How much charge a shape will hold, before anything is charged

Take two pieces of metal, hold them near each other, and put charge $+Q$ on one and $-Q$ on the other — each one’s charge arranging itself so the other’s field ends on it. A potential difference $V$ appears between them. Double the charge and the potential difference doubles exactly; halve it and it halves. The ratio

$$C = \frac{Q}{V}$$

does not depend on how much charge was used. It is fixed before any charge is delivered, by nothing but the shapes of the two conductors and how they are arranged.

That the ratio is constant follows from linearity: the field of a charge distribution is proportional to the charge, the potential difference is an integral of the field, so doubling every charge doubles every field and every potential difference and changes no ratio anywhere. What is less obvious, and is the argument of this rung, is that the constant is a piece of geometry — a number with units of length times a constant of nature, computable from a drawing.

The plane result, and where it comes from #

For two parallel plates of area $A$ separated by $d$, the answer is

$$C = \frac{\varepsilon_0 A}{d},$$

and every step of the derivation has already appeared on this site.

The field of a charged plane is $\sigma/2\varepsilon_0$ and does not weaken with distance. Two oppositely charged planes therefore give $\sigma/\varepsilon_0$ between them and zero outside, the contributions adding in one region and cancelling in the other. The potential difference is the field times the gap, $V = \sigma d/\varepsilon_0$. The charge is $Q = \sigma A$. Divide.

The formula therefore says three things, and the third is the useful one.

Bigger plates hold more. Doubling the area doubles the charge at the same potential, because the same surface charge density covers twice the surface.

Closer plates hold more. This is the counter-intuitive half. Halving the gap does not change the field for a given surface density; it halves the potential difference, because there is half as much distance to integrate the field over. Half the voltage for the same charge is twice the capacitance.

Nothing else enters. Not the metal, not the thickness of the plates, not how the charge was delivered. The permittivity of free space, an area and a length.

The numbers on that curve are worth registering, because they explain the whole industry. Two hand-sized plates half a millimetre apart make 177 picofarads. Reaching one farad by that route needs an area of roughly $10^8$ square metres, which is a hundred square kilometres of metal. That every phone contains capacitors of a farad or more says that something other than two flat plates is going on, and the something is dealt with below.

The edges, and the honest version of the picture #

The standard drawing of a capacitor shows a set of parallel lines between two plates, with a note that edge effects have been ignored. The hero figure ignores nothing: it traces the field from the charges themselves, so the edges appear whether or not they are wanted.

The measured uniformity is the figure’s own assertion, and it moves the way it should: 92 per cent of the central field at nine-tenths of the way out when the gap is a tenth of the width, 80 per cent at a fifth, 72 per cent at a third, 68 per cent at six-tenths. The idealisation is not a lie that becomes a truth at some threshold; it is a limit approached as $d/w \to 0$, and the aspect ratio is the parameter that says how close.

Two practical consequences follow directly.

Real capacitors are wide and thin because the formula demands it, not because it is convenient. A film capacitor is metres of foil separated by micrometres of plastic, rolled up — an aspect ratio of a million, which puts the fringing field somewhere in the eighth decimal place.

A guard ring buys the last few digits. Surrounding a measuring plate with a separate, co-planar ring held at the same potential moves every edge in the apparatus out to the ring, so the measured plate sees only the uniform middle. That is how the parallel-plate formula becomes a standard rather than an approximation, and it is the same trick as the one that makes the shielding argument exact in a real box: arrange for the awkward geometry to happen somewhere that is not being measured.

Why the charge sits where it does #

The plates in the figure carry charge on their inner faces, and it is worth asking what put it there, because nobody arranged it.

Inside a conductor the field is zero, which forces the charge to the surface, and the surface density then adjusts until the field it produces satisfies that condition everywhere. On an isolated plate the charge would spread over both faces. Bring an oppositely charged plate up close and the attraction pulls almost all of it to the inner faces — the two sheets of charge hold each other in place.

Why the charge sits where it does is decided by the same condition. Solve for the surface charge on a conductor and almost everything ends up on the inner faces of two facing plates, because that is where the opposite charge is to hold it. The residue on the outer surfaces is what the fringing field is made of — a small fraction of the total, and the whole of the difference between the idealised capacitor and a real one.

This is why capacitance is a property of the pair and not of either conductor. A single isolated sphere does have a capacitance — $C = 4\pi\varepsilon_0 R$, its second plate being the sphere at infinity — and for a sphere the size of the Earth that comes to 710 microfarads, which is a small capacitor. Capacitance is a relationship, and the closer the partner, the larger it is.

What is stored, and where #

Charging a capacitor takes work, because every additional charge has to be pushed against the potential difference already established by the charge already there. Integrating that gives

$$U = \tfrac12 CV^2 = \tfrac12 QV = \frac{Q^2}{2C},$$

and the factor of a half is the average potential difference over the charging, not a fudge.

The more interesting question is where the energy is. Two answers are available: on the charges, or in the field between the plates. Electrostatics cannot distinguish them, because the two bookkeepings give the same total for every arrangement.

The field version writes the energy density as $u = \tfrac12\varepsilon_0 E^2$ at every point of space, and integrating that over the volume between the plates reproduces $\tfrac12 CV^2$ exactly. It is the version that turns out to be right, and the evidence is not electrostatic at all: a changing field detaches from its source and eventually radiates away as light, carrying energy with it to places where no charge exists. Energy that can leave the charges behind was never on the charges.

What is stored, and where, is the question a field picture answers and a charge picture does not. Drawn with its charges, the energy looks as though it belongs to them. The case that decides it is the one where the source is removed and the field carries on existing — a radiated field carries energy away from a charge that no longer has it — so the energy is in the field, and a capacitor is a device for putting it there in a small volume.

That reading also explains a piece of engineering that is otherwise arbitrary. Since $u \propto E^2$, energy density rises with the square of the field, so the way to store more in a given volume is to run a stronger field rather than to use more space. Which is why the limit on every capacitor is the field at which its insulator breaks down, and why the whole design problem is a competition between permittivity and dielectric strength.

Where the model stops #

The vacuum assumption. Everything above has empty space between the plates. Real capacitors have an insulator, whose molecules polarise in the field and partly cancel it — reducing the potential difference for a given charge, and multiplying the capacitance by the relative permittivity, a factor of 2 to 3 for plastics and of several thousand for the ceramics used in the small blue components on any circuit board. The formula becomes $C = \varepsilon_r\varepsilon_0 A/d$, and the physics of $\varepsilon_r$ is a separate subject.

The breakdown field. Making $d$ small increases the capacitance and also increases the field for a given voltage. Air breaks down at about 3 MV/m, so a millimetre gap cannot hold more than about 3 kV whatever else is true. The two halves of the design pull in opposite directions, which is why capacitors are specified with a voltage rating rather than a capacitance alone, and why exceeding it destroys them rather than merely misreading.

Static only. A capacitance is defined from an electrostatic potential difference. At high enough frequency the wavelength becomes comparable to the component and there is no single potential difference across it — the same failure of the potential concept that induction introduces, arriving from a different direction. Every real capacitor also has an inductance, and above its self-resonant frequency it behaves as an inductor, which is a piece of information that has ended more designs than it has saved.

The double layer. The supercapacitor gets to farads by making $d$ molecular. Charge on an electrode attracts ions in a liquid to within a nanometre or so, and the separation in the formula becomes the thickness of that layer, with an area given by porous carbon at hundreds of square metres per gram. The parallel-plate formula still describes it — an area of $10^8$ square metres does appear, folded into a few grams — but nothing about the arrangement resembles two plates, and the voltage limit is now set by the chemistry of the electrolyte rather than by a breakdown field.

Geometry as the whole of the answer #

The recurring idea here is worth stating plainly, because it appeared two rungs ago in a different guise.

The falloff exponent of a field turned out to be a statement about the shape of the source and not about electricity. Capacitance is the same kind of quantity: strip out the constant of nature and what is left, $A/d$, is a length. Every capacitance is $\varepsilon_0$ times a length that is read off the drawing — $4\pi R$ for a sphere, $A/d$ for parallel plates, $2\pi\ell/\ln(b/a)$ for a coaxial cable.

That last one earns its place because it is the geometry of a cable, and it says the capacitance per metre depends only on the ratio of the two diameters. Scaling a coaxial cable up or down changes nothing about its electrical behaviour per unit length, which is why the same 50-ohm specification covers cables from a millimetre to a hand’s width across.

The shape whose capacitance is only a length #

If every capacitance is $\varepsilon_0$ times a length read off a drawing, the obvious question is whether there is an arrangement in which that length is a single measurable dimension and nothing else — no ratio to get right, no area to determine, no gap to know.

There is, and it was found in 1956 by Thompson and Lampard while they were looking for something else. Take four long electrodes arranged as the quarters of a closed cylinder, separated by narrow gaps, and measure the capacitance between one opposite pair — the cross-capacitance. Their theorem says that for a cylinder of any cross-sectional shape whatever, the two cross-capacitances per unit length satisfy a relation which, for a symmetric arrangement, forces each of them to

$$\frac{C}{\ell} = \frac{\varepsilon_0 \ln 2}{\pi} \approx 1.953\ \text{picofarads per metre}.$$

Nothing about the cross-section appears. Not its diameter, not its shape, not the width of the gaps provided they are small. Make the cylinder round or square or lopsided and the answer per unit length is the same number, fixed by $\varepsilon_0$ and a logarithm.

That makes it a standard rather than a curiosity. The only dimension that has to be measured is the length, and even that is arranged as a difference: a movable guard electrode slides along the assembly and the change in capacitance between two of its positions is compared with the distance it moved, measured by an optical interferometer. An absolute capacitance is thereby realised from a wavelength of light.

For decades that chain was how the electrical units were tied to the mechanical ones. A calculable capacitor gave the farad; a series of impedance bridges walked from the farad to the ohm; and comparing that ohm with the quantised Hall resistance gave a value of the fine-structure constant, at uncertainties of parts in a hundred million. All of it resting on the claim this essay opens with — that a capacitance is decided by geometry, and by nothing that has to be charged to find out.

Reading a distance off the number #

The same claim, used in the opposite direction, makes capacitance the instrument of choice for measuring small displacements.

If $C = \varepsilon_0 A/d$ and $A$ is fixed, then a fractional change in capacitance is a fractional change in the gap, with a minus sign and nothing else in it. A bridge that resolves a part in a million of a capacitance across a hundred-micrometre gap is therefore resolving a tenth of a nanometre of motion, and it does so with no calibration against any material property: the sensitivity is set by the gap, which is a length, and the reading drifts only as fast as the fixture holding the plates expands.

That is why the technique is everywhere something small has to move and be believed. A condenser microphone is a stretched diaphragm forming one plate of a capacitor. A MEMS accelerometer is a silicon proof mass between two fixed plates, read differentially so that the two gaps’ errors cancel. Machine-tool metrology, mirror positioning in a large telescope and the gap in a scanning probe are all read the same way.

The inversion worth noticing is the touchscreen. Everything above treats the fringing field at the edges as the error term — the part the idealisation throws away and the guard ring exists to banish. A projected- capacitance touch sensor is a grid of electrodes whose mutual capacitance is carried almost entirely by that fringing field, arching up out of the glass; a finger arriving in it provides a path to ground and takes some of it away. The signal is a change of a fraction of a picofarad in the one part of the field this essay has been at pains to exclude.

The jar, and the eighteenth-century version of the same argument #

The first capacitor was built in 1745 and nobody understood it for years, and the way it was misunderstood is instructive because the correct answer is the one this page arrives at by another route.

The Leyden jar is a glass jar with metal foil inside and out, and it stores a shock large enough to be dangerous. The immediate assumption was that the electricity was in the water the early jars were filled with — a fluid in a container, which is what the apparatus looks like and what the word “charge” still suggests. That reading survived a first test: pouring the water into a second jar was expected to move the charge, and it did not, which was blamed on the pouring.

Franklin’s experiment settled it by dismantling the jar. He discharged a jar, took it apart, tested the two foils and found them nearly neutral, then reassembled it with fresh foils and found the shock still there. The charge had not been on the metal and had not been in the water. It was associated with the glass — that is, with the insulating region between the two conductors, which is where the field is.

That is the eighteenth-century form of the energy-in-the-field claim made above, arrived at with no field concept available and by a straightforwardly decisive experiment. It also identifies the right variable: what a capacitor does is decided by the gap and not by the plates, which is why the modern formula has the separation in the denominator and no property of the metal anywhere.

What the figure cannot show #

The traced field in these figures is exact for the arrangement it draws, and it draws a two-dimensional slice of a pair of infinite strips. Three things are missing from it, and the third is the one that matters.

The third dimension. The bowing at the ends of a real rectangular plate happens at four edges and four corners, and corner fields are stronger than edge fields for the same reason a point on a conductor concentrates charge. Breakdown starts at corners, which is why high-voltage electrodes are made with rounded rims.

The dynamics. Nothing here shows the charge arriving. The field pictured is the settled state, reached in the time it takes light to cross the plate — of order a picosecond for hand-sized plates — which is why the electrostatic picture is adequate for anything slower than a few gigahertz and useless above it.

The number that is not on the drawing. The capacitance itself is nowhere in the traced field. It has to be computed by integrating the field to get a potential difference and dividing by the charge; the picture shows the field that produces the number, not the number. A reader can see from the figure that closer plates give a shorter path and therefore less potential difference for the same field — which is the whole of $C \propto 1/d$ — but the seeing is an inference, not an observation, and the figure would look identical if the arithmetic were wrong.

Where the ladder goes next #

The rungs beyond this one: dielectrics, and what polarisation does to the field; the method of images, which solves a charge near a conductor by replacing the metal with a fictitious charge; capacitors in series and parallel, and why the combination rules are opposite to the resistor ones; the energy density integrated over all space for a point charge, which diverges and is one of the first hints that a point charge is not a consistent object; the coaxial and spherical geometries in full; and electrostatic force from energy, where differentiating $\tfrac12 CV^2$ with respect to a coordinate gives the attraction between the plates without any force law being invoked.

The claim to carry forward is the first sentence. A capacitance exists before any charge does. It is a number attached to a shape, and charging the thing merely reveals it.