it converts an unmeasurably small force into a measurable angle. A fine silver wire hangs from a fixed head and carries a light horizontal needle with a small pith ball at one end, all sealed inside a glass cylinder to keep out draughts. A second, fixed ball is brought in and both are charged, and the repulsion swings the needle round until the twist in the wire balances it. Coulomb calibrated the wire separately by hanging known small torques on it, so the deflection angle read directly as a force. The trick, which is the whole art of the thing, is that the restoring torque of a thin wire can be made almost arbitrarily weak by making the wire long and fine, so a force of a few micronewtons becomes a swing of tens of degrees.

His published numbers deserve to be looked at rather than summarised. Starting with the balls 36 degrees apart, he halved the separation to 18 degrees and found the torsion needed rose from 36 units to 144, a factor of exactly four for a halving of distance. Then he pressed to 8.5 degrees, where an inverse square predicts $36 \times (36 / 8.5)^{2} = 646$ units, and measured 575.5, about 11 per cent low. He attributed the shortfall to charge leaking away from the balls during the minutes the reading took, which was almost certainly right, and it is worth noticing that the law was announced on the strength of two good points and one poor one. The law is believed today because of the far more sensitive experiments described at the end of this lesson, not because of Coulomb's data.

The law

For two point charges $q_{1}$ and $q_{2}$ separated by a distance $r$ , the force each exerts on the other has magnitude

F = k \frac{| q_{1} q_{2} |}{r^{2}}

and is directed along the line joining them, repulsive if the charges have the same sign and attractive if they do not. The two forces are equal in magnitude and opposite in direction, which is Newton's third law and not an extra assumption: it comes out of the symmetry of the expression under swapping the labels.

"Point charge" is doing real work in that sentence. The law as written is exact only for charges of no size, and applies to spheres of finite size only because of a theorem proved in the third lesson, the electrical twin of the shell theorem that lets a planet be treated as a point.

The constant depends on the units. In SI,

k = 8.9875 \times 10^{9} {N m² C}^{- 2}

which is usually written instead as $k = 1 / (4 π ε_{0})$ , with $ε_{0} = 8.8542 \times 10^{- 12}$ farads per metre called the permittivity of free space. Moving the $4 π$ into the constant looks like gratuitous ugliness at this stage, and at this stage it is. The payoff arrives in the third lesson, where the $4 π$ cancels the surface area of a sphere and leaves Gauss's law clean. Physics chose to put the awkwardness in the force law rather than in the field equations, on the grounds that the field equations get used more.

Example. A charge of $+ 3.0$ μC and a charge of $- 5.0$ μC sit 20 cm apart in air. What force does each feel?

$F = k | q_{1} q_{2} | / r^{2} = (8.9876 \times 10^{9}) (3.0 \times 10^{- 6}) (5.0 \times 10^{- 6}) / (0.20)^{2}$ . The numerator is $8.9876 \times 10^{9} \times 1.5 \times 10^{- 11} = 0.1348$ , and dividing by $0.04$ gives $F = 3.37$ N. The signs are opposite, so it is an attraction, and both charges feel 3.37 N pulling them together.

Now you. A charge of $+ 2.0$ μC and a charge of $+ 7.0$ μC sit 15 cm apart. What force does each feel?

Answer

$F = (8.9876 \times 10^{9}) (2.0 \times 10^{- 6}) (7.0 \times 10^{- 6}) / (0.15)^{2} = (8.9876 \times 10^{9}) (1.4 \times 10^{- 11}) / 0.0225 = 5.59$ N, repulsive, since both charges are positive.

Superposition

Coulomb's law covers two charges, and the world contains more. The additional experimental fact, and it is a fact rather than a deduction, is superposition: the force on a charge from several others is the vector sum of the forces each would exert alone, unaffected by the presence of the rest. A third charge placed between two others does not screen them or weaken their interaction.

That is not guaranteed. A theory in which the force between two charges depended on what else was nearby would be perfectly consistent, and gravity in general relativity is exactly such a theory, which is why gravitational fields do not simply add. Electromagnetism is linear to the limits of measurement, and every calculation in this course leans on it.

In practice superposition means resolving into components and adding. On a line it means keeping track of signs.

Example. Three charges sit on the $x$ axis: $q_{1} = + 5.0$ nC at $x = 0$ , $q_{2} = + 3.0$ nC at $x = 4.0$ cm, and $q_{3} = - 8.0$ nC at $x = 10.0$ cm. What is the net force on $q_{2}$ ?

Take rightwards as positive. The charge $q_{1}$ is positive and 4.0 cm to the left, so it pushes $q_{2}$ rightwards: $F_{1} = (8.9876 \times 10^{9}) (5.0 \times 10^{- 9}) (3.0 \times 10^{- 9}) / (0.040)^{2} = 8.43 \times 10^{- 5}$ N, positive. The charge $q_{3}$ is negative and 6.0 cm to the right, so it pulls $q_{2}$ rightwards too: $F_{3} = (8.9876 \times 10^{9}) (3.0 \times 10^{- 9}) (8.0 \times 10^{- 9}) / (0.060)^{2} = 5.99 \times 10^{- 5}$ N, also positive. Both point the same way, so the net force is $1.44 \times 10^{- 4}$ N in the $+ x$ direction.

Now you. Three charges sit on the $x$ axis: $q_{1} = + 6.0$ nC at $x = 0$ , $q_{2} = + 2.0$ nC at $x = 5.0$ cm, and $q_{3} = + 9.0$ nC at $x = 10.0$ cm. What is the net force on $q_{2}$ ?

Answer

Both neighbours are positive, so they push $q_{2}$ in opposite directions. From $q_{1}$ : $(8.9876 \times 10^{9}) (6.0 \times 10^{- 9}) (2.0 \times 10^{- 9}) / (0.050)^{2} = 4.31 \times 10^{- 5}$ N to the right. From $q_{3}$ : $(8.9876 \times 10^{9}) (2.0 \times 10^{- 9}) (9.0 \times 10^{- 9}) / (0.050)^{2} = 6.47 \times 10^{- 5}$ N to the left. The net is $2.16 \times 10^{- 5}$ N in the $- x$ direction, towards $q_{1}$ .

How strong is it, really

Put two charges of one coulomb each a metre apart and the force between them is $8.99 \times 10^{9}$ N, the weight of nine hundred thousand tonnes. That tells you immediately that a coulomb is an enormous amount of static charge, and that the situation described never occurs: the charges would tear themselves off the objects holding them long before you assembled them.

A better feel comes from asking how little charge separation is needed to matter. One gram of copper contains $6.022 \times 10^{23} / 63.546 = 9.48 \times 10^{21}$ atoms, each with 29 electrons, so $2.75 \times 10^{23}$ electrons in total. Strip away one electron in every $10^{9}$ and the sphere is left with $2.75 \times 10^{14}$ elementary charges, which is $4.40 \times 10^{- 5}$ C. Two such spheres a metre apart repel with $k q^{2} / r^{2} = 17.4$ N, about the weight of a 1.8 kg brick, from removing a billionth of the available electrons from a gram of metal. Matter is neutral to a staggering precision because it has no choice: the smallest imbalance produces forces that undo it.

The comparison with gravity is the sharpest statement of this. In a hydrogen atom, with the electron at the Bohr radius of 52.9 pm, the electrical attraction is

F_{e} = \frac{k e^{2}}{r^{2}} = 8.24 \times 10^{- 8} N

while the gravitational attraction between the same two particles is $G m_{e} m_{p} / r^{2} = 3.63 \times 10^{- 47}$ N. The ratio is $2.27 \times 10^{39}$ , and note that it does not depend on $r$ , since both fall as the inverse square: it is $k e^{2} / (G m_{e} m_{p})$ , a pure number built from constants. Gravity is weaker than electricity by thirty-nine orders of magnitude.

Which raises the obvious question of why gravity, and not electricity, shapes the solar system. The answer is the two signs. Gravity has only one, so mass accumulates and its effect grows without limit; charge has two, so any large accumulation attracts its own neutraliser and is cancelled. The Sun holds about $10^{57}$ protons and very nearly exactly $10^{57}$ electrons, and the residue is what matters. Astronomy is gravitational not because gravity is strong but because electricity is so strong that it is never left over.

How well do we know the exponent

Writing $F \propto r^{- 2}$ commits to the exponent being exactly 2, and that is a strong claim about a measured quantity. Coulomb's own data would support $r^{- 2.06}$ as happily as $r^{- 2}$ .

The good bound does not come from measuring forces at all, but from a null experiment described in the third lesson: an inverse square law, and only an inverse square law, implies that the electric field inside a hollow closed conductor is exactly zero. Any charge placed inside must then move entirely to the outside surface. Henry Cavendish did this in 1773 with concentric spheres and, finding no charge inside, concluded the exponent was $2 \pm 0.02$ . Maxwell repeated it with better instruments in 1873 and tightened the bound to about one part in twenty thousand. The modern version, done by Williams, Faller and Hill in 1971 with a five-shell apparatus and a lock-in amplifier, writes the law as $F \propto r^{- (2 + q)}$ and reports $| q | < 3 \times 10^{- 16}$ .

What is being tested is more than a curve fit. In modern theory the exponent is exactly 2 if and only if the photon has exactly zero rest mass, and a massive photon would give a force falling off exponentially with a range set by that mass. The null result therefore converts into a bound on the photon mass, currently below about $10^{- 18}$ electronvolts in mass-energy, which is roughly $10^{- 54}$ kilograms. This is one of the places where a laboratory measurement in a basement constrains a fundamental constant of the universe.

Example. Two identical positive charges are fixed on the $x$ axis, $+ 9.0$ μC at $x = 0$ and $+ 4.0$ μC at $x = 1.00$ m. Where on the line between them does a test charge feel no net force?

Let the point be at distance $r$ from the 9 μC charge, so it is $1.00 - r$ from the other. Setting the magnitudes equal, $k (9) q / r^{2} = k (4) q / (1 - r)^{2}$ , and both $k$ and the test charge $q$ cancel, which is why the answer does not depend on either. Taking the square root of both sides gives $3 / r = 2 / (1 - r)$ , so $3 - 3 r = 2 r$ and $r = 0.60$ m. The null point sits closer to the smaller charge, as it must.

Now you. Two positive charges are fixed on the $x$ axis, $+ 16$ μC at $x = 0$ and $+ 4.0$ μC at $x = 1.50$ m. Where between them is the net force on a test charge zero?

Answer

$16 / r^{2} = 4 / (1.5 - r)^{2}$ , so $4 / r = 2 / (1.5 - r)$ , giving $6 - 4 r = 2 r$ and $r = 1.00$ m from the 16 μC charge, that is 0.50 m from the 4.0 μC charge.

What the law does not say

Coulomb's law is a complete account of what two charges do to each other, and it contains a silence at its centre. Nothing in $F = k q_{1} q_{2} / r^{2}$ says how the second charge learns that the first one is there, or across what, or how quickly. Written as it stands, the force depends on the separation now, which means that moving one charge would change the force on the other instantaneously, at any distance. That is action at a distance, and Newton, who had the same problem with gravity, called the idea so great an absurdity that no competent thinker could fall into it, while continuing to use it because it worked.

The repair, developed by Faraday in the 1830s and made mathematical by Maxwell, is to split the interaction into two steps. A charge fills the space around it with a condition, and a second charge responds to the condition where it sits, knowing nothing about what produced it. The condition is the electric field, and it will turn out to have energy, to carry momentum, to take a definite time to propagate, and eventually to exist entirely on its own with no charge anywhere near it. That last case is light. Building the field is the next lesson.