The force that lives where the model is not

Hold a sheet of glass at the open mouth of a charged parallel-plate capacitor and let go. It is pulled in.

This is an easy demonstration and an awkward one to explain, because the standard model of a parallel-plate capacitor contains no force capable of doing it. Between the plates the field is uniform and perpendicular to them. The slab’s induced charges sit on its faces, top and bottom, and the field pushes them apart — perpendicular to the plates, which is perpendicular to the direction the slab is free to move.

Within the model, nothing pulls the slab anywhere. And the model’s energy, differentiated, gives the pull to three figures.

The energy method, twice #

Capacitance is a geometrical quantity. With the slab inserted a fraction $x$ of the way, the capacitor is two capacitors in parallel — one filled, one empty — so

$$C(x) = \frac{\varepsilon_0 w L}{d}\left[\varepsilon_r x + (1-x)\right],$$

which rises linearly with insertion. Everything below is a consequence of that one line and of how the capacitor is being held.

Held at constant charge, with the capacitor charged and then disconnected, the stored energy is $Q^2/2C$. As the slab enters, $C$ rises and the energy falls. A system whose energy falls as a coordinate increases is being pushed in that direction, with force $-\mathrm{d}U/\mathrm{d}x$, and the force is inward.

Held at constant voltage, with the battery still attached, the stored energy is $\tfrac12 CV^2$. As the slab enters, $C$ rises and the energy rises. Read naively, that is a system climbing uphill, and the force should be outward.

It is not, and the resolution is that the battery is part of the system and has been left out of the accounting.

Keeping the voltage fixed while the capacitance rises requires moving more charge onto the plates: $\Delta Q = V\,\Delta C$. The battery does work $V\,\Delta Q = V^2 \Delta C$ in doing so. The field’s energy rises by $\tfrac12 V^2 \Delta C$. The difference, another $\tfrac12 V^2\Delta C$, has gone somewhere, and the only place available is mechanical work on the slab.

So the battery supplies exactly twice what the field stores, and the surplus is the pull. Both accounts give

$$F = \frac{\varepsilon_0 w (\varepsilon_r - 1) V^2}{2d},$$

inward, which for plates $10$ cm wide separated by a millimetre at a kilovolt with a relative permittivity of $4$ is $1.33$ millinewtons — the weight of about a seventh of a gram.

The factor of two is not a coincidence of this geometry. Any capacitor held at fixed voltage while its capacitance changes has the same ledger, because $\Delta W_{\text{batt}} = V^2\Delta C$ and $\Delta U = \tfrac12 V^2 \Delta C$ whatever the geometry, and the mechanical work is always the remainder. What is being minimised at fixed voltage is therefore not the field energy but the field energy minus the battery’s contribution — a different function, and the one whose gradient is the force.

Why the two ledgers have to agree #

The agreement of the two calculations is not an accident of this geometry either, and the reason is a general one about which function to differentiate.

At fixed charge the natural variable is $Q$ and the energy $U(Q, x) = Q^2/2C(x)$ is a function of it; the force is $-\partial U/\partial x$ at fixed $Q$. At fixed voltage the natural variable is $V$, and the function to differentiate is not $U$ but $U$ minus the work the source has done — the Legendre transform of $U$, which comes out as $-\tfrac12 CV^2$. Differentiating that at fixed $V$ gives $+\tfrac12 V^2\,\mathrm{d}C/\mathrm{d}x$, inward, the same as before.

This is the same manoeuvre thermodynamics makes when it moves between internal energy and free energy. A system held at fixed temperature does not minimise its energy; it minimises its energy less the heat the reservoir supplies, which is the Helmholtz free energy. A capacitor held at fixed voltage does not minimise its field energy; it minimises the field energy less the work the battery supplies. In both cases the reservoir is part of the system and forgetting it flips a sign.

The lesson generalises past electrostatics: whenever a system is held at a fixed intensive quantity by a reservoir, the potential whose gradient is the force is not the one whose value is the stored energy.

Where the force actually is #

The energy method has now given a number and no location. That is its characteristic strength and its characteristic silence, and this system is a good place to notice both.

The force is not distributed over the slab. It acts almost entirely at the leading edge, over a region of the order of the plate separation, where the field is not uniform.

Field lines bow because there is nothing to keep them straight, and the same bowing at the mouth of a capacitor is what supplies a component along the plates. That is where the force actually lives: not in the uniform region the model describes, but in the fringe the model deletes. The parallel-plate idealisation is exact about the capacitance and silent about the only part of the field that pulls.

At the mouth of the capacitor the lines bow outward. A line that bows has a component along the plates, and it is that component acting on the polarisation charge at the slab’s leading face that produces the pull. Move the edge of the slab well inside, so that the fringing region is entirely within the dielectric, and the force is unchanged — because the fringe has moved with it.

That is the reason the energy method works despite the model containing no such force. The energy it differentiates is the energy of the whole field, fringe included, and the fringe’s energy depends on where the slab is even though the parallel-plate formula does not mention it. Differentiating a total that contains the mechanism gives the right answer without ever displaying the mechanism.

Where equipotentials crowd the field is strong, and where they curve it has a component in an unexpected direction. Both happen at the edge and neither happens in the middle — so a picture of the idealised interior contains no information about the force, and the surfaces have to be drawn right out to the rim before anything mechanical is visible.

The same force with no plates in sight #

A dielectric object in a non-uniform field is pulled toward the strong field, and the capacitor is a special case of that rather than the other way round.

The same force appears with no plates in sight. A dielectric sphere polarised by an applied field acquires an induced dipole along that field, and a dipole in a gradient feels a net force even though a dipole in a uniform field feels only a torque. That is the whole mechanism in its simplest setting, and it explains why the slab is pulled in rather than pushed: the field is stronger inside than outside, so the gradient points inward.

An applied field polarises the object, inducing a dipole moment proportional to the field. A dipole in a uniform field feels a torque and no net force, because the pull on one end matches the push on the other. In a gradient the two do not match, and the net force is

$$\mathbf{F} = (\mathbf{p}\cdot\nabla)\mathbf{E},$$

which for an induced dipole $\mathbf{p} = \alpha\mathbf{E}$ becomes $\tfrac12\alpha\nabla E^2$ — always up the gradient, for a particle more polarisable than its surroundings, and independent of the sign of the field.

That expression is worth staring at, because it is the same one that appears in three subjects. It is why a stream of water bends toward a charged rod, which is the standard demonstration and is usually explained badly — the water is neutral throughout, nothing has been transferred to it, and the bending is a polarisability meeting a gradient. It is dielectrophoresis, used to sort cells and to trap viruses between microfabricated electrodes. And at optical frequencies it is the gradient force that holds a particle in a focused laser beam — the same $\alpha\nabla E^2$, the same independence of sign, the same requirement of a non-uniform field.

The capacitor’s slab is that force integrated over a fringing field, and the reason the answer is so clean is that the energy method has already done the integral.

The numbers, and why the demonstration is hard #

The force scales as the square of the voltage and inversely as the gap, so it is easy to make large on paper and awkward in a laboratory. At a kilovolt across a millimetre the field is $10^6$ V/m, a third of the breakdown field of air, and the pull on a ten-centimetre-wide slab is $1.33$ mN. Doubling the voltage quadruples the force and puts the field past what dry air will hold.

Working in a liquid dielectric changes the arithmetic favourably — transformer oil breaks down at about ten times the field of air, and $\varepsilon_r - 1$ is larger for the slab relative to a liquid than to air only if the slab is more polarisable than the liquid, which for glass in oil it barely is. That last observation is not a detail. The force depends on the difference in permittivity between the slab and what it displaces, so a body immersed in a medium more polarisable than itself is pushed out of the strong field rather than pulled in, and the sign of a dielectrophoretic force is routinely reversed by changing the suspending fluid rather than the particle.

What the model is doing when it says the field is uniform #

The uniform-field model is not wrong; it is answering a different question, and how much charge a shape will hold is the question it was built for. It computes the capacitance, which depends on the total field energy for a given charge and is dominated by the region between the plates. The fringe contributes a small correction to the capacitance and the whole of the force, so the same approximation is excellent for one and useless for the other.

That is a general pattern worth naming. A quantity dominated by the bulk of a system tolerates a model that discards the edges. A quantity that vanishes in the bulk by symmetry does not, and every part of it is at the edge by construction — which is the same trap a field line drawn as an object sets, one level up. Deciding which kind of quantity is being asked for is the whole of knowing whether an approximation may be used.

The energy of a capacitor can be computed from the charges or from the field, and the two agree — but only the second contains the fringe. That is why different derivations of the same force look as though they disagree about where it comes from: the charge-based account puts it on the plates and the field-based account puts it in the space at the edge, and both arrive at the same number because the two energies are equal.

What a fringe is worth, measured #

It is fair to ask how big the neglected part of the field is, since the whole argument turns on a region the model calls negligible.

The fringing region extends about one plate separation beyond the edge, so for plates of area $A$ and separation $d$ it occupies a volume of order $Pd^2$ where $P$ is the perimeter. Against the working volume $Ad$ the ratio is $Pd/A$ — for a $10 \times 10$ cm plate at a millimetre, about four parts in a thousand. The capacitance computed by the parallel-plate formula is therefore right to that accuracy, which is why nobody bothers with the correction.

The force is a different matter, and the comparison is instructive. The force is not a small correction to a large force; it is the whole of a quantity whose leading term is exactly zero. A four-parts-in-a-thousand piece of the field energy, differentiated with respect to a coordinate the rest of the energy does not depend on, produces one hundred per cent of the answer.

That inversion — the negligible region supplying the entire effect — is worth recognising as a shape rather than as a curiosity of capacitors. It recurs wherever a symmetry makes the dominant contribution cancel: the lift on a wing lives in the circulation rather than in the pressure, the force between two current loops lives in the field gradient rather than in the field, and the drag on a slowly moving sphere lives in a region of the flow that the leading approximation gets wrong.

Where it stops #

The slab is assumed rigid and incompressible. It is not: the same field that pulls it in also squeezes it, because the induced charges on the two faces attract one another. That is electrostriction, and it changes the density of the dielectric, which changes its permittivity, which changes the force. The correction is small for a solid and large for a liquid — a dielectric liquid drawn up between vertical plates rises to a height set by a balance in which electrostriction is not negligible.

And the force density is not unique. There are several expressions in the literature for the force per unit volume in a polarised medium, and they disagree with one another while giving the same total force on a body. The reason is that they differ by terms that integrate to zero over a closed body — gradients of quantities that vanish outside it — so no experiment on the body as a whole can distinguish them. Experiments that measure the stress inside a dielectric can, and the resulting question of which expression is correct occupied a long argument in the twentieth century that is still not entirely settled for moving media.

At a dielectric surface the normal component of one field is continuous and the tangential component of the other, and it is the mismatch between them that produces a stress on the boundary. That is where this stops being a story about capacitors: the force on any polarisable body in any field comes from the same boundary conditions, and the capacitor is simply the case where the geometry is simple enough to compute.

The permittivity has to be a constant. At high fields it is not, and in any case a real dielectric breaks down: the field at the leading edge is larger than the uniform field between the plates, so the failure begins exactly where the force is, which is a nuisance for anybody designing an electrostatic actuator.

And nothing here is dynamic. A slab released at the mouth accelerates, overshoots, and oscillates, dissipating energy by whatever mechanism is available; the equilibrium the energy argument finds is the end of a process rather than the process.

A charge above a conducting plane is pulled toward it, and it is the same kind of force — a body polarised by a field being drawn toward the field’s own source. The image-charge construction makes that computable, and the attraction it gives grows without limit as the charge approaches, which is the same edge-dominated behaviour seen once more.

The experiment, and what it is easy to get wrong #

The demonstration is old and the standard way of doing it hides the interesting part.

Charge the capacitor, disconnect it, and offer the slab: it is pulled in, and the pull can be measured on a balance. Leave the battery connected and it is pulled in with the same force, which is the surprise, since the stored energy has gone the other way. Anybody expecting a force to point downhill in the field energy will predict the wrong sign in the second case, and the two arrangements are otherwise indistinguishable to look at.

There is a third arrangement that catches out the arithmetic rather than the intuition. Insert the slab first, then charge, then withdraw it: at fixed charge the withdrawal costs work $Q^2/2$ times the change in $1/C$, which is the same integral traversed backwards, so the energy returns exactly. Do the same at fixed voltage and the battery is charged on the way out, recovering half of what it supplied on the way in. An accounting that tracks only the capacitor’s energy finds a discrepancy of exactly a factor of two in one direction and misses it entirely in the other.

None of this is subtle physics. It is a reminder that “the energy of the system” is a phrase that has to be finished, and that the battery is not scenery.

The stress tensor, which is the other way to do it #

There is a formulation that gives the location as well as the total, and it is worth naming because it is what a numerical solver actually uses.

Maxwell’s stress tensor expresses the electromagnetic force on any region as an integral of a stress over the surface enclosing it: field lines pull along their own length and push sideways, with a tension and a pressure both equal to the energy density. Wrap a surface around the slab, integrate the stress over it, and the answer is the same $1.33$ millinewtons, arrived at without differentiating anything and with the distribution over the surface displayed.

Doing it that way makes the mechanism unavoidable. The parts of the enclosing surface deep between the plates contribute nothing along the direction of motion, because the field there is perpendicular and the stress is symmetric; every bit of the answer comes from the part of the surface that crosses the fringing region. The energy method and the stress method are two spellings of one fact, and the second spells out where.

The ladder from here #

Later rungs on this anchor: the force on a dielectric liquid, where electrostriction and the free surface both enter and the height risen is measurable; the Kelvin and Korteweg–Helmholtz force densities compared properly, with the experiment that distinguishes them; dielectrophoresis in an alternating field, where the sign of the force depends on frequency because the particle’s polarisability does; and the Casimir force, which is what remains when the applied field is removed altogether and only the vacuum’s own fluctuations are left to pull the plates together.

The neighbouring ladders are the field the matter takes away, which is the polarisation this force acts on; where the energy of a field actually is, which is the quantity being differentiated; and the light that pulls rather than pushes, where the same gradient force appears at $10^{14}$ hertz and holds a bacterium still.