---
title: Capacitor Energy and Force
module: Capacitance
moduleNumber: 4
lessonNumber: 3
order: 403
summary: |
  Charging a capacitor takes work, because every increment of charge is pushed through
  the voltage the earlier charge already established. We total that work three
  equivalent ways, $U=Q^2/(2C)=Q\Delta V/2=C(\Delta V)^2/2$, locate it in the field as
  a density $u=\tfrac12\epsilon_0E^2$, then let the plates move. Differentiating the
  stored energy at fixed charge, or the coenergy at fixed voltage, gives the mechanical
  force; the two boundaries differ only by the work the source supplies. We work the
  parallel-plate attraction and its electrostatic pressure in full, and follow the same
  gradient into pull-in, tilt, comb drives, and traceable force calibration.
topics: [Capacitance]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 24 — Capacitance and Dielectrics; §24-4"
---

## Charging work and stored electric-field energy

Energy enters a capacitor while charge is moved from one conductor to the other. For
a linear capacitor, the potential difference rises from zero to its final value
during the charging process. When charge $q$ is already separated, the increment
$\d q$ requires work

$$
\d U=V(q)\,\d q=\frac{q}{C}\,\d q.
$$

Integration from an uncharged state to final charge $Q$ gives

$$
U=\int_0^Q\frac{q}{C}\,\d q
=\frac{Q^2}{2C}=\frac12Q\Delta V=\frac12C(\Delta V)^2.
$$

The factor one-half is a consequence of the rising voltage. The charge-potential
graph is a straight line for a linear capacitor, and stored energy is the triangular
area beneath that line. Replacing the integral by $Q\Delta V$ would assume that
every increment crossed the final voltage and would overstate the energy by a factor
of two. The three final forms require stated boundary variables: use $Q$ when free
charge is held and $\Delta V$ when a source maintains terminal voltage.

$$
% caption: Charging a linear capacitor. Terminal voltage rises in proportion to the charge already separated, so the work stored in the field is the shaded triangular area under the voltage-charge line, which is half of the final charge times the final voltage rather than the full rectangle.
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In a vacuum parallel-plate capacitor with area $A$, gap $x$, and negligible
fringing, $C=\epsilon_0A/x$. Substitution into the voltage form gives

$$
U=\frac12\epsilon_0\left(\frac{\Delta V}{x}\right)^2Ax.
$$

The corresponding field-energy density is $u=\epsilon_0E^2/2$. Multiplying by the
central field volume $Ax$ recovers the terminal energy. This equality checks the
geometry and units. Finite plates also store energy in fringing space beyond the
rectangular central volume.

## Differential work and mechanical coordinate choices

If a capacitor geometry changes, capacitance becomes a function of a mechanical
coordinate. Let positive $x$ increase plate separation. The force is obtained from
a derivative only after the electrical boundary condition has been stated. For an
isolated capacitor, free charge is fixed and the appropriate stored energy is

$$
U(Q,x)=\frac{Q^2}{2C(x)},
\qquad
F_x=-\left(\frac{\partial U}{\partial x}\right)_Q.
$$

With $C(x)=\epsilon_0A/x$,

$$
F_x=-\frac{Q^2}{2\epsilon_0A}.
$$

The negative sign means the force reduces the gap. In this ideal fixed-charge model,
field magnitude and attractive force remain constant as the plates move because
surface charge density remains constant. Finite plates, supports, and nonuniform
surface charge modify the exact value, but the fixed-charge derivative states the
correct system boundary.

$$
% caption: Attraction between charged parallel plates. The nearly uniform field between the plates pulls each plate toward the other; for an isolated charged capacitor the fixed-charge energy derivative gives a force that reduces the gap, independent of which plate is labeled positive.
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$$

A held-voltage capacitor is a different system because a source supplies or absorbs
charge as $C(x)$ changes. The capacitor field energy is still $CV^2/2$, but
differentiating that expression at fixed voltage while ignoring source work gives an
incomplete mechanical balance. Electrical coenergy provides a convenient held-voltage
quantity,

$$
W'(V,x)=\frac12C(x)V^2,
\qquad
F_x=\left(\frac{\partial W'}{\partial x}\right)_V.
$$

For ideal plates,

$$
F_x=-\frac{\epsilon_0AV^2}{2x^2}.
$$

At held voltage, force grows as the gap becomes smaller because charge increases as
capacitance increases. This is the origin of electrostatic pull-in in compliant plate
systems: a small gap decrease raises attraction, which can decrease the gap further
if mechanical restoring force is insufficient. Using the held-charge formula while a
low-impedance source remains connected predicts the wrong variation with gap.

## Source work and system boundaries

The source resolves the apparent difference between fixed-charge and held-voltage
energy changes. At held voltage, an incremental capacitance change causes
$\d Q=V\,\d C$, so source work is $\d W_{\rm src}=V\,\d Q=V^2\d C$. The field energy changes
by $\d U=V^2\d C/2$. The remainder is mechanical work delivered by the electric force
in a quasistatic insertion or plate-motion process. For a displacement against the
electric attraction, the external mechanical agent supplies work and the source can
absorb the appropriate charge-related energy. Signs follow from the chosen work-on or
work-by convention; the complete source-field-mechanical boundary must be named.

> **Worked example (held-voltage plate force).** Vacuum plates of area
> $A=1.00\times10^{-2}\ \mathrm{m^2}$ at gap $x=1.00\ \mathrm{mm}$ are held at
> $V=100\ \mathrm V$. Then
>
> $$
> C=\frac{\epsilon_0A}{x}=88.5\ \mathrm{pF},\qquad
> U=\tfrac12CV^2=0.443\ \mathrm{\mu J},\qquad
> F=\frac{\epsilon_0AV^2}{2x^2}=4.43\times10^{-4}\ \mathrm N.
> $$
>
> Halving the gap at held voltage doubles capacitance and doubles field energy; the
> source supplies more than that increase because the electric force also does
> mechanical work. Disconnect the source before the motion and the charge is fixed
> instead, so the ideal force keeps its initial magnitude. Both results hold only below
> breakdown and while the plates stay approximately parallel.

The same numerical geometry illustrates the fixed-charge contrast. At the initial
held-voltage state, charge is $Q=CV=8.85\ \mathrm{nC}$. Disconnect the source while
that charge is present, then change the gap quasistatically. The ideal force magnitude
is $Q^2/(2\epsilon_0A)=0.443\ \mathrm{mN}$, equal to the initial held-voltage force
but independent of subsequent gap in the uniform-field approximation. At twice the
gap, capacitance is half its initial value, voltage doubles, and stored energy doubles
because the external mechanical agent has pulled against the electric attraction. The
opposite energy trend follows from the source boundary chosen before motion began.
It identifies a different physical system, not an algebraic inconsistency.

Force measurements require the same boundary documentation as energy calculations.
Record whether the source was connected during displacement, source output impedance,
initial charge or voltage, gap coordinate, plate area used in the model, and the
direction used for reported force. A force sensor connected through a compliant mount
can introduce a mechanical restoring force and a position-dependent gap; the measured
equilibrium then follows from the sum of mechanical and electrical forces, not from
the electric pressure alone. Compare force at several gaps under both held-charge and
held-voltage protocols. The distinct constant-versus-inverse-square trends provide an
independent check that the intended electrical boundary was actually realized.

At a fixed gap, the vacuum pressure $\epsilon_0E^2/2$ provides a local stress
check. Multiplying it by central area recovers the ideal total force, while spatial
variation near edges warns against using one average pressure for a flexible or
patterned electrode. A field calculation and a terminal-energy calculation should
therefore agree only when they describe the same geometry, charge state, and system
boundary.

## Electric pressure and geometry-dependent force

The pressure form follows from the same energy balance as the force derivative. In a
vacuum region where the electric field is normal to a conductor, the normal stress
magnitude is

$$
p_{\rm e}=\frac{\epsilon_0E^2}{2}.
$$

For wide parallel plates, the central field is nearly uniform, so
$F=p_{\rm e}A$. The stress acts on each facing conductor toward the other one.
That direction does not depend on the sign chosen for terminal voltage: reversing the
charge signs reverses the electric field but leaves $E^2$, energy density, and
attractive pressure unchanged. The sign of a reported force still depends on the
mechanical coordinate; a gap coordinate and an overlap coordinate assign opposite
signs to the same attractive tendency.

Finite electrodes need a spatial calculation. Near an edge, field lines bow outward,
and the normal stress varies across the surface. A flexible membrane can therefore
deflect most strongly near a patterned edge even when the area-averaged force agrees
with a parallel-plate estimate. Surface roughness, conductor curvature, and nearby
grounded structures also change the local field. The terminal relation $C(x)$
already includes those geometric effects if it has been measured or calculated with
the complete environment. Differentiating a simplified $C(x)$ while comparing with
a fixture containing guard rings and supports mixes two different systems.

$$
% caption: Stored energy versus gap for the same parallel-plate geometry under two electrical boundaries. Held charge gives energy proportional to the gap; held voltage gives energy inversely proportional to the gap. The opposite slopes still produce the same attractive force once the source work is included.
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$$

The pressure picture also distinguishes mechanical force from electrical breakdown.
A high field can produce a modest total force over a tiny area while exceeding the
dielectric strength of the gap material. Conversely, a large-area capacitor can
produce a substantial force at a field well below breakdown. Mechanical design needs
both the maximum local field and the integrated force. The limiting condition may be
dielectric failure, plate buckling, spring travel, thermal loss in the source, or
surface contamination rather than the energy formula itself.

A conductor entering between capacitor plates at held voltage gains capacitance as
the field-filled volume changes. Let $\ell$ be the overlap length of plates
of width $b$ and gap $g$, with negligible fringing. Then

$$
C(\ell)=\frac{\epsilon_0b\ell}{g},
\qquad
F_\ell=\frac12V^2\frac{\d C}{\d\ell}
=\frac{\epsilon_0bV^2}{2g}.
$$

The positive sign corresponds to motion toward greater overlap when positive $\ell$
denotes increased overlap. This geometry gives a direct force check because the ideal
force is constant over the overlap range away from entry and exit edges. A measured
slope with overlap can identify fringing, a changing gap, or a source whose voltage
falls as the capacitance increases.

$$
% caption: A variable-overlap capacitor turns an electrical boundary condition into a lateral force. At held voltage the capacitance grows linearly with overlap length, so the coenergy derivative predicts a pull toward greater overlap except near the entry and exit edges where fringing changes the geometry.
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$$

## Equilibrium and pull-in of a compliant support

Electrostatic attraction becomes a stability problem when one plate is mounted on a
spring, membrane, or flexure. Let the undeformed gap be $g_0$, let deflection
$x$ reduce that gap to $g=g_0-x$, and let a linear support exert restoring force
$kx$. At held voltage, equilibrium requires

$$
kx=\frac{\epsilon_0AV^2}{2(g_0-x)^2}.
$$

An equilibrium is stable only while a small added deflection produces a restoring
net force. The electrical force increases as the gap narrows, whereas the spring
force rises linearly. At the limiting point, the force slopes match:

$$
k=\frac{\epsilon_0AV^2}{(g_0-x)^3}.
$$

Combining the two conditions gives $x=g_0/3$, so the ideal gap at pull-in is
$2g_0/3$. Beyond that point, no nearby static balance exists in the ideal model.
The moving plate accelerates toward contact until another mechanism, such as a
mechanical stop, a nonlinear support, charge loss, or dielectric breakdown, changes
the system. Pull-in follows the coupled electrical boundary and mechanical stability
condition.

$$
% caption: Mechanical restoring force and held-voltage attraction versus plate deflection. A low-deflection crossing is stable because the spring slope exceeds the electrical-force slope; where the curves become tangent marks the pull-in limit, beyond which attraction grows faster than mechanical recovery.
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$$

> **Worked example (pull-in voltage).** A square MEMS plate of side $200\ \mathrm{\mu m}$,
> so $A=4.0\times10^{-8}\ \mathrm{m^2}$, hangs on a suspension of stiffness
> $k=0.50\ \mathrm{N/m}$ at initial gap $g_0=2.0\ \mathrm{\mu m}$. Equating the two
> equilibrium conditions at $x=g_0/3$ gives the pull-in voltage
>
> $$
> V_{\rm PI}=\sqrt{\frac{8k g_0^3}{27\epsilon_0A}}
> =\sqrt{\frac{8(0.50)(2.0\times10^{-6})^3}{27(8.854\times10^{-12})(4.0\times10^{-8})}}
> =1.8\ \mathrm V.
> $$
>
> Below $1.8\ \mathrm V$ the plate settles at a stable deflection under $g_0/3$; above
> it no static balance exists and the plate snaps to contact. The threshold scales as
> $g_0^{3/2}$, so a design with wider travel needs a sharply higher drive voltage.

Measure force with an independently calibrated load cell or displacement balance,
then compare it with both a capacitance derivative and a field-based stress estimate.
Record plate planarity, gap calibration, source voltage at the terminals, lead and
guard configuration, temperature, and the mechanical stiffness of the mount. A
voltage measured at the source output can exceed the capacitor terminal voltage if
the leads carry series resistance during a dynamic sweep. A force sensor can change
the gap through its own compliance. These are model inputs, not negligible apparatus
details.

An uncertainty budget for the ideal held-voltage force
$F=\epsilon_0AV^2/(2g^2)$ gives relative contributions from area, voltage, and
gap. The voltage contribution enters twice and the gap contribution enters twice with
opposite sign in a differential calculation. A one-percent gap error therefore
contributes approximately two percent to force uncertainty before edge-field and
alignment effects are included. Report repeatability separately from the model
discrepancy revealed by force-versus-gap residuals. Agreement at one gap cannot
validate an inverse-square trend.

## Nonlinear charge--voltage relations

The familiar forms $U=Q^2/(2C)$ and $U=CV^2/2$ require a linear relation between
charge and voltage at the selected geometry. A measured device can depart from that
relation through voltage-dependent permittivity, field-dependent geometry, trapped
charge, or an active circuit connected across its terminals. The energy calculation
then returns to its differential definition:

$$
U(Q,x)=\int_0^Q V(q,x)\,\d q.
$$

The area under a measured voltage-versus-charge curve gives the field energy only
when the charge state and mechanical coordinate are specified. A local slope
$\d Q/\d V$ is differential capacitance. It can differ from the ratio $Q/V$ when the
curve is nonlinear. Reporting only one capacitance value without its voltage range
can therefore hide the energy and force relevant to the operating point.

The held-voltage quantity has a corresponding integral form:

$$
W'(V,x)=\int_0^VQ(v,x)\,\d v,
\qquad
F_x=\left(\frac{\partial W'}{\partial x}\right)_V.
$$

The two areas differ whenever the charge--voltage curve is nonlinear, just as they
differ by a source-work term in the linear case. The choice follows the boundary:
isolate charge before displacement for the fixed-$Q$ derivative; retain the voltage
source and include its transferred charge for the fixed-$V$ derivative. A
quasistatic measurement can determine either curve by stepping voltage, waiting for
settling, and recording charge with a calibrated electrometer or integrating source
current over the charging interval.

$$
% caption: A nonlinear charge-voltage record has a voltage-dependent slope. The shaded area under voltage versus charge is the stored energy for a stated charge, while the area under charge versus voltage is the held-voltage coenergy used for a force derivative; a single ratio of charge to voltage replaces neither area away from the operating point.
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Hysteresis needs an additional distinction. If a voltage sweep follows one
charge--voltage branch on the way up and another on the way down, the enclosed loop
area represents energy dissipated or transferred to internal degrees of freedom per
cycle. A conservative electrostatic model cannot assign one unique energy function to
the same pair $(Q,x)$ when the prior history changes the result. Record sweep rate,
temperature, prebias, dwell time, and the direction of the voltage sweep. Those
conditions determine whether a reported capacitance describes an equilibrium state or
a rate-dependent device response.

## Work paths and quasistatic operation

Quasistatic motion means that electrical and mechanical variables remain close enough
to equilibrium for the selected $Q$- or $V$-dependent relation to apply at every
step. It does not mean that power vanishes. A small displacement $\d x$ at finite
speed transfers mechanical power $P_{\rm mech}=F_x\dot x$. A source exchanges power
$P_{\rm src}=V I$. The field energy changes at the difference required by the
chosen system boundary, with additional terms for resistive heating, radiation, and
material loss.

Consider a capacitor charged through a resistor and then used to pull a plate. During
the initial charging transient, part of the source energy is stored and part becomes
heat in the resistor. During later isolated motion, charge is fixed and an external
agent that separates the plates increases field energy. Holding voltage during that
same motion requires charge to flow back to the source. Combining those stages into
one sentence such as “the capacitor gains energy” erases the distinct sources and
destinations of work. Draw the electrical leads, mechanical coordinate, and loss
element before assigning a sign to a measured energy change.

Dynamic operation adds circuit time scales. A plate motion faster than the resistance
times capacitance can occur with nearly fixed charge even though a voltage source is
still physically connected through a large resistor. A low-impedance source at a
slow mechanical rate maintains nearly fixed voltage. Compare the mechanical
frequency with the electrical relaxation rate $1/(RC)$ before assigning a boundary
condition. At intermediate rates, charge and voltage both vary and a coupled circuit
equation is required. The fixed-charge and held-voltage formulas then provide
limiting checks rather than complete descriptions.

## Force-data residuals and model selection

Plot measured force against the variable predicted by the model. For ideal
held-voltage parallel plates, $F$ should be linear in $V^2/g^2$ with slope
$\epsilon_0A/2$. For fixed charge, force should be independent of gap. Residuals
against gap, voltage, overlap, and sweep direction locate missing physics. A residual
that grows at small gap can indicate fringing, tilt, or incipient pull-in. A residual
that changes sign on reversing a voltage sweep indicates hysteresis or charge
relaxation. Random scatter of the expected size supports the stated uncertainty
model, but a small scatter around a biased curve does not.

Keep the displacement coordinate convention with the force record. Reversing the
sign of a coordinate reverses a reported derivative sign, not the measured direction
of attraction or restoration. Record whether points were acquired while the gap was
increasing or decreasing; backlash and support compliance can separate those two
traces before any electrostatic correction is considered.

## Generalized force and torque from capacitance gradients

The mechanical coordinate need not be a plate gap. Let $q$ denote any controlled
coordinate: overlap length, lateral offset, rotation angle, membrane displacement, or
the position of a movable dielectric boundary. Under a held terminal voltage and a
linear capacitance model, the generalized force is

$$
F_q=\frac12V^2\frac{\d C}{\d q}.
$$

The units determine the physical name of the result. If $q$ is a distance,
$F_q$ has units of newtons. If $q$ is an angle in radians, the derivative gives
torque in newton metres. The sign follows the coordinate definition. A capacitance
that increases with overlap pulls a sliding electrode toward greater overlap at held
voltage. A capacitance that increases with rotation produces torque toward the
orientation of larger overlap.

Comb-drive actuators provide a specific capacitance-gradient geometry. Interleaved
conductor teeth form many side-by-side capacitors. In the central travel range, tooth overlap
changes while the lateral gap remains nearly constant, so total capacitance varies
approximately linearly with travel. The ideal held-voltage force is nearly constant
there. Near the ends of travel, fringing and incomplete overlap change the slope.
Lateral misalignment can make the force uneven across the two side gaps and pull the
moving comb sideways; an axial one-coordinate model cannot predict that instability.

$$
% caption: A lateral comb capacitor has many parallel tooth gaps. Greater tooth overlap raises capacitance at a nearly constant lateral gap, producing a held-voltage pull along the travel coordinate; the side gaps cancel lateral force only while the moving teeth stay centered.
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$$

Rotary capacitors use the same derivative with an angular coordinate. Consider two
sector electrodes separated by a fixed gap. As the moving sector turns into greater
overlap with the stationary sector, capacitance rises. At held voltage, the torque
tends to increase the overlap. A torsion spring exerts opposing torque
$\kappa\theta$. Equilibrium occurs where the electrical torque equals the spring
torque, and stability requires that the spring's torque slope exceed the electrical
torque slope near that angle. The same pull-in logic used for a closing gap can arise
in rotation when electrostatic torque increases faster than the mechanical restoring
torque.

$$
% caption: A rotary overlap capacitor turns an angular capacitance gradient into torque. At held voltage the moving sector is driven toward greater common area with the fixed sector; a torsion support balances the electrical torque only where its restoring slope is large enough for a stable angle.
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Capacitance-gradient measurements provide a direct route to force prediction when
the geometry is too complex for a closed-form field calculation. Measure $C(q)$
with the same terminals, guard, frequency, and mechanical fixture used during force
operation. A centered finite difference,

$$
\left.\frac{\d C}{\d q}\right|_{q_i}
\simeq
\frac{C(q_i+\delta q)-C(q_i-\delta q)}{2\delta q},
$$

estimates the local slope. The displacement interval must be large enough to exceed
capacitance-meter noise and small enough that curvature does not bias the derivative.
Repeat the measurement with several intervals. A derivative that changes strongly
with interval signals insufficient resolution, hysteresis, or a geometry whose slope
varies over the chosen range.

The predicted held-voltage force can then be compared with an independent load-cell
measurement. Propagate uncertainty from voltage, the two capacitance readings,
position calibration, and correlation between shared bridge corrections. A
finite-difference derivative amplifies noise because it subtracts nearby values.
Fitting a physically justified smooth $C(q)$ curve and differentiating that fit can
reduce random noise, but the fit residuals must remain available to reveal structure
the model has smoothed away.

### Local stress and the integrated force

The parallel-plate force formula is an area integral in disguise. At a conductor
surface in vacuum, the electric field immediately outside is normal to the surface,
and the local normal stress has magnitude

$$
p_n(\vec r)=\frac{\epsilon_0}{2}E_n^2(\vec r).
$$

With a rigid plate, the differential force is $\d\vec F=p_n\,\d A\,\hat n$
with the inward-facing normal chosen toward the opposing electrode. A
wide central region has nearly constant $E_n$, whereas the field spreads outward
near an edge and changes direction through the surrounding space. The local stress
there cannot be obtained by inserting one central gap into the uniform-field formula.
The total force remains the surface integral of the computed stress, but its spatial
distribution can differ markedly from the uniform-pressure picture.

This distinction matters when the electrode is compliant. A stiff plate may transmit a
nonuniform pressure pattern to its support and still show a total load close to
$\epsilon_0AV^2/(2g^2)$. A thin membrane can bend under the same pattern, changing
the local gap and hence the local field. Edge regions can then influence the measured
force indirectly through deformation even if their direct contribution to the
integrated force is modest. A force calculation that assumes a prescribed flat gap is
not self-consistent once the resulting stress noticeably changes that gap.

The field must be evaluated with the physical surroundings present. An open edge next
to free space, an edge next to a grounded frame, and an edge next to a guard electrode
have different stress distributions even when the central plate dimensions match.
Guard structures can make the central region more nearly uniform, but their own
charges and mechanical supports belong to the modeled system. A guarded capacitance
measurement and force comparison require the same connected guard geometry.

$$
% caption: Local electric stress on finite parallel plates. The central region carries a nearly uniform normal pressure while the edge field spreads outward and changes the local stress; the net load is the surface integral of that stress, not one pressure taken at the center.
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$$

### Plate tilt, lateral alignment, and torque

Parallelism is a mechanical condition, not a default property of two mounted plates.
Let a rectangular plate of width $b$ extend along coordinate $y$, and let a small
tilt make its local gap $g(y)=g_0+\alpha y$. At held voltage, the central-field
approximation gives the local pressure

$$
p(y)\simeq\frac{\epsilon_0V^2}{2[g_0+\alpha y]^2}.
$$

The narrower side has the larger pressure. Its excess load produces a torque about
the plate centre,

$$
\tau\simeq\int_{-b/2}^{b/2} y\,p(y)\,L\,\d y,
$$

where $L$ is the perpendicular plate dimension. For a positive tilt, this torque
acts in the direction that decreases the already small gap. A support that is merely
soft in rotation can therefore lose alignment before a one-coordinate gap model
predicts an instability. The integral remains informative even when the small-tilt
pressure approximation is only qualitative: total force alone can miss the mode that
actually limits travel.

Lateral displacement creates a different failure of the one-coordinate model. When
two plates are shifted sideways, the overlap area changes and edge fields are no
longer paired symmetrically. The energy gradient with respect to lateral coordinate
can generate a side force, while unequal gaps on opposite sides can generate an
additional side pull. A centered fixture may have zero net lateral force by symmetry, yet that
equilibrium need not be restoring. Displace the movable plate slightly in either
direction and evaluate the sign of $\d F_{\rm side}/\d s$; a force that increases the
displacement indicates lateral instability.

Alignment measurements should therefore include more than one gap reading. Measure
the gap near all accessible corners, or infer tilt from several displacement probes,
before interpreting a capacitance or force value as a uniform-gap result. The average
gap does not determine the local pressure because the dependence on gap is nonlinear.
A small narrow region can dominate both maximum stress and torque while contributing
little to the average separation reported by one central sensor.

$$
% caption: Tilted-plate electromechanics. A smaller local gap gives a larger held-voltage pressure, so the pressure distribution carries a torque that tends to close the narrow side; corner-gap readings separate this tilt from a single average-gap force model.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
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$$

### Force measurement with residual diagnostics

Use a force protocol that separates electrical load from mechanical preload. With the
plates at a surveyed gap, record the load-cell output at zero voltage, then apply a
sequence $0,+V,0,-V,0$ while monitoring the voltage at the capacitor terminals.
The electrostatic attraction has the same direction for $+V$ and $-V$, so the
two nonzero-voltage force changes should agree after the zero readings are removed.
Their average gives a current-even force estimate; their difference is a sensitive
check for offsets, charge injection, or drift that is synchronized with the source
polarity rather than with $V^2$.

At each voltage, wait long enough for the position and load sensor to settle, then
record terminal voltage, actual gap, force, and source current. The measured gap is
essential: a compliant mount can close slightly under attraction, and inserting the
initial gap into $\epsilon_0AV^2/(2g^2)$ then creates an apparent high-voltage
excess. Repeat the voltage sequence at several nominal gaps. A direct held-voltage
comparison uses

$$
r_i=F_{\rm meas,i}-\frac{\epsilon_0A V_i^2}{2g_i^2},
$$

or replaces the ideal term with a force computed from a measured capacitance gradient
when edge geometry is important. Plot residuals against voltage squared, gap, and
acquisition order; one plot rarely identifies every failure mode.

A residual that is nearly constant across voltage often indicates an imperfect load
zero or a gravitational change in the fixture. Positive curvature with voltage can
come from gap closure, source-voltage error, or unmodeled edge geometry. A residual
that appears only after repeated sweeps can indicate heating or slow charge motion on
nearby insulating surfaces. None of these patterns is random merely because the force
points are individually repeatable. Repeatability at one operating point tests sensor
noise; a controlled sweep tests the physical model.

Choose a voltage range below the onset of mechanical instability and any electrical
rating for a numerical run. Randomize or interleave the order of gap points
to avoid confusing monotonic drift with a geometry effect. Report the preload removal,
voltage meter uncertainty, gap calibration, active area or complete field model,
source connection, and the covariance of repeated readings. A force residual becomes
evidence only after those boundary and coordinate details have been retained.

### Reading a force-data disagreement

The first response to a disagreement should be a check of the boundary, not an
arbitrary correction to the force formula. Confirm that the source actually held the
listed terminal voltage during motion and that the voltmeter reference was connected
to the two capacitor plates rather than to distant supply terminals. Confirm that the
load-cell direction and the reported electric-force direction use the same sign
convention. A sign mismatch can otherwise be hidden by reporting only force
magnitudes.

Next separate geometry changes from force-sensor error. Repeat the zero-voltage gap
survey after a high-voltage sweep. A changed corner reading indicates support creep or
rotation, which changes the electric stress distribution even if the central gap
appears unchanged. Measure capacitance at the same mechanical coordinates when that
is practical. A capacitance shift that accompanies the force residual is evidence of
a changed electric geometry; a force shift with unchanged capacitance points more
strongly toward the load path, preload, or sensor electronics.

Keep residuals alongside any adjusted area or gap. An effective area may summarize
one operating range, but it has little diagnostic value if it changes with voltage,
alignment, or sweep direction. Refine the physical model with a measured gap map, a
complete capacitance-gradient record, or a field calculation that includes the
fixture. That sequence preserves the distinction between a known geometric effect and
an unexplained discrepancy.

### Source control and finite source impedance

The phrases “held voltage” and “fixed charge” are limiting descriptions of a circuit,
not labels that any connected source automatically earns. An ideal voltage source
connected directly to a capacitor holds its terminal voltage while charge changes with
geometry. A source with series resistance $R_s$ holds that voltage only to the
extent that the capacitor current produces a negligible drop. During a motion with
capacitance $C(q)$, the terminal relation is

$$
V_s=V_c+R_s\frac{\d[C(q)V_c]}{\d t}.
$$

The derivative contains both terms $C\,\d V_c/\d t$ and
$V_c(\d C/\d q)\,\d q/\d t$. Even with a constant commanded source voltage, a moving
electrode draws or returns current through the resistance. If the mechanical motion
is slow compared with the electrical relaxation time and the source has adequate
current capacity, $V_c$ remains close to $V_s$. If it does not, force must be
computed from the measured capacitor voltage, not from the source setting. The force
can then lag the motion or follow a rate-dependent path despite a capacitance that is
otherwise linear.

Leakage gives another departure from the ideal boundaries. With a leakage resistance
$R_\ell$ across the capacitor, the source current is

$$
\frac{V_s-V_c}{R_s}=\frac{\d(CV_c)}{\d t}+\frac{V_c}{R_\ell}.
$$

At a stationary dc operating point, leakage makes $V_c$ smaller than $V_s$ when
the series resistance is appreciable. Conversely, an isolated precharged capacitor is
only approximately fixed charge: its charge decays through leakage and instrumentation
input currents. The relevant time scale is set by the actual leakage path, including
the electrometer, humid surfaces, and cable insulation. State the duration between
disconnection and force reading when invoking a fixed-charge result.

An ideal current source has a different role. While it is active, it specifies
$\d Q/\d t=I_s$, not a constant value of $Q$. The terminal voltage evolves according
to the changing capacitance and current history. A current source can charge a device
to a known charge by integrating delivered current, after which disconnection can
approximate a fixed-charge experiment. It cannot be treated as a held-charge boundary
throughout a continuing mechanical motion unless feedback actively adjusts its output
to cancel every geometry-induced charge change.

These distinctions can be checked in data rather than inferred from wiring alone.
Record $V_c(t)$, source current, and mechanical coordinate during a programmed
motion. A voltage trace that departs from the command as speed increases indicates a
finite-impedance boundary. Slowing the same trajectory until the trace converges tests
whether the intended held-voltage approximation is justified.

### Quasistatic work cycles and loss accounting

Quasistatic electromechanical motion permits the electrical state and the mechanical
coordinate to be regarded as a sequence of nearby equilibria. It does not imply that
every cycle is lossless. The electrical work entering the capacitor terminals is

$$
W_{\rm el}=\int V_c\,\d Q,
$$

while the electrical force does work through $\int F_q\,\d q$. For a reversible
cycle in which the source, field, and mechanical store all return to their initial
states, the signed work exchanges balance. The area enclosed by a path in a
voltage-charge plot represents net electrical work for that cycle, provided the path
uses the actual terminal voltage and charge rather than nominal source settings.

Series resistance converts part of that work into heat,

$$
W_R=\int i^2R_s\,\d t.
$$

Mechanical damping, support friction, and internal structural hysteresis can consume
another part. They appear as a difference between mechanical work during the forward
and reverse portions of a coordinate cycle. A slow cycle reduces some rate-dependent
losses, but it does not remove dry friction or a support with path-dependent strain.
The term quasistatic therefore describes the absence of significant inertial and
electrical-lag effects, not an assurance that the work path retraces itself.

A work-cycle experiment drives the coordinate from $q_1$ to $q_2$ at a controlled
terminal boundary and returns it while recording $V_c$, current, and position. The
charge record follows from current integration with a stated initial charge, or from
a calibrated charge measurement. Compare the forward and reverse voltage-charge
paths. A separated pair of paths signals net electrical work per cycle. To assign the
loss, compare it with the independently measured mechanical work and with the
calculated $\int i^2R_s\,\d t$. This bookkeeping prevents a source-resistance loss from
being misidentified as an unusual capacitor force law.

Mechanical loading can also change the electrical trajectory. A compliant support
changes coordinate as voltage changes, so a voltage sweep is not necessarily a sweep
at fixed geometry. If the aim is to measure a capacitance curve at one coordinate,
the support must be locked or its displacement recorded and included. If the aim is
energy conversion, the displacement is part of the cycle and the source work must be
reported together with the mechanical work extracted or supplied.

### Traceable capacitance-gradient force calibration

With a held terminal voltage verified at the device, force calibration is based on
the local capacitance gradient rather than on an assumed plate area:

$$
F_q=\frac12V_c^2D,
\qquad D=\frac{\d C}{\d q}.
$$

Measure $C$ at $q-h$, $q$, and $q+h$ with a bridge whose reference standard,
test frequency, cable correction, and guard configuration are documented. The centered
estimate is $D\simeq[C(q+h)-C(q-h)]/(2h)$. The same terminals and fixture must be
used in the force experiment; changing cable routing or guard connection can alter
the measured gradient even when the movable structure itself has not changed.

For independent capacitance readings, the derivative variance from bridge noise is

$$
\sigma_{D,C}^2=\frac{\sigma_{C+}^2+\sigma_{C-}^2}{(2h)^2}.
$$

Shared reference and cable corrections can correlate the two readings. Include the
covariance term $-2\mathrm{cov}(C_+,C_-)/(2h)^2$ when it is known; common
offsets can cancel in a difference, whereas unrecognized drift does not. Coordinate
calibration adds approximately $(D\sigma_h/h)^2$ to the derivative variance.
Increasing $h$ reduces differencing noise but can bias the result when $C(q)$ is
curved, so repeat the calibration at several step sizes and document the selected
range.

> **Worked example (gradient force calibration).** Suppose
> $h=0.1000\pm0.0001\ \mathrm{mm}$, the two capacitance readings differ by
> $0.360\ \mathrm{pF}$, and each bridge reading has standard uncertainty
> $3\ \mathrm{fF}$. The centered gradient is
> $D=0.360\ \mathrm{pF}/0.2000\ \mathrm{mm}=1.80\ \mathrm{pF/mm}$. The bridge
> contribution is $\sqrt{3^2+3^2}\ \mathrm{fF}/(0.2000\ \mathrm{mm})
> =0.0212\ \mathrm{pF/mm}$ and the coordinate contribution is
> $0.0018\ \mathrm{pF/mm}$, combining to $0.0213\ \mathrm{pF/mm}$, or $1.18\%$.
>
> At $V_c=100.0\pm0.2\ \mathrm V$, the predicted force is
> $F_q=\tfrac12V_c^2D=9.00\ \mathrm{\mu N}$, with relative standard uncertainty
>
> $$
> \left(\frac{\sigma_F}{F_q}\right)^2=
> \left(2\frac{\sigma_V}{V_c}\right)^2+\left(\frac{\sigma_D}{D}\right)^2
> =(1.25\%)^2,
> $$
>
> or about $0.11\ \mathrm{\mu N}$. A load-cell comparison carries its own calibration
> and repeatability uncertainty and should not be folded into the prediction before the
> two are compared: voltage metrology, capacitance metrology, coordinate calibration,
> and force sensing each keep a visible contribution to the final claim.

### Maintaining the calibration chain

The gradient is a local property of a stated mechanical configuration. Do not measure
it once at a relaxed position and apply it without qualification after the support has
been loaded, repositioned, or thermally cycled. A reference displacement reading
before and after the capacitance sequence establishes whether the mechanical datum has
returned. If it has not, repeat the gradient measurement at the operating coordinate
rather than treating the shift as a small correction to force alone.

Traceability also requires a clear distinction between calibration values and control
values. The bridge standard establishes capacitance scale; the displacement reference
establishes the numerator separation in the gradient; the terminal voltmeter
establishes the voltage actually appearing across the device. A supply-panel display
controls operation but is not automatically a voltage calibration at the capacitor
after cable and series-resistance drops. Similarly, a stage encoder can command a
motion without establishing the electrode coordinate unless its zero and scale have
been related to the physical geometry.

At the force-comparison stage, keep the prediction and the force-sensor record
separate until both uncertainty budgets are complete. The prediction contains voltage,
capacitance-difference, coordinate, and source-boundary terms. The force-sensor record
contains its scale calibration, zero repeatability, alignment, and any load-transfer
correction. Combining them only for the final comparison prevents a favorable
force-sensor reading from masking a weak gradient calibration, or the reverse.

Archive raw bridge readings, reference-standard certificates, coordinate calibration
data, terminal-voltage traces, source current, and the exact derivative formula. The
record must also state the time between capacitance readings
and force operation, since slow mechanical relaxation can make the two nominally same
coordinates physically different. This record leaves the force prediction
independently recalculable from the complete measurement boundary and more than one
instrument display.