One number for every point, and nothing at all is lost

A field is a vector at every point of space: three numbers, everywhere, forever. That is an uncomfortable amount of information, and the previous rung on this ladder made a point of how uncomfortable.

It can be reduced to one number per point, with nothing lost. Not compressed, not summarised — the three components are recoverable exactly from the single scalar, and every prediction the field made is still available. That reduction is the most economical trade in the subject, and this rung is about why it is possible at all.

What the number is #

Potential is potential energy per unit charge. Put a test charge $q$ at a point, ask how much work was needed to bring it there from far away, divide by $q$, and the answer is a property of the point rather than of the charge — which is the same manoeuvre that made the field itself a property of space rather than of the test charge.

For a single point charge the answer falls out of the inverse-square law:

$$V = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r},$$

and the exponent is worth a moment. The field falls as $1/r^2$ and the potential as $1/r$, because the potential is what has to be integrated to get from one to the other. A quantity and its rate of change differ by one power of the variable, always, and here that is the entire relationship between the two descriptions.

Why one number can be enough #

The reduction works because of a property of the electrostatic field, and it would fail for a field without that property.

The work done moving a charge between two points is independent of the route taken. That is the same statement as: the work around any closed loop is zero. And a quantity whose change between two points does not depend on the path is a difference of two numbers, one for each point — which is precisely what having a potential means.

So the potential exists for the same reason a hill has a height. Every route from the bottom of a hill to the top gains the same altitude, so altitude is a property of position; if some routes gained more altitude than others, no contour map could be drawn.

The recovery of the field from the potential is the slope. The field points in the direction of steepest decrease of $V$, with magnitude equal to that steepness:

$$\mathbf{E} = -\nabla V.$$

Three numbers from one, by differentiation. No information was lost because the three components were never independent — the condition that made the potential exist is exactly the condition that constrains them, and a general triple of functions is not an electrostatic field.

The right angle #

The most visible feature of the hero figure is that the two families of curves are perpendicular, everywhere, and the reason is a single sentence.

Moving along a contour of constant potential does no work, by definition. Work is force times displacement along the direction of motion, so zero work with a non-zero force means the force is perpendicular to the motion. The field is therefore perpendicular to every equipotential surface, at every point.

That exception is worth naming because it is where the picture’s structure is most informative. Contours of a function cross only at a saddle, and a saddle is where the gradient vanishes. So the crossing point in the like-charge figure is the null of the field, arrived at from the potential side; the arrow picture shows it as arrows shrinking to nothing and the line picture shows it as an absence of lines. Three representations, three completely different visual signatures, one physical fact.

The perpendicularity has an immediate practical consequence, and it is the one most often used. A conductor at equilibrium is an equipotential — if it were not, charges would move along its surface until it was — so field lines must meet a conductor’s surface at exactly ninety degrees. That single statement determines the shape of the field around every wire, plate and electrode in existence, and it does so without any calculation.

What the scalar buys #

The reduction is not only tidy; it makes the subject calculable, in four separate ways.

Superposition becomes addition. The potential of several charges is the ordinary arithmetic sum of their potentials — no components, no angles, no vector diagram. Adding fields requires resolving each into components and recombining; adding potentials requires adding numbers. For a continuous distribution, the difference is between three integrals and one.

Energy becomes immediate. The energy of a charge at a point is $qV$, so the whole apparatus of energy landscapes and turning points transfers to electrostatics unchanged. An electron accelerated through 100 volts arrives with 100 electronvolts of energy, which is the definition of the unit and the reason accelerator physicists count in volts rather than in joules.

It is measurable. No instrument reports the electric field at a point directly. Voltmeters, on the other hand, are everywhere, and what they report is a potential difference. The quantity chosen as fundamental in the theory and the quantity that instruments actually deliver are not usually the same, and here they are — which is why practical electricity is entirely conducted in volts and the field is left to the physicists.

And boundary conditions become statable. Nearly every real electrostatics problem is set by conductors held at fixed potentials — an electrode at 12 volts, another at earth. Stated in terms of the field those are awkward conditions; stated in terms of the potential they are simply values on a surface, and the resulting problem is one well-studied equation with well-understood solutions.

Reading a contour map as a field #

Since the potential is a landscape, everything that can be read off a landscape can be read off it, and the translations are worth listing because they turn a picture into a calculation.

Crowded contours mean a strong field, because the potential is changing rapidly over a short distance. That is the same encoding field-line density uses, and it has the same weakness: the eye judges spacing badly, and a two-dimensional slice misrepresents a three-dimensional density.

A flat region means no field. A region where the contours are far apart or absent is a region where nothing pushes, which is why the interior of a conductor shows as a single unbroken area of one value.

A closed contour surrounds a charge, or a null, which Gauss’s law states as a counting result. Contours cannot simply stop, any more than the height on a map can, so a contour that closes on itself has something inside it.

And the value carries a sign. Positive contours surround positive charge and negative surround negative, so the dipole figure shows two colours where the field-line figure shows one family of lines. That extra bit of information — the sign at each point, rather than only the direction of the arrow — is present in the scalar picture and absent from the vector one, which is the one respect in which the reduction to a single number gives more than it took.

What it costs #

Three charges, and the third one is severe enough to be its own section.

The zero is arbitrary. Only differences of potential mean anything. Setting $V = 0$ far away is a convention, convenient for isolated charges and useless for an infinite line of charge, whose potential diverges at infinity and must be referenced somewhere else. Every voltage ever quoted is a difference against something, usually left unstated, and “the voltage at a point” is not by itself a physical claim.

Recovering the field costs a derivative. Going from field to potential is an integration, which smooths; going back is a differentiation, which sharpens. Any error or noise in a measured or computed potential is amplified in the field derived from it, and the amplification grows with how finely the differentiation is done. Numerical electrostatics routinely solves for the potential — the equation is easier and the boundary conditions are natural — and then pays for it at the last step.

And the direction is only implicit. A contour map does not show which way is downhill until it is read carefully, and a beginner reading equipotentials often takes the contours themselves for lines of force, which is exactly ninety degrees wrong. The picture at the top of this page draws both families for that reason.

Where it fails outright #

The whole construction rests on the work around a closed loop being zero, and there is a circumstance in which it is not.

A changing magnetic field induces an electric field whose loops close on themselves, with no charge at either end. Carrying a charge around such a loop does work — that is what induction is for — so the work around a closed path is not zero, so no potential function exists. The reduction to one number per point simply stops being available.

This is not an exotic edge case. It is what every transformer, motor and generator runs on — lines that close on themselves with no charge to end on — and it has a consequence that reliably astonishes anyone meeting it for the first time: in the presence of a changing magnetic field, the voltage between two points depends on how the voltmeter’s leads are routed.

The demonstration is a loop of wire with two unequal resistors in it, threaded by a changing magnetic field. Two voltmeters connected to the same two points, with their leads passing on opposite sides of the region where the field is changing, read different values — different in magnitude, and of opposite sign. Both are correct. There is no single number that is “the voltage between those points”, because the quantity being measured is a line integral and the two instruments are integrating along different lines.

Everything that makes circuit theory work — that a node has a voltage, that Kirchhoff’s loop rule holds — is therefore a statement that no significant flux is changing inside the circuit. That assumption is nearly always true and it is an assumption, and the cases where it fails are exactly the ones that ordinary circuit intuition gets wrong.

The crowding at a point, and why lightning rods are pointed #

One consequence of the potential picture is worth following through, because it explains a piece of everyday hardware and because the reasoning is short.

Take two conducting spheres of different radii, connected by a thin wire. Being connected, they are one conductor, so they share a single potential. For a sphere of radius $R$ carrying charge $Q$, that potential is $Q/4\pi\varepsilon_0 R$, so equal potentials require the charges to be in the ratio of the radii — the big sphere takes most of the charge.

But the field at a sphere’s surface is $Q/4\pi\varepsilon_0R^2$, which is the potential divided by $R$. Equal potentials therefore mean the field at the small sphere’s surface is larger, in inverse proportion to its radius. Making the radius ten times smaller makes the surface field ten times stronger, while the charge it carries falls.

So sharp regions of a conductor have intense fields around them, and the sharper the more intense. Air breaks down at about three million volts per metre — a threshold that is a statement about the tail of a distribution as much as about the field, so a sufficiently sharp point on a charged conductor ionises the air near it and begins to leak charge — corona discharge, visible as a faint glow and audible as a hiss.

That is the whole of a lightning rod, and the folklore about it is half wrong. Its main function is not to be struck; it is to bleed charge into the air continuously and reduce the potential difference that would otherwise build. When it is struck, its second function — an unbroken conducting path to earth — takes over. Franklin argued for points and Wilson argued for blunt ends, the dispute became entangled with the American revolution, and George III had the rods on his palace changed to blunt ones as a matter of loyalty. The physics was on Franklin’s side, and the potential picture is the shortest route to seeing why.

The same crowding is what limits high-voltage engineering everywhere: every edge is rounded, every conductor is thicker than it needs to be for current, and toroidal rings are fitted around terminals for no reason except to keep the local radius of curvature large.

Where the potential came from #

The idea arrived from gravity rather than from electricity, and it arrived because a scalar was easier to compute with.

Lagrange used a function of this kind in celestial mechanics in the 1770s; Laplace developed it into the equation now named after him; and Poisson extended it to the case with sources present, giving the pair of equations that remain the working tools of the subject. The word potential was Green’s, in an essay of 1828 that he published privately in Nottingham, having taught himself mathematics while running a windmill.

Green’s essay was almost entirely unread. Fifty copies were printed by subscription, and it was rediscovered only after his death, by William Thomson, who found a copy in 1845 and had it reprinted — at which point it turned out to contain the theorem now called Green’s, the concept of the potential function, the method of images, and most of the analytic framework that Maxwell would later use. Maxwell was explicit about the debt.

What is worth extracting from that is not the romance of the self-taught miller but the shape of the contribution. Green did not discover a new phenomenon. He introduced a quantity — a way of writing down what was already known — and the quantity turned out to make problems solvable that had previously been merely stateable. That kind of contribution is much harder to notice at the time, which is a fair part of why the essay sat unread for seventeen years.

The ladder from here #

Later rungs: Poisson’s and Laplace’s equations, and what “solve for the potential” actually involves. Uniqueness — why specifying the potential on the boundary determines it everywhere inside, which is what makes the boundary-value formulation legitimate. The method of images, which replaces a conductor by a fictitious charge. Capacitance as the ratio of charge to potential difference. Energy stored in a field, computed as the work of assembly. The multipole expansion, which is the potential of a complicated distribution written as a series with the monopole first. The magnetic case, where no scalar potential exists in general and a vector one takes its place. And the gauge freedom that goes with that vector potential, which is the same arbitrariness as the zero of $V$, promoted into one of the organising principles of modern physics.