Forces on Dielectrics

Rub a comb through your hair and it lifts paper scraps that carry no net charge. A charged capacitor, disconnected from everything, will physically swallow a slab of glass placed at its mouth. A focused laser beam — nothing but light — holds a plastic bead pinned at its focus firmly enough to stretch a single DNA molecule tied to it. All three are the same force: a neutral, polarizable body pulled toward the place where the field is strongest. The paradox is where that pull comes from. Inside an ideal capacitor the field is uniform, and Torque and Force on a Dipole in a Field proved that a uniform field exerts torque but zero net force on a dipole. Every dipole in the slab's interior feels nothing. The entire force is delivered by the fringing field — the messy, bowing, nonuniform fringe at the plate edges that we spend the rest of the course pretending isn't there. Computing that fringe honestly is miserable. The escape is one of the prettiest moves in electrostatics: write the total energy as a function of the slab's position, differentiate, and the fringe hands you its answer without ever being drawn. That is the energy method, and this article builds it carefully — including the famous sign trap, where the stored energy rises at fixed voltage yet the slab is pulled in anyway, because the battery pays exactly double. Master this and you own the design equation of MEMS actuators, dielectrophoretic cell sorters, electrowetting lenses, and optical tweezers — a Nobel Prize that is, at heart, this article at optical frequency.

Key takeaways

A linear dielectric in a nonuniform field is pulled toward stronger field: induced dipoles lower the system's energy fastest by migrating up the gradient of $E^2$ .
In a uniform field there is torque but no net force; for a slab entering a capacitor the entire pull is exerted by the nonuniform fringing field at the plate edges.
The energy method sidesteps the fringe: $F=-\partial W/\partial x$ at fixed charge, $F=+\partial W/\partial x$ at fixed voltage — opposite signs in the formula, the same physical force.
At fixed voltage the battery supplies exactly twice the stored-energy increase (the double-payment theorem of Energy Stored in Dielectric Systems); the surplus half is the mechanical work.
Flagship result: a slab of width $w$ in a plate gap $d$ at voltage $V$ is pulled in with the constant force $F=\varepsilon_0\chi_e wV^2/2d$ , independent of insertion depth.
The same gradient force, with the Clausius–Mossotti factor of Dielectric Sphere in a Uniform Field, becomes dielectrophoresis and optical tweezers.
Key idea: a dielectric is pulled toward stronger field because polarizing it lowers the system energy there; the force is found not from local fields but from $F=\pm\,\partial W/\partial x$ (sign set by the constraint), or $F=\tfrac{1}{2}V^2\,\mathrm{d}C/\mathrm{d}x$ at fixed voltage.

The core insight

A dielectric is pulled toward stronger field because polarizing it lowers the system energy there; quantitatively the force is most easily found not from local fields but from $F=\pm\,\partial W/\partial x$ (sign set by the constraint) or, equivalently, $F=\tfrac{1}{2}V^2\,\mathrm{d}C/\mathrm{d}x$ at fixed voltage.

Where this lives today

The slab formula is a design equation now. MEMS comb drives — the actuators that steer your phone camera's autofocus and deploy airbag accelerometer self-tests — are precisely interdigitated capacitor fingers exploiting $F=\tfrac12V^2\,\mathrm{d}C/\mathrm{d}x$ : a hundred finger pairs, $20\ \mathrm{\mu m}$ deep with $2\ \mathrm{\mu m}$ gaps at $30\ \mathrm{V}$ , deliver $\sim8\ \mathrm{\mu N}$ of perfectly controllable force. Electrowetting-on-dielectric puts the same energy derivative to work on droplets, reshaping liquid lenses in milliseconds. In microfluidics, dielectrophoresis runs the sphere version: with $\nabla E^2\sim10^{16}\ \mathrm{V^2/m^3}$ from microelectrodes, a $5\ \mathrm{\mu m}$ cell with $|K|\sim0.5$ feels $\sim60\ \mathrm{pN}$ , enough to sort live from dead cells label-free — and cells less polarizable than the buffer ( $K<0$ ) are pushed to field minima instead, a sign flip you can dial with frequency. And in a thousand biophysics labs, Arthur Ashkin's single-beam gradient trap — Example 2 with better optics — holds beads tethered to single DNA molecules, kinesins, and ribosomes, measuring their piconewton lives one step at a time. The 2018 Nobel Prize in Physics was, at bottom, awarded for the force in Equation (2) evaluated at optical frequency, Clausius–Mossotti factor and all.

Origins · 1 minhermann von helmholtz and how this was discovered

hermann von helmholtz

1821–1894 · the physiologist-physicist who made energy an audit rule, then priced the force on matter in fields

Helmholtz arrived at electromagnetism from medicine: measuring the speed of nerve impulses taught him to trust energy bookkeeping over mechanism-guessing. His 1847 memoir Über die Erhaltung der Kraft — written at 26, initially rejected by Poggendorff's Annalen as too speculative — turned energy conservation into physics' universal audit rule, the very rule this article uses in place of impossible fringe-field integrals. Three decades later, as Berlin's imperial professor of physics, he derived how a dielectric fluid is stressed by a field: the Helmholtz force density of Equation (4), electrostriction term and all, still the standard against which the Kelvin density is compared. His students carried the program everywhere — Hertz to the discovery of electromagnetic waves, Boltzmann to statistical mechanics. The energy method for forces is Helmholtz's habit of mind made into a formula: never ask what pushes; ask what the books say when it moves.

History

1837

— Faraday finds dielectrics raise capacitance and are pulled toward charged bodies: force on neutral matter needs explaining

1847

— Helmholtz's Über die Erhaltung der Kraft makes energy conservation the universal bookkeeping rule

1853

— Kelvin's absolute electrometer turns the force between capacitor plates into a voltage standard: field forces become metrology

1873

— Maxwell's Treatise recasts field forces as stresses — tension along field lines, pressure across them

1881

— Helmholtz derives the force density on a dielectric fluid, electrostriction included; the Kelvin-versus-Helmholtz density debate opens

1986

— Ashkin's single-beam gradient trap: the dielectric gradient force, at optical frequency, holds micron beads — optical tweezers

2018

— Nobel Prize to Ashkin: the slab-in-capacitor force, ninety years on, measures molecular motors in piconewtons

The physical picture

Start with one molecule. The field polarizes it, so its dipole moment points along the local $\mathbf{E}$ , and Torque and Force on a Dipole in a Field showed that such a dipole in a nonuniform field feels the net force $(\mathbf{p}\cdot\nabla)\mathbf{E}$ — the charge sitting in the stronger field is pulled harder than its partner in the weaker field. (That article also settled the $\nabla(\mathbf{p}\cdot\mathbf{E})$ versus $(\mathbf{p}\cdot\nabla)\mathbf{E}$ bookkeeping: for a rigid dipole in a curl-free electrostatic field the two agree.) Since an induced dipole has $\mathbf{p}\parallel\mathbf{E}$ , the force always points toward larger $E^2$ , never away — a dielectric cannot be electrostatically repelled by a bare field gradient. Summing over the molecules of a chunk of material turns $\mathbf{p}$ into the polarization density and the single-dipole force into the bulk force density $(\mathbf{P}\cdot\nabla)\mathbf{E}$ . Now look at where that density is nonzero. Deep inside a capacitor the field is uniform: every interior dipole feels torque at most, and contributes nothing to the net pull. Equivalently, in the bound-charge picture the slab's interior is neutral and its two polarized faces sit against the plates, pulled symmetrically up and down. The asymmetry — the thing that actually drags the slab inward — lives at the entrance, where the slab's leading portion pokes into the fringing field. There the field bows outward, its magnitude falls off with distance from the gap, and the induced dipoles in the protruding material feel a genuine gradient pointing into the capacitor. The force is real, local, and entirely a fringe effect.

Figure

Where the pull actually acts: the fringing field at the mouth of the capacitor

Here is the miracle this article turns into a tool. The fringe geometry is hopeless to integrate directly — it depends on plate thickness, edge rounding, gap aspect ratio. But energy conservation does not care about the route the force takes. If we can write the system's total energy $W(x)$ as a function of the slab's insertion depth $x$ — and for a capacitor we can, because capacitance is cheap to compute in the fringe-free approximation — then the force is a derivative. Every detail of the fringe drops out of the bookkeeping, yet the answer is the fringe force, exact in the limit where the fringe region is small compared to the plates. Griffiths calls this out as one of the subtlest points in the subject (§4.4.4): the uniform-field idealization hides the very field that does the pulling, and the energy method quietly does the fringe integral for you. One warning before the derivations: which derivative you take depends on what you hold fixed. An isolated capacitor keeps its charge $Q$ ; a battery-connected one keeps its voltage $V$ . The two protocols give energy functions that move in opposite directions as the slab enters — and the same force. Sorting that out cleanly is the heart of the subject, and Energy Stored in Dielectric Systems already did the hard accounting for us.

One force, two ledgers

The system's stored energy, from Energy Stored in Dielectric Systems, is $W=\tfrac12\int\mathbf{D}\cdot\mathbf{E}\,\mathrm{d}\tau$ , which for a capacitor collapses to the circuit forms $W=Q^2/2C=\tfrac12CV^2$ . Suppose the dielectric sits at position $x$ and we let it move quasistatically by $\mathrm{d}x$ . Whatever agent holds the slab feels the electrical force $F$ along $x$ ; energy conservation ties $F$ to the rate at which the energy accounts change. The result is a pair of formulas that look contradictory and are not.

F=-\left.\frac{\partial W}{\partial x}\right|_{Q}=+\left.\frac{\partial W}{\partial x}\right|_{V} \qquad (1)

where: $F$ — force on the dielectric along the displacement coordinate $x$ (positive = pulled toward larger $x$ ); $Q$ — total free charge (fixed: isolated capacitor); $V$ — terminal voltage (fixed: battery attached). Read it physically: at fixed $Q$ the field energy is the only account, so the force spends it directly; at fixed $V$ a battery joins the books, and the sign flips because the battery overpays — the derivations below make both statements precise. For the battery-connected case the working form is worth its own number, because $C(x)$ is usually the only thing you need to compute:

F=\tfrac{1}{2}V^2\,\frac{\mathrm{d}C}{\mathrm{d}x} \qquad (2)

where: $C(x)$ — capacitance as a function of the dielectric's position; $V$ — the fixed voltage. Since inserting dielectric always increases $C$ , the force is always inward — reversing the polarity squares away: $V^2$ doesn't care about sign.

Derivation AThe energy method at fixed charge

We derive the force on a dielectric from energy conservation alone, in the cleanest setting: an isolated capacitor, total free charge $Q$ frozen, no battery anywhere. The strategy is to treat the stored energy $W(x)=Q^2/2C(x)$ as a potential energy for the slab coordinate $x$ and let mechanical energy bookkeeping produce the force. The one subtlety to watch: we must be sure the field energy is a function of $x$ through the charge configuration alone, which is exactly what fixed $Q$ buys us — Gauss's law for D pins $\mathbf{D}$ to the free charge regardless of where the matter sits.

Step 1. Let the slab move quasistatically from $x$ to $x+\mathrm{d}x$ while an external hand supplies force $-F$ to keep it from accelerating (so $F$ is the electrical force on the slab). No charge flows anywhere — the capacitor is isolated — so no external electrical work is done.

Step 2. Energy conservation for the step: the field energy changes by $\mathrm{d}W$ , and the hand receives work $F\,\mathrm{d}x$ from the slab. With no other accounts open, $\mathrm{d}W+F\,\mathrm{d}x=0$ , i.e. $F=-\mathrm{d}W/\mathrm{d}x$ at fixed $Q$ . Why is quasistatic essential? If the slab accelerates, kinetic energy opens a third account and the force you compute is still correct but the bookkeeping needs the extra term; slow motion keeps the ledger two-entry.

Step 3. Write the stored energy in the charge form, $W=Q^2/2C(x)$ , and differentiate with $Q$ constant. This gives $F=-\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{Q^2}{2C}\right)=+\dfrac{Q^2}{2C^2}\dfrac{\mathrm{d}C}{\mathrm{d}x}$ .

Step 4. Read the sign. Dielectric entering means $\mathrm{d}C/\mathrm{d}x>0$ , so $F>0$ : the slab is pulled in, and the stored energy falls — the field pays the slab's mechanical work out of its own account. That is precisely the $0.500\to0.100\ \mathrm{\mu J}$ collapse audited in Energy Stored in Dielectric Systems, whose missing $0.400\ \mathrm{\mu J}$ we have now located as the integral of this force. Verified: dimensions — $Q^2/C^2\cdot\mathrm{d}C/\mathrm{d}x$ carries $\mathrm{V^2\cdot F/m=J/m=N}$ ; and integrating Step 3 from $C_0$ to $\varepsilon_rC_0$ reproduces $\Delta W=\tfrac{Q^2}{2}(1/C_0-1/\varepsilon_rC_0)$ , the fixed-charge energy drop, exactly.

Go deeper · Derivation B · 1 minFixed voltage gives the same pull

Derivation BFixed voltage gives the same pull

Now attach the battery and hold $V$ fixed. The stored energy $\tfrac12CV^2$ rises as the slab enters — yet the slab is still pulled in. We prove the force is identical to the fixed-charge result, by opening the battery's account and using the double-payment theorem proved in Energy Stored in Dielectric Systems: at constant voltage, the battery delivers exactly twice any increase in stored energy. Watch how the "wrong" sign in $F=+\partial W/\partial x|_V$ is really the battery's fingerprint.

Step 1. Recall the theorem (proved, not re-proved, here): for a capacitance change $\mathrm{d}C$ at fixed $V$ , the battery delivers $\mathrm{d}W_{\text{bat}}=V^2\,\mathrm{d}C$ while the stored energy gains only $\mathrm{d}W=\tfrac12V^2\,\mathrm{d}C$ . The unaccounted half is mechanical.

Step 2. Energy conservation with three accounts: $\mathrm{d}W_{\text{bat}}=\mathrm{d}W+F\,\mathrm{d}x$ . Substituting Step 1: $V^2\,\mathrm{d}C=\tfrac12V^2\,\mathrm{d}C+F\,\mathrm{d}x$ , so $F=\tfrac12V^2\,\dfrac{\mathrm{d}C}{\mathrm{d}x}=+\left.\dfrac{\partial W}{\partial x}\right|_V$ .

Step 3. Check equivalence with the fixed-charge route at the same instant. The configuration (slab at $x$ , plates carrying $Q=CV$ ) is identical in both protocols, so substitute $Q=CV$ into Derivation A's result: $F=\dfrac{Q^2}{2C^2}\dfrac{\mathrm{d}C}{\mathrm{d}x}=\dfrac{(CV)^2}{2C^2}\dfrac{\mathrm{d}C}{\mathrm{d}x}=\tfrac12V^2\dfrac{\mathrm{d}C}{\mathrm{d}x}$ — the same force, term for term. Why must they agree? The force is exerted by the fields on the matter, here and now; it cannot know whether a battery is attached three feet away. Only the bookkeeping of where the work comes from differs: at fixed $Q$ the field account pays; at fixed $V$ the battery pays double, the field banks half, and the slab gets the other half.

Step 4. The formal summary is a Legendre transform: the mechanical force is always $-\partial/\partial x$ of the appropriate thermodynamic potential — $W$ itself at fixed $Q$ , but $\tilde W=W-QV$ at fixed $V$ , and $-\partial\tilde W/\partial x|_V=+\partial W/\partial x|_V$ . The sign flip is the transform, not new physics. Verified: integrating both protocols over full insertion of an $\varepsilon_r=5$ slab into a $100\ \mathrm{pF}$ , $100\ \mathrm{V}$ capacitor gives mechanical work $0.400\ \mathrm{\mu J}$ (fixed $Q$ ) and $2.000\ \mathrm{\mu J}$ (fixed $V$ ) — exactly the two ledgers of Energy Stored in Dielectric Systems, Worked Example 1.

The two formulas earn a dispatch table:

you know	protocol	use	energy account
$Q$ (isolated)	charge frozen	$F=+\dfrac{Q^2}{2C^2}\dfrac{\mathrm{d}C}{\mathrm{d}x}$	field energy falls, pays the work
$V$ (battery on)	voltage frozen	$F=+\tfrac12V^2\dfrac{\mathrm{d}C}{\mathrm{d}x}$	battery pays double; field banks half
either	same configuration	same $F$	only the bookkeeping differs

The slab force, earned end to end

Time to cash the method on the classic geometry — the one every exam, every MEMS datasheet, and every electrowetting paper ultimately reduces to.

Go deeper · Derivation C · 1 minThe slab force from the capacitance function

Derivation CThe slab force from the capacitance function

A dielectric slab of susceptibility $\chi_e=\varepsilon_r-1$ is inserted a depth $x$ into a parallel-plate capacitor: plate length $\ell$ , width $w$ , gap $d$ , battery holding voltage $V$ . We compute $C(x)$ in the fringe-free approximation, differentiate, and apply the fixed-voltage formula. The punchline to anticipate: $x$ will cancel — the pull is constant throughout insertion — and the answer will contain only the fringe-free interior quantities, even though the fringe does all the work.

Step 1. Model the partly filled capacitor as two capacitors in parallel — filled area $wx$ , empty area $w(\ell-x)$ , both with gap $d$ : $C(x)=\dfrac{\varepsilon_0\varepsilon_r wx}{d}+\dfrac{\varepsilon_0 w(\ell-x)}{d}=\dfrac{\varepsilon_0 w}{d}\big[\ell+\chi_e x\big]$ . Why parallel and not series? The two regions share the same plates and the same voltage $V$ , side by side; charge distributes between them. (Stacked layers along the field would be series — the sandwich geometry of Linear Dielectrics, Susceptibility and Permittivity.)

Step 2. Differentiate: $\dfrac{\mathrm{d}C}{\mathrm{d}x}=\dfrac{\varepsilon_0 w\,\chi_e}{d}$ — a constant, independent of $x$ . The capacitance grows linearly as dielectric replaces vacuum.

Step 3. Apply Equation (2): $F=\tfrac12V^2\dfrac{\mathrm{d}C}{\mathrm{d}x}$ , giving the flagship result $F=\dfrac{\varepsilon_0\chi_e\,w\,V^2}{2d}$ , directed inward. Constant force: the slab is pulled with the same strength whether it has barely entered or is nearly home — a flat force plateau, ending only when the slab fills the gap (or when its trailing edge reaches the fringe region and the parallel-capacitor model itself expires).

Step 4. Locate the force physically. Nothing in Steps 1–3 mentioned the fringing field, yet the interior uniform field exerts zero net force — so the constant $F$ we computed is the fringe force, delivered at the mouth of the capacitor on the slab's protruding portion. The energy method has evaluated the messy fringe integral without drawing a single fringe line. That is its entire magic, and also its honesty debt: if someone asks where the force acts, the answer is the fringe, always. Verified: for $\varepsilon_r=5$ , $w=1\ \mathrm{cm}$ , $d=1\ \mathrm{mm}$ , $V=1000\ \mathrm{V}$ : $F=8.854\times10^{-12}\cdot4\cdot0.01\cdot10^6/(2\times10^{-3})=1.77\times10^{-4}\ \mathrm{N}$ — matching Griffiths Eq. 4.64 and the provenance benchmark.

Whose force density is it, anyway

The energy method delivers total forces impeccably. Push one level deeper — how is that force distributed through the material? — and you walk into a century-old argument, which we state honestly rather than hide. The dipole picture suggests the Kelvin force density: sum $(\mathbf{p}\cdot\nabla)\mathbf{E}$ over molecules to get, per unit volume,

\mathbf{f}_{\text{K}}=\rho_f\mathbf{E}+(\mathbf{P}\cdot\nabla)\mathbf{E} \qquad (3)

where: $\rho_f\mathbf{E}$ — the ordinary force on any free charge present; $(\mathbf{P}\cdot\nabla)\mathbf{E}$ — the summed dipole force, nonzero only where the field is nonuniform. This picture localizes force wherever polarized matter sits in a gradient. Energy variation (vary $W=\tfrac12\int D^2/\varepsilon\,\mathrm{d}\tau$ under a virtual displacement of the material, as Landau & Lifshitz §15 do) instead produces the Helmholtz force density:

\mathbf{f}_{\text{H}}=\rho_f\mathbf{E}-\tfrac{1}{2}\varepsilon_0 E^2\,\nabla\varepsilon_r+\nabla\!\left[\tfrac{\varepsilon_0}{2}E^2\,\rho\,\frac{\partial\varepsilon_r}{\partial\rho}\right] \qquad (4)

where: the middle term concentrates force where the permittivity varies — for a rigid solid, a delta-function spike at its surfaces, pointing from low $\varepsilon_r$ to high (into the dielectric, toward strong field); the last, electrostriction term involves the mass density $\rho$ and prices the material's squeeze under field — being a pure gradient, it integrates to zero for any rigid body and redistributes pressure, not net force, in a fluid. The uncomfortable truth: $\mathbf{f}_{\text{K}}$ and $\mathbf{f}_{\text{H}}$ disagree pointwise — Kelvin spreads the force through the polarized bulk, Helmholtz piles it onto the surfaces — yet for any rigid body surrounded by vacuum they integrate to the same total force, which is also what the energy method and Equation (2) deliver. Statics experiments measure total forces (or, in fluids, pressure distributions where electrostriction genuinely enters), so the choice between densities went unresolved for a century and, for deformable media, is still argued in the liquid-dielectrics literature. The systematic referee is the Maxwell stress tensor, which converts field momentum flux through any surface in vacuum into an unambiguous enclosed force — machinery we build in the Conservation Laws chapter, where it belongs. For everything in this article, the energy method already gives the full, correct answer.

Limiting cases

case	result	meaning
uniform field	$\mathbf{F}=0$	dipoles feel torque only; no gradient, no pull — recall Torque and Force on a Dipole in a Field
dilute sample, $\chi_e\ll1$ , volume $\mathcal{V}$	$\mathbf{F}\approx\tfrac12\varepsilon_0\chi_e\mathcal{V}\,\nabla E^2$	the gradient-force seed of dielectrophoresis: force $\propto\nabla E^2$ , toward field maxima
slab in capacitor, fixed $V$	$F=\varepsilon_0\chi_e wV^2/2d$ , constant in $x$	flat plateau: same pull at every insertion depth until the geometry model expires
$\varepsilon_r\to1$	$F\to0$	no permittivity contrast, no force — the "dielectric" is vacuum in disguise
sphere in a host medium, $\varepsilon_p<\varepsilon_m$	$K<0$ : pushed toward weak field	the Clausius–Mossotti factor $K=\dfrac{\varepsilon_p-\varepsilon_m}{\varepsilon_p+2\varepsilon_m}$ can flip sign — negative dielectrophoresis (an air bubble in water has $K=-0.17$ )

Common trapThe uniform interior field does none of the pulling

Two traps, both graded ruthlessly. First: do not try to compute the slab force from the interior field — it is uniform and exerts exactly zero net force; the entire pull is the fringing field's, and the energy method is how you get it without integrating the fringe. Second: at fixed voltage the stored energy increases as the slab enters, and students conclude the slab must be pushed out. Wrong — the battery delivers twice the stored increase (the double-payment theorem), so the total system energy still falls along the direction of motion. Track what is held fixed, use $F=-\partial W/\partial x|_Q$ or $F=+\partial W/\partial x|_V$ , and never mix the two ledgers in one line.

Worked examples

Example 1 — the slab that closes an old ledger (confidence builder). Energy Stored in Dielectric Systems audited a $C_0=100\ \mathrm{pF}$ capacitor at $V_0=100\ \mathrm{V}$ swallowing an $\varepsilon_r=5$ slab, and found $0.400\ \mathrm{\mu J}$ (fixed $Q$ ) or $2.000\ \mathrm{\mu J}$ (fixed $V$ ) of mechanical work whose agent it deferred to this article. Give the capacitor geometry — gap $d=1\ \mathrm{mm}$ , square plates — and find the force in both protocols.

Step 1. Geometry. $C_0=\varepsilon_0 w\ell/d$ with $w=\ell$ gives $w=\ell=\sqrt{C_0d/\varepsilon_0}=\sqrt{10^{-10}\times10^{-3}/8.854\times10^{-12}}=10.6\ \mathrm{cm}$ . Then $\mathrm{d}C/\mathrm{d}x=\varepsilon_0 w\chi_e/d=8.854\times10^{-12}\times0.1063\times4/10^{-3}=3.76\ \mathrm{pF}$ per millimetre of insertion.

Step 2. Fixed voltage. $F=\tfrac12V_0^2\,\mathrm{d}C/\mathrm{d}x=\tfrac12(100)^2(3.764\times10^{-9}\ \mathrm{F/m})=18.8\ \mathrm{\mu N}$ — constant at every depth. Work over full insertion: $F\ell=18.8\ \mathrm{\mu N}\times0.1063\ \mathrm{m}=2.000\ \mathrm{\mu J}$ . The fixed-V ledger closes exactly.

Step 3. Fixed charge. Now $F(x)=\dfrac{Q_0^2}{2C(x)^2}\dfrac{\mathrm{d}C}{\mathrm{d}x}$ with $C(x)=C_0(1+4x/\ell)$ : the force starts at $18.8\ \mathrm{\mu N}$ (same configuration, same force — Derivation B's point, live) but falls as the slab enters and the voltage sags, ending at $18.8/\varepsilon_r^2=0.75\ \mathrm{\mu N}$ , a factor of $25$ weaker.

Step 4. Fixed-charge work. $\displaystyle\int_0^\ell F\,\mathrm{d}x=\frac{Q_0^2}{2}\left(\frac{1}{C_0}-\frac{1}{\varepsilon_rC_0}\right)=0.400\ \mathrm{\mu J}$ . Both of the older article's mechanical entries are now accounted for, force by force.

Figure

The force plateau at fixed V versus the fading pull at fixed Q — Example 1 numbers

Example 2 — optical tweezers, the slab force at light frequency (exam level). A polystyrene bead of radius $a=0.25\ \mathrm{\mu m}$ ( $n_p=1.57$ ) sits in water ( $n_m=1.33$ ) near the focus of a $P=100\ \mathrm{mW}$ laser focused to a Gaussian waist $w_0=0.6\ \mathrm{\mu m}$ . Estimate the trapping force.

Step 1. The polarizability of a sphere in a host. Dielectric Sphere in a Uniform Field gives the induced moment through the Clausius–Mossotti combination; in a host medium the factor generalizes to $K=\dfrac{\varepsilon_p-\varepsilon_m}{\varepsilon_p+2\varepsilon_m}$ . At optical frequency $\varepsilon=n^2$ : $\varepsilon_p=2.465$ , $\varepsilon_m=1.769$ , so $K=0.696/6.00=0.116$ .

Step 2. Intensity and its gradient. Peak intensity $I_0=2P/\pi w_0^2=2(0.1)/\pi(0.6\times10^{-6})^2=1.77\times10^{11}\ \mathrm{W/m^2}$ . The transverse profile $I=I_0e^{-2r^2/w_0^2}$ has its steepest slope at $r=w_0/2$ : $|\nabla I|_{\max}=(2/w_0)I_0e^{-1/2}=3.58\times10^{17}\ \mathrm{W/m^3}$ .

Step 3. Gradient force. In the Rayleigh (small-bead) regime the time-averaged gradient force is $F=\dfrac{2\pi n_m a^3}{c}\,K\,\nabla I$ . Numerically: $\dfrac{2\pi(1.33)(0.25\times10^{-6})^3}{3\times10^8}(0.116)(3.58\times10^{17})\approx1.8\times10^{-11}\ \mathrm{N}\approx18\ \mathrm{pN}$ — comfortably above the piconewton scale of molecular motors, which is why tweezers revolutionized single-molecule biophysics.

Step 4. Trap stiffness and honesty. Near the axis the trap is a spring: $k=4K(2\pi n_ma^3/c)I_0/w_0^2\approx99\ \mathrm{pN/\mu m}$ , i.e. about $5\ \mathrm{pN}$ at a $50\ \mathrm{nm}$ displacement. Honesty clause: at $a\sim\lambda/4$ the Rayleigh approximation is at the edge of its licence — quantitative trap design uses Mie theory — but the scaling ( $F\propto a^3K\nabla E^2$ ) and the order of magnitude are exactly this calculation. Trap force $\sim18\ \mathrm{pN}$ , stiffness $\sim0.1\ \mathrm{pN/nm}$ .

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §4.4.4 (Energy in dielectric systems) and §4.4.3 (force on a dielectric) · Jackson, Classical Electrodynamics, 3rd ed., §4.7 (Electrostatic energy in dielectric media) and §6.8 (Maxwell stress) · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 10 · Zangwill, Modern Electrodynamics, §6.7–6.8 (forces and stresses in dielectrics) · Landau & Lifshitz, Electrodynamics of Continuous Media, §15–16 (body forces and electrostriction)

Provenance. Equations cross-checked against Griffiths Introduction to Electrodynamics 4th ed. §4.4.3–4.4.4 (dipole force, slab force $F=\tfrac12V^2\,\mathrm{d}C/\mathrm{d}x$ , fixed-Q vs fixed-V equivalence), Jackson 3rd ed. §4.7 and §6.8 (energy, stress tensor), and Landau-Lifshitz ECM §15–16 (Helmholtz force density with electrostriction). A verifier should: (1) confirm $C(x)$ for a parallel slab gives $\mathrm{d}C/\mathrm{d}x=\varepsilon_0w(\varepsilon_r-1)/d$ ; (2) check the numerical slab-force example ( $\approx1.8\times10^{-4}$ N for the stated parameters); (3) verify $-\partial_x(Q^2/2C)$ equals $+\partial_x(\tfrac12CV^2)$ via $Q=CV$ ; (4) confirm the body-force derivation reproduces $-\tfrac12\varepsilon_0E^2\nabla\varepsilon_r$ . Constant used: $\varepsilon_0=8.854\times10^{-12}$ F/m (CODATA).

Builds on: Torque and Force on a Dipole in a Field · Linear Dielectrics, Susceptibility and Permittivity · The Electric Displacement D · Boundary Conditions for D and E Leads to: Maxwell Stress Tensor · Dielectrophoresis and Optical Trapping · Electrostriction

If the energy method never mentions the fringing field, how can its answer be the fringe force?

Because energy conservation is indifferent to where the work is done. The method computes the work the electrical system delivers per unit displacement of the slab — a global statement. Separately, we know by elimination that the only region able to exert net force is the fringe (the interior field is uniform, and uniform fields exert zero net force on neutral matter). So the global number and the local agent must match: the derivative is the integrated fringe force. You get the total exactly, while remaining honestly ignorant of its detailed distribution — for that distribution you need the stress tensor, not more differentiating.

Does reversing the battery polarity push the slab out?

No. The force goes as $V^2$ (or $Q^2$ ), so it is blind to sign: reverse the plates and the induced bound charges reverse too, leaving the attraction intact. This is the same reason a comb picks up paper whichever way you charged it, and why dielectrophoresis works fine with AC fields — indeed AC is standard in DEP, because it defeats electrode screening while the force, depending on the square of the field, survives averaging.

At fixed Q the energy falls, at fixed V it rises — so which is it physically?

Both, and neither is "the" energy of the force. The configuration and the force are identical at the matching instant (same slab position, same charge on the plates). What differs is the ledger: with the battery absent, the field's account is the only source, so it drains; with the battery present, it pumps in twice the field's gain and the surplus does the mechanical work — the double-payment theorem of Energy Stored in Dielectric Systems. Ask "who pays?" and the protocols differ; ask "what force?" and they cannot.

Should I use grad of p dot E or p dot grad of E for the dipole force here?

For everything in this article they agree. Torque and Force on a Dipole in a Field owns the distinction: $(\mathbf{p}\cdot\nabla)\mathbf{E}$ is the honest sum of forces on the two charges, and it equals $\nabla(\mathbf{p}\cdot\mathbf{E})$ for a rigid $\mathbf{p}$ in a curl-free field — both conditions hold in electrostatics with permanent-magnitude induced moments. For an induced dipole there is a further subtlety worth remembering: the energy is $-\tfrac12\alpha E^2$ (half, because you build the moment as you go), which is exactly why the gradient force on a bead carries the factor arrangement it does.

Why does insulating oil climb up between charged capacitor plates against gravity?

Same force, liquid version. The fringe pulls oil into the gap until the electrical pull balances the weight of the raised column: $h=\varepsilon_0\chi_eE^2/2\rho_m g$ . Transformer oil ( $\varepsilon_r=2.2$ , $\rho_m=880\ \mathrm{kg/m^3}$ ) in a $1\ \mathrm{MV/m}$ gap rises $h=8.854\times10^{-12}\times1.2\times10^{12}/(2\times880\times9.81)\approx0.6\ \mathrm{mm}$ — small but easily visible, and measuring $h$ is a classic way to determine a fluid's susceptibility. In fluids, note, the electrostriction term of Equation (4) genuinely contributes to the pressure distribution, though not to the height once the books are done correctly.

Kelvin and Helmholtz force densities disagree — which one is real?

For rigid bodies, the question has no experimental content: both integrate to the same total force, and total force is what a rigid body lets you measure. The densities differ in where they say the force sits (bulk gradient regions for Kelvin, permittivity jumps for Helmholtz), which matters only when the material can deform — liquid surfaces, elastic solids — and there electrostriction must be carried along, after which correctly done experiments (pressure profiles in dielectric fluids) are consistent with the Helmholtz form. The clean, assumption-free machinery is the Maxwell stress tensor evaluated in the vacuum outside the body; that is Conservation Laws material, and it adjudicates every such dispute by construction.

Do optical tweezers really work by this electrostatic formula even though light oscillates at hundreds of terahertz?

Yes, with two translations. Replace static permittivities by their optical values $\varepsilon=n^2$ (dispersion is the business of the electromagnetic-waves chapter), and time-average the square of the oscillating field, which supplies the factors relating force to intensity. The structure survives untouched: the bead is a dielectric sphere in a nearly uniform local field, its response is the Clausius–Mossotti factor $K$ from Dielectric Sphere in a Uniform Field, and the force is proportional to $a^3K\nabla E^2$ — the dilute-sample line of our limiting-cases table wearing optical units. High-index beads ( $K>0$ ) are pulled to the bright focus; a bubble ( $K<0$ ) is expelled, which is why trapping hollow particles needs beam-shaping tricks.

If the force on the slab is constant during insertion, what happens if I let it go — does it oscillate forever?

On frictionless rails, yes, and it is a lovely exam problem. The slab accelerates under the constant force until fully inside; symmetry then reverses the roles (the trailing edge is now in the fringe on the far side, if the plates are longer than the slab, or the slab overshoots out the other side of a short capacitor) and it decelerates, stops, and returns — a periodic oscillation powered by no net energy loss. Any real slab has friction and radiates a little, so it settles centered, fully inserted: the energy minimum at fixed charge, exactly where $C$ is maximal. The plateau force with sharp turnarounds also tells you the period scales as $\sqrt{m\ell d/\varepsilon_0\chi_e w}/V$ — fast for light slabs and high voltage.