The attraction that needs no charge

A water molecule carries no net charge, so Gauss’s law is unambiguous about it: the flux of the electric field through any closed surface drawn around that molecule is exactly zero, and no cleverness in choosing the surface changes the answer. The molecule nonetheless has a field, and one nanometre away along its own axis that field is $1.1 \times 10^8$ V/m.

Everything below is where that $1.1 \times 10^8$ comes from, what the arrangement costs in falloff, and where the expansion behind it stops being one.

What is left when the monopole cancels #

The potential of a set of charges $q_i$ at positions $\mathbf{r}_i$ is a sum of Coulomb terms, and is a scalar rather than a vector, which is why the expansion is done on it. Seen from a distance $r$ much greater than the size of the set, each term expands in powers of $r_i/r$:

$$V = \frac{1}{4\pi\varepsilon_0}\left[\frac{Q}{r} + \frac{\mathbf{p}\cdot\hat{\mathbf{r}}}{r^2} + \frac{\text{quadrupole}}{r^3} + \cdots\right].$$

The first term carries the total charge $Q = \sum q_i$ and nothing else. The second carries the dipole moment $\mathbf{p} = \sum_i q_i \mathbf{r}_i$, a vector with the dimensions of charge times length. The third needs five numbers rather than three, and so on upward.

For a neutral object the first term is identically zero. The series does not end there; it starts there. The dipole moment is promoted from a correction to the leading term, and becomes the first thing a distant observer can learn about the object at all.

The promotion comes with a property that makes $\mathbf{p}$ a fact about the object rather than about the bookkeeping. Shifting the origin by $\mathbf{a}$ replaces every $\mathbf{r}_i$ by $\mathbf{r}_i - \mathbf{a}$, so $\mathbf{p}$ changes by $-\mathbf{a}Q$ — which is zero when $Q$ is. A neutral object’s dipole moment does not depend on where the origin was put, whereas a charged object’s does. Hence the dipole moment of an ion is a quantity nobody quotes and that of a molecule is tabulated to four figures.

What is left when the monopole cancels is worth stating exactly, because the cancellation is complete and the consequence is not. Draw a closed surface round a neutral object and every field line entering it leaves again, so the crossings cancel and the flux is zero. Gauss’s law is exactly true here and reports one number, and that number happens to be zero. The field at each point of that surface is neither zero, nor small, nor constrained at all — which is the ordinary situation with an integral law, and the reason this essay needs something finer than flux.

For water, $p = 6.2 \times 10^{-30}$ C·m, or 1.86 debye in the unit the subject uses. Dividing by the elementary charge gives 0.039 nm, less than half the 0.096 nm O–H bond length — so the molecule is not two unit charges 0.039 nm apart. It is a smear of partial charges across some 0.28 nm whose first moment equals that of such a pair, and the expansion says that from far enough away nothing else about it survives.

The field of a dipole, and the two directions that differ by two #

The gradient of the dipole term gives the field in spherical components, with $\theta$ measured from the direction of $\mathbf{p}$ and $k = 1/4\pi\varepsilon_0$:

$$E_r = \frac{2kp\cos\theta}{r^3}, \qquad E_\theta = \frac{kp\sin\theta}{r^3}.$$

Two directions bracket everything between them. Along the axis, $\theta = 0$, the field is $2kp/r^3$ and points along $\mathbf{p}$: for water at 1 nm, $1.11 \times 10^8$ V/m. Dry air ionises at about $3 \times 10^6$ V/m, so one neutral molecule maintains, a nanometre out, a field thirty-seven times the one that tears air apart. On the perpendicular bisector, $\theta = 90°$, the field is $kp/r^3$ — half as large at the same radius — and points opposite to $\mathbf{p}$: $5.6 \times 10^7$ V/m. No direction gives more than twice that, and none gives zero.

The consequence is one everybody has met. In a sodium chloride crystal the nearest ions sit 0.282 nm apart, bound by $8.18 \times 10^{-19}$ J, or 5.11 eV — 198 times the thermal energy at room temperature, so nothing at 300 K pulls that apart. Immerse the crystal in water and the same pair, now separated by a medium of relative permittivity 80, are bound by 0.064 eV: 2.5 times thermal energy, comparable and therefore beatable. Salt dissolves in water and not in oil because of a number built out of $6.2 \times 10^{-30}$ C·m repeated $3.3 \times 10^{28}$ times per cubic metre.

The tower of falloffs #

The exponents form a ladder, and each rung is bought with one more order of cancellation.

A monopole’s potential goes as $1/r$ and its field, being a gradient, as $1/r^2$. The force between two charges is that field times a charge: $1/r^2$.

A dipole’s potential goes as $1/r^2$, one power steeper, because the two contributions nearly cancel and what survives is their difference. Its field goes as $1/r^3$.

The force on a permanent dipole needs a gradient, since a uniform field gives none. In the field of a point charge, $E \sim 1/r^2$, the gradient — and hence the force — is $1/r^3$. Between two permanent dipoles the inducing field is itself $1/r^3$, its gradient $1/r^4$, and the force $1/r^4$.

The force on an induced dipole loses one power more, because the moment is not fixed but proportional to the field it sits in: $p_{\text{ind}} = \alpha E \sim 1/r^2$ next to a charge, and multiplying by a gradient of $1/r^3$ gives $1/r^5$.

One force law, and five arrangements that turn it into exponents running from two to five. Nothing in the sequence mentions how strong any charge is. The exponent is decided by how many cancellations stand between the source and what is left of it at a distance; strength enters only as a prefactor.

The counting argument cannot produce that exponent, and it is worth seeing why. Shells at one, two and three times the radius have areas in the ratio 1 : 4 : 9, and dividing a fixed flux between them gives the inverse square — but a neutral object has no flux to divide. The $-3$ is not a statement about area at all; it is what near-cancellation between two opposite charges leaves behind, and it has to be computed from the pair rather than counted off a sphere.

The inverse square is a geometrical fact about surfaces in three dimensions and can be had by counting. The inverse cube cannot: it is the leading survivor of a near-cancellation, and its exponent counts how nearly the cancellation succeeded.

The torque that aligns, and the force that is not there #

Put a dipole in a perfectly uniform field. The force on the positive end is $+q\mathbf{E}$ and on the negative end $-q\mathbf{E}$, and their sum is exactly zero — not approximately, not to leading order. A uniform field cannot move a neutral object at all. What it can do is turn it, because the two forces act at different places and so make a couple:

$$\boldsymbol{\tau} = \mathbf{p} \times \mathbf{E}, \qquad U = -\mathbf{p}\cdot\mathbf{E}.$$

For water in a laboratory field of $10^6$ V/m the maximum torque is $6.2 \times 10^{-24}$ N·m, and aligned differs from anti-aligned by $1.24 \times 10^{-23}$ J. Thermal energy at 300 K is $4.14 \times 10^{-21}$ J, or 25.9 meV — half a $k_BT$ for every way of moving, of which rotation offers three. The ratio $pE/k_BT$ is $1.5 \times 10^{-3}$.

An orientation energy that small aligns nothing. Weighting orientations by the Boltzmann factor and expanding for small argument gives the Langevin result $\langle\cos\theta\rangle \simeq pE/3k_BT = 5.0\times 10^{-4}$: five parts in ten thousand. The molecules in a glass of water in an ordinary field remain randomly oriented to any reasonable approximation, the alignment being a bias of a twentieth of a per cent on top of chaos.

That bias is what a dielectric constant is made of. Multiplying it by the moment and the number density, $\chi = np^2/3\varepsilon_0 k_BT$, gives a susceptibility of 11.7 and a relative permittivity of 12.7. The measured value for water at room temperature is 80. The mechanism is the right one and accounts for a sixth of the answer; the missing factor of 6.3 is that neighbouring water molecules are hydrogen-bonded and turn in correlated groups rather than independently, which Onsager in 1936 and Kirkwood in 1939 supplied.

The moment the field makes for itself #

Most matter has no permanent moment. A hydrogen atom, an argon atom, a scrap of paper: place any of them in a field and the charge distribution distorts, the positive and negative centroids separate slightly, and a moment appears where there was none, $\mathbf{p}_{\text{ind}} = \alpha\mathbf{E}$.

The polarisability $\alpha$ is small: $7.42\times 10^{-41}$ C·m²/V for a hydrogen atom. In $1.11 \times 10^8$ V/m — the field a water molecule maintains at 1 nm — the induced moment is $8.3 \times 10^{-33}$ C·m, smaller than water’s permanent moment by a factor of 750; in an ordinary laboratory field of $10^6$ V/m it is $7.4\times 10^{-35}$ C·m, smaller by 84,000. Induced moments are feeble, and they still run most of the everyday electrostatics anybody has seen.

Their energy is not $-\mathbf{p}\cdot\mathbf{E}$ but half of it, because half the work went into the distortion:

$$U = -\tfrac{1}{2}\alpha E^2 \quad\Longrightarrow\quad F = -\frac{\mathrm{d}U}{\mathrm{d}r} = -\frac{2\alpha q^2}{(4\pi\varepsilon_0)^2 r^5}$$

next to a point charge $q$. There is the $1/r^5$, and something stranger sitting beside it.

The force depends on $q^2$. It cannot know the sign. A rubbed comb picks up paper whether the rubbing left it positive or negative, and a neutral object is attracted to a charge of either sign with identical strength. That is the one-sentence reason the trick looks like magic: every other electrostatic force anybody meets comes in a repulsive version, and this one has no partner.

The reason is not an accident of the algebra. The polarisability of any system in its ground state is positive — second-order perturbation theory guarantees it, every excited state contributing a term of the same sign — so $U$ is negative and grows more negative wherever the field is stronger. A polarisable object always seeks high field, and the field is always higher nearer the charge.

Magnetism is not like this. Magnetic susceptibility can be negative: a diamagnet is pushed towards low field, which is why a live frog was levitated in a 16 T magnet at Nijmegen in 1997 and why no dielectric has ever been levitated electrostatically. Earnshaw’s theorem of 1842 forbids the electrical version outright — in a charge-free region $|E|^2$ has no local maximum, so a high-field-seeker has nowhere stable to sit. The asymmetry belongs to the field that has no ends.

Putting the two halves together gives the result the essay was after, and it explains the missing repulsion. The induced moment is proportional to the field, and the force on a moment is proportional to the gradient of the field — so the force goes as the field times its own gradient. Reverse the sign of the field and both factors reverse, so their product does not: the force is towards the stronger field whichever way the rod is charged. That is why an uncharged scrap of paper is always attracted and never repelled, and why the effect has no counterpart to be symmetric with.

The fifth power gives the comb trick its character. Take a comb carrying 10 nC, giving $1.0\times 10^5$ V/m at 3 cm, and a 5 mm scrap of 80 gsm paper modelled as a dielectric sphere of relative permittivity 3, so $\alpha = 7.0\times 10^{-19}$ C·m²/V. At 3 cm the attraction is $4.6\times 10^{-7}$ N against a weight of $2.0\times 10^{-5}$ N: forty-three times too small, and the paper does not stir. At 1.4 cm the two are equal. At 5 mm the attraction is $3.6\times 10^{-3}$ N, 180 times the weight, and the scrap leaps. A factor of six in distance is $6^5 = 7{,}776$ in force, which is why the effect has no gentle approach and appears to switch on.

What it costs #

The dielectric constant became an instrument. Debye’s 1912 paper turned the arithmetic above into a measurement: the orientational susceptibility carries a $1/T$ and the induced part does not, so plotting $\chi$ against $1/T$ for a gas gives a line of slope $p^2/3\varepsilon_0 k_B$ and intercept $\alpha$. Both fall out of a capacitance measured at several temperatures, and that is still how gas-phase dipole moments are obtained. The unit named after him is worth $3.336\times10^{-30}$ C·m.

A gradient has to be manufactured, and gradients are expensive. The force on neutral matter goes as $\nabla|E|^2$, so dielectrophoresis — sorting cells, trapping viruses, steering droplets on a chip — needs not a strong field but a rapidly varying one. Electrodes 10 μm across at 10 V give $10^6$ V/m and a gradient of $10^{11}$ V/m²: shrinking the electrode buys more than raising the supply.

The range is a nanometre and not a metre. A $1/r^5$ force falls by $10^5$ over one decade of distance, and the fluctuating version below falls faster still, so anything built on induced moments — molecular recognition, adhesion, a gecko’s foot — is a technology of contact.

Water’s permittivity sets the chemistry of a planet. Change the 80 to the 2 of oil and the 5.11 eV binding sodium to chloride becomes 2.6 eV instead of 0.064 — a hundred times thermal energy rather than a few, and nothing dissolves.

Where the model stops #

The expansion is an expansion, and it is a statement about being far away. For two opposite charges separated by $d$, the exact axial field exceeds the point-dipole formula $2kp/r^3$ by 12.1 per cent at $r = 2d$, by 2.0 per cent at $r = 5d$, and by 0.50 per cent at $r = 10d$. The error falls as $(d/r)^2$ rather than $(d/r)$, because the even moments vanish by symmetry and the next surviving term is the octupole. Two separations away, the dipole approximation is wrong in the first significant figure.

For real water the relevant length is the molecule’s own extent, around 0.1 nm, so at the 1 nm quoted throughout the dipole term is good to about a per cent, while at the 0.3 nm of molecular contact the higher moments contribute of order 10 per cent. Every serious water model in simulation therefore carries an explicit quadrupole.

Linear polarisability fails at fields comparable with the atomic one. An electron at the Bohr radius sits in $5.1\times10^{11}$ V/m. The $1.1\times10^8$ V/m of a water molecule at 1 nm is $2.2\times10^{-4}$ of that, so $\mathbf{p} = \alpha\mathbf{E}$ is excellent there. At the focus of a laser delivering $10^{16}$ W/cm² the field reaches $2.7\times10^{11}$ V/m, comparable with the atomic field; the induced moment is then proportional to nothing, and the whole of nonlinear optics lives in the correction terms.

None of this explains cohesion. Two argon atoms have no permanent moment, and one polarising the other’s already-induced moment is far too weak to account for argon condensing at 87 K. The real attraction comes from moments that fluctuate: each atom’s charge distribution flickers, the flickers correlate, and the correlation lowers the energy. London worked this out in 1930, irreducibly quantum-mechanically, giving $U \sim -1/r^6$ and a force as $1/r^7$. What transfers is not the exponent but the sign argument — the moment is again proportional to the field that induced it, so the interaction is again quadratic and again cannot repel, which is why dispersion forces are attractive between all matter and why a liquid has a surface tension at all. The exponent differs because the inducing field is a dipole’s $1/r^3$ rather than a charge’s $1/r^2$ — this essay’s arithmetic one rung along.

The same expansion in gravity, and in a nucleus #

The multipole expansion was not invented for electricity. Legendre and Laplace built it in the 1780s to compute the gravitational attraction of a spheroid, and its functions still carry Legendre’s name.

Gravity gets a different first term, and the difference is instructive. Mass has one sign, so the monopole never cancels — $GM$ is always there — and the dipole term, far from being promoted, is identically zero once the origin is taken at the centre of mass, by the same origin-shift identity used above. The first thing surviving beyond a point mass is the quadrupole.

For the Earth that quadrupole is measured to five figures and called $J_2 = 1.0826\times10^{-3}$: one part in a thousand of the monopole, and the 21.4 km by which the equatorial radius exceeds the polar one. Nothing about a distant object’s shape is knowable except through these moments, and $J_2$ is known so precisely because a satellite orbit integrates it — the bulge torques the orbital plane and makes the node regress, about 5° a day for the International Space Station. Sun-synchronous satellites are flown near 98° inclination so that the regression instead runs backwards at 0.9856° a day, matching the Earth’s motion round the Sun, and the orbit therefore crosses the equator at the same local time for years. A whole class of Earth-observation missions rests on one term of a series.

Gravitational radiation sharpens the pattern. The second time derivative of the mass dipole is the rate of change of total momentum, zero for an isolated system, so there is no gravitational dipole radiation at all. The leading term is the quadrupole — which is why a passing wave stretches one transverse direction while squeezing the other instead of pushing everything one way, as the dipole radiation of an accelerated charge does.

The series reaches into nuclei too. A nucleus in a state of definite parity has no electric dipole moment, so the first shape information is again the quadrupole. The deuteron’s was measured at $2.86\times10^{-31}$ m² by Kellogg, Rabi, Ramsey and Zacharias in 1939–40, and its not being zero was a real discovery: a deuteron in a pure S state would be spherical, so a non-spherical one proves the nuclear force is not purely central. Four per cent of D state accounts for the value — a structural fact about the strong interaction, read off the same expansion that says what a comb does to paper.

What the picture cannot show #

The figures are two-dimensional and the field is not. Lines from a dipole spread over a sphere, and their planar density falls one power more slowly, so counting lines in the hero figure gives the wrong exponent — as it does for any line drawing of a three-dimensional field.

They draw two separated point charges, not a dipole. Everything quantitative here concerns $r \gg d$ and the figures show $r \sim d$, which is where the expansion fails by the 12 per cent computed above. The pictures are of the object; the arithmetic is about its shadow at a distance the frame does not reach.

No figure shows a force on anything neutral. That force goes as the gradient of the square of the field — a derivative of a derivative of what is drawn — and the eye reads neither. The plate figure comes closest, and only because the generator prints the 69 per cent.

Nothing is at a temperature. The most consequential number here is the $5\times10^{-4}$ of alignment, an average over molecules pointing every way at once. A figure showing a dipole lying neatly along the field would misrepresent it by three orders of magnitude, so no figure here shows one.

The ladder from here #

Later rungs on this anchor: the quadrupole term, and why shape at that order needs five numbers rather than three. Polarisation as a field in its own right, with bound charge and the displacement vector that lets Gauss’s law be written for free charge alone. The dipole that radiates once it is shaken. Screening in an electrolyte, and the Debye length beyond which a charge’s field stops being Coulombic. The image dipole a neutral atom induces in a nearby wall, which is the same $q^2$ argument with the charge supplied by the metal.

The neighbouring ladders are close by. The exponent that belongs to the arrangement is where the tower of falloffs properly lives, and the potential is what the expansion is performed on. Conductors are the limiting case of infinite polarisability, and how much charge a shape will hold asks this essay’s geometrical question of a conductor instead of a molecule. Induced dipoles that oscillate rather than sit still are why the sky is blue: a molecule polarised by a passing wave re-radiates as the dipole described here, and the $\omega^4$ in its power is the whole of the colour. And the vector field before any lines were drawn on it is where this ladder began.

Faraday measured specific inductive capacity in 1837 with no idea what was happening inside the glass. Debye explained it in 1912 with an average over orientations that comes to five parts in ten thousand — turning a curiosity of capacitors into a way of weighing the asymmetry of a molecule.