The field the matter takes away

A conductor placed in an electric field ends up with no field inside it at all. That result is exact, it does not depend on the metal, and it follows from one sentence about mobile charge: if any field remained, charges would still be moving, and they are not.

An insulator placed in the same field does something less absolute and considerably more interesting. Its charges cannot leave their molecules, but they can shift within them, and the shifted charge makes a field of its own. What is left inside is neither the applied field nor nothing.

The number that appears there, one third, is not a property of any material. It is a property of a sphere, and changing the shape changes it over the whole range from nought to one.

The circularity, and the one shape that resolves it #

The difficulty in a dielectric is stated in a sentence and it is genuine. A molecule polarises in proportion to the field it sits in. The field it sits in is the applied field plus the field of every other polarised molecule. So the polarisation depends on itself.

For an arbitrary shape that is a coupled problem with no closed solution, and the reason is that the polarisation comes out non-uniform: the field near a corner differs from the field at the centre, so the molecules there polarise differently, so the field changes again. There is exactly one family of shapes for which the self-consistency collapses to a single number — the ellipsoids — and for them the internal field of a uniform polarisation is itself uniform.

The sum rule is worth pausing on because it is not a convention. Apply the same field successively along the three principal axes of an ellipsoid and the three depolarising factors must add to exactly one, since the three applications together amount to surrounding the body — and a shape cannot depolarise more than the whole of what it is given. A needle spends its whole unit on its two transverse axes and has nothing left for the long one; a slab spends everything on the one axis across itself.

So the field inside a uniformly polarised ellipsoid is

$$\mathbf{E}_{\text{in}} = \mathbf{E}_0 - \frac{N\mathbf{P}}{\epsilon_0},$$

and with a linear material, $\mathbf{P} = \epsilon_0\chi \mathbf{E}_{\text{in}}$, the two lines solve one another in a step:

$$\mathbf{E}_{\text{in}} = \frac{\mathbf{E}_0}{1 + N\chi}.$$

Three shapes, three answers #

The consequence is that the material and the geometry enter only through their product.

The familiar textbook result — that the field inside a dielectric is the applied field divided by $\epsilon_r$ — is the slab case and only the slab case. It is the case a parallel-plate capacitor presents, which is why it became the familiar one, and it is the most extreme of the three.

A needle is the opposite extreme and is worth an example, because it makes the shape-dependence unmistakable. Water has a susceptibility near 79. A sphere of water in a field keeps $3/82$ of it, about 4 per cent. A thin cylinder of the same water lying along the field keeps essentially 100 per cent. A drop and a jet of the same substance in the same laboratory field have internal fields differing by a factor of twenty-five, and no measurement on the material could have told anyone that.

This is also why a susceptibility quoted for a substance is a property of the substance and a measurement of it is a property of the sample. The standard geometry for measuring one is a slab between plates precisely because that geometry has $N = 1$ and is the least ambiguous, and a measurement made on a rod has to be corrected before it means anything.

A conductor is the same problem at infinite susceptibility. As $\chi$ grows without bound, $1/(1 + N\chi)$ goes to zero for any nonzero depolarising factor — so the interior field vanishes whatever the shape, and the shape dependence that dominates a dielectric disappears entirely. That limit is worth having because it explains why conductors are easy and dielectrics are not: the answer stops depending on geometry precisely when the response becomes infinite.

The bound charge is real charge #

Nothing above requires anything exotic. The polarisation of a body means each molecule has its positive centre displaced slightly from its negative one, and inside the body every molecule’s positive end sits against its neighbour’s negative end and the two cancel. At the surface there is no neighbour, and what is left over is an uncancelled sheet.

Its density is $\sigma_b = \mathbf{P}\cdot\hat{\mathbf{n}}$, largest where the surface faces along the polarisation and zero where it lies parallel — which is why the surface charge on the sphere in the hero figure follows a cosine, and why the field it makes is uniform inside.

Gauss’s law counts all charge, bound and free alike, and does not distinguish them. That is inconvenient rather than wrong — the bound charge is real charge, it produces a real field, and the only reason to separate it is bookkeeping. The whole apparatus of $\mathbf{D}$ and susceptibility exists to let one write down the free charge, which is the part an experimenter controls, and absorb the rest into a constant.

The distinction between $\mathbf{E}$ and $\mathbf{D}$ is entirely this bookkeeping and is worth stating without reverence. $\mathbf{E}$ is the field: it is what exerts force on a charge, it is what appears in the energy density, and it is what the molecules respond to. $\mathbf{D}$ is a device for solving problems in which the free charge is known and the bound charge is not, and it is convenient exactly because $\nabla\cdot\mathbf{D} = \rho_{\text{free}}$. Neither of the two is more fundamental, and the temptation to treat $\mathbf{D}$ as the “real” field in matter should be resisted for the same reason that treating field lines as objects should be.

Outside a uniformly polarised sphere the field is exactly a dipole’s — not approximately, not far away, but at every point outside the surface. That is a stronger statement than it looks: the bound surface charge of a uniformly polarised sphere produces precisely the field of a point dipole at the centre, with no corrections at any order, which is why the sphere is the one shape this subject can be done in closed form.

At the surface of a dielectric the potential is continuous and its normal derivative is not, and the jump is the bound surface charge. That discontinuity is the whole of what the matter has done — the field inside is reduced because a layer of charge has appeared on the boundary, and every result in this essay is a way of computing how much.

The third that comes back #

The sphere’s factor turns up in a second place that looks unrelated, and the connection is one of the better arguments in classical electromagnetism. Ask what field an individual molecule sits in. It is not the macroscopic average field, because the molecule is not in the average position — it is at a place where its own contribution has to be excluded and its neighbours’ included.

Carve out a small sphere around it. The material outside that sphere can be treated as a continuum, and it contributes the macroscopic field plus the field of the cavity’s bound surface charge, which is $+\mathbf{P}/3\epsilon_0$ — the sphere’s depolarising factor with the sign reversed, because the charge is on the inside of a hole rather than the outside of a body. The molecules within the sphere contribute nothing on average in a cubic lattice or a liquid, by symmetry — the same cancellation that makes a shell of charge exert nothing on what is inside it.

The local field is therefore larger than the macroscopic one, and following the arithmetic through gives

$$\frac{\epsilon_r - 1}{\epsilon_r + 2} = \frac{n\alpha}{3\epsilon_0},$$

the Clausius–Mossotti relation, connecting a bulk permittivity to a single molecule’s polarisability.

The divergence has a name — the polarisation catastrophe — and it is not a prediction. It is the place where treating a molecule’s neighbours as a uniform sphere of continuum stops being defensible, and a genuine ferroelectric transition, which really does polarise itself, has to be derived some other way. What the relation does capture correctly is that the approach to it is steep: a modest increase in density can move a substance a long way up the curve, which is why the permittivity of a gas rises linearly with pressure and that of a liquid does not.

Measuring a molecule with a thermometer #

The relation above has a variant that turned it from a consistency check into one of the most productive instruments in physical chemistry, and the extra ingredient is temperature.

Two quite different things can polarise a molecule. Its electron cloud can be distorted, which is what the polarisability $\alpha$ describes and which does not care how hot the sample is. Or the molecule can already have a permanent dipole and be rotated into line — which is opposed by thermal agitation, so it contributes a susceptibility proportional to $\mu^2/3k_BT$, in the same way and for the same reason that a paramagnet’s alignment falls as one over the temperature.

Put both into the left-hand side of the relation and the result is a straight line in $1/T$:

$$\frac{\epsilon_r - 1}{\epsilon_r + 2}\cdot\frac{M}{\rho} = \frac{N_A}{3\epsilon_0}\left(\alpha + \frac{\mu^2}{3k_BT}\right).$$

Measure a gas’s permittivity at a series of temperatures, plot, and the slope gives the permanent dipole moment and the intercept gives the electronic polarisability, separately. The two mechanisms that a single measurement could never disentangle are separated by the temperature dependence alone.

That measurement is how the dipole moments of small molecules were first obtained, and the numbers are structural information of a kind nothing else at the time could give. A molecule with a moment of zero is symmetric; carbon dioxide’s zero says it is linear with the carbon in the middle, while water’s 1.85 debye says its two bonds are not opposite. The shape of a molecule, read off a capacitor in an oven.

It also explains the caution the previous section ended on. The derivation of the local field assumed the neighbours contribute nothing on average, which is a statement about a distorted electron cloud sitting in a lattice. A liquid of rotating permanent dipoles has neighbours that correlate with one another, and the assumption fails — which is why water sits so awkwardly on the curve.

The catastrophe is an artefact, and what replaced it #

The divergence in the relation is worth one more sentence, because the repair is instructive about what the local-field argument was doing wrong.

The trouble is that the cavity construction gives a molecule credit for the field its own dipole induces in the surrounding medium. That induced polarisation does produce a field back at the molecule, but it points along the molecule’s own moment by construction — so it can never exert a torque on it, and it can never help to orient it. Including it in the orienting field is what drives the denominator to zero and manufactures a spontaneous polarisation out of nothing.

Separating the two — the cavity field, which comes from outside and does orient, and the reaction field, which comes from the molecule’s own influence on its surroundings and does not — removes the divergence entirely. The resulting expression fits polar liquids far better and predicts no transition, which is correct: water does not spontaneously polarise, and a theory saying it nearly does is a theory with a defect rather than a warning.

The lesson generalises past dielectrics. A self-consistent calculation has to be careful about which part of a system’s environment is genuinely external to it, and a term that a body produces in response to itself is not a term that can act on it as a driver. The same error, made in other subjects, produces spurious instabilities in exactly the same way.

What it is for #

A dielectric in a capacitor increases its capacitance — it raises how much charge a given shape will hold at a given voltage — and the reason is now sayable exactly: the material’s bound charge partly cancels the free charge on the plates, so the same charge produces less voltage.

The more interesting consequence is that a dielectric is pulled into a capacitor, and the direction of the force is the same whether the capacitor is held at fixed charge or fixed voltage even though the stored energy moves in opposite directions in the two cases.

A slab slid into a parallel-plate capacitor is where the sign confuses people, and it is worth settling. Held at fixed charge, the stored energy falls as the slab goes in, and the fall is the work done on the slab. Held at fixed voltage the energy rises, and the battery supplies twice that rise — half going into the field and half into the mechanical work. The force is inward and the same size in both cases, which is the only thing that could have been true, and the two energy accounts differ because they are accounting for different systems.

That force is a fringing-field effect: the uniform region between the plates exerts nothing along the plates, and everything happens at the edge, where the field bulges out and pulls the polarised material toward the region of higher field. It is the same mechanism by which a neutral object is attracted to any charged one, scaled up and put to work.

What this says about shielding #

One practical conclusion falls straight out of the screening figure and is worth stating, because it is a question that gets asked in the wrong terms.

A conductor excludes a static field completely, whatever its shape. A dielectric does not exclude it at all: the best a slab of susceptibility $\chi$ can do is reduce the interior field by a factor of $1 + \chi$, and any other shape does worse. A material with a relative permittivity of eighty — which is as high as ordinary substances get — reduces the field inside a flat sheet of it by eighty and inside a rod of it by nothing.

So there is no such thing as electrostatic shielding by an insulator, however impressive its permittivity. Reducing a field by a factor of a thousand needs a susceptibility of a thousand and the most favourable geometry; reducing it by a factor of a million is not available at any thickness. The only material property that does the job is mobile charge, and the only reason it works is that mobile charge redistributes itself until the interior field is zero exactly rather than merely small.

The magnetic case runs the other way and it is worth the comparison. There is no magnetic conductor — nothing whose interior field must be zero — so magnetic shielding has to be done with the insufficient mechanism: a high-permeability material, in the most favourable geometry, buying a factor of a few thousand at best. That is why an electrostatic screen is a sheet of any metal and a magnetic screen is several nested cans of a carefully annealed alloy, and why the second is expensive and the first is not.

Where the model stops #

Only ellipsoids have a depolarising factor. A cube does not: its polarisation is non-uniform, the internal field varies from face to centre, and the single number people quote for it — about a third — is an average that no point in the cube actually experiences. Every “N” for a real component is an approximation by the nearest ellipsoid.

The material is assumed linear and isotropic. A crystal responds along a tensor rather than a scalar, so the polarisation need not be parallel to the field at all, and the two indices a birefringent crystal shows are that tensor made visible. Which components of that tensor a given symmetry permits is a question about crystal classes and belongs elsewhere.

Nothing here is a force on a charge in the material. The field computed is the macroscopic average, and a charge placed inside a real dielectric sits in the local field of whatever molecule it is nearest, which is enormous and has nothing to do with the average. The averaging is legitimate for anything measured over many molecules and illegitimate for anything that is not, and the boundary between the two cases is where the field concept itself needs restating.

And it is assumed instantaneous. A permanent dipole has to rotate to respond, which takes picoseconds in water and much longer in a solid, so ε_r falls with frequency in steps as each mechanism is left behind. The 80 that water shows at low frequency is 1.77 at optical frequencies, and the difference between those two numbers is the whole reason water is transparent and also an excellent microwave absorber.

What the pictures cannot show #

The hero figure draws the bound charge as discrete marks and there is no such thing. The surface charge is a continuous density, and drawing it as countable dots imports a granularity the calculation does not have — though it is granular at a scale ten decades below the drawing, which the drawing also cannot show.

Nor can any figure show the self-consistency being solved. What is drawn is the answer: a uniform polarisation and the uniform field it makes. The iteration by which the polarisation and the field settle on each other is an algebraic step with no spatial picture, and the temptation to imagine it happening in time should be resisted — nothing about the final state records how it was reached.

Where this ladder goes next #

This is the first rung of a new anchor, and what it establishes is that the field inside matter is a question whose answer contains a shape. The rungs above it are the ones this essay declined.

The energy of a polarised body, which is not the integral of ε₀E²/2 and differs from it by exactly the work done on the material — a distinction that decides whether a dielectric is pulled in or pushed out of a non-uniform field. The frequency dependence, where a permittivity becomes complex, the imaginary part is the absorption, and the two are not independent. And the boundary conditions themselves, which are what all of this reduces to at an interface, and which govern reflection, refraction and everything that happens where the medium changes.

The habit worth carrying away is not about dielectrics. When a body’s response depends on its own response, the answer is a division, and the divisor carries the geometry. The same structure — one over one plus a shape factor times a susceptibility — governs the magnetisation of a magnet, the screening of a plasma, the settling of a feedback loop and the effective stiffness of a compliant support. Recognising it saves working the problem out again, and the shape factor is always the part that has to be looked up.