Capacitor Energy and Force

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 = 0 \int Q \frac{q}{C} d q = \frac{Q ^{2}}{2 C} = \frac{1}{2} Q Δ V = \frac{1}{2} C (Δ 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 $Q Δ V$ 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 $Δ V$ when a source maintains terminal voltage.

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.

In a vacuum parallel-plate capacitor with area $A$ , gap $x$ , and negligible fringing, $C = ϵ_{0} A / x$ . Substitution into the voltage form gives

U = \frac{1}{2} ϵ_{0} (\frac{Δ V}{x})^{2} A x .

The corresponding field-energy density is $u = ϵ_{0} E^{2} /2$ . Multiplying by the central field volume $A x$ 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}}{2 C ( x )}, F_{x} = - (\frac{\partial U}{\partial x})_{Q} .

With $C (x) = ϵ_{0} A / x$ ,

F_{x} = - \frac{Q ^{2}}{2 ϵ _{0} A} .

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.

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.

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 $C V^{2} /2$ , 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) = \frac{1}{2} C (x) V^{2}, F_{x} = (\frac{\partial W ^{'}}{\partial x})_{V} .

For ideal plates,

F_{x} = - \frac{ϵ _{0} A V ^{2}}{2 x ^{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 $d W_{src} = V d Q = V^{2} d C$ . 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.

The same numerical geometry illustrates the fixed-charge contrast. At the initial held-voltage state, charge is $Q = C V = 8.85 nC$ . Disconnect the source while that charge is present, then change the gap quasistatically. The ideal force magnitude is $Q^{2} / (2 ϵ_{0} A) = 0.443 mN$ , 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 $ϵ_{0} E^{2} /2$ 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_{e} = \frac{ϵ _{0} E ^{2}}{2} .

For wide parallel plates, the central field is nearly uniform, so $F = p_{e} A$ . 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.

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.

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 $ℓ$ be the overlap length of plates of width $b$ and gap $g$ , with negligible fringing. Then

C (ℓ) = \frac{ϵ _{0} b ℓ}{g}, F_{ℓ} = \frac{1}{2} V^{2} \frac{d C}{d ℓ} = \frac{ϵ _{0} b V ^{2}}{2 g} .

The positive sign corresponds to motion toward greater overlap when positive $ℓ$ 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.

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.

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 $k x$ . At held voltage, equilibrium requires

k x = \frac{ϵ _{0} A V ^{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{ϵ _{0} A V ^{2}}{( g _{0} - x ) ^{3}} .

Combining the two conditions gives $x = g_{0} /3$ , so the ideal gap at pull-in is $2 g_{0} /3$ . 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.

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.

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 $F = ϵ_{0} A V^{2} / (2 g^{2})$ 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 $U = Q^{2} / (2 C)$ and $U = C V^{2} /2$ 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) = 0 \int 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) = 0 \int V Q (v, x) d v, F_{x} = (\frac{\partial W ^{'}}{\partial x})_{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.

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.

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 $P_{mech} = F_{x} \overset{x}{˙}$ . A source exchanges power $P_{src} = V I$ . 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 $1/ (R C)$ 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 $ϵ_{0} A /2$ . 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} = \frac{1}{2} V^{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.

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.

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 $κ θ$ . 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.

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.

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,

\frac{d C}{d q}_{q_{i}} ≃ \frac{C ( q _{i} + δ q ) - C ( q _{i} - δ q )}{2 δ 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} (r) = \frac{ϵ _{0}}{2} E_{n}^{2} (r) .

With a rigid plate, the differential force is $d F = p_{n} d A \overset{n}{^}$ 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 $ϵ_{0} A V^{2} / (2 g^{2})$ . 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.

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.

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 $g (y) = g_{0} + α y$ . At held voltage, the central-field approximation gives the local pressure

p (y) ≃ \frac{ϵ _{0} V ^{2}}{2 [ g _{0} + α y ] ^{2}} .

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

τ ≃ - b /2 \int 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 $d F_{side} / d s$ ; 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.

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.

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 $ϵ_{0} A V^{2} / (2 g^{2})$ then creates an apparent high-voltage excess. Repeat the voltage sequence at several nominal gaps. A direct held-voltage comparison uses

r_{i} = F_{meas, i} - \frac{ϵ _{0} A V _{i}^{2}}{2 g _{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 $R_{ℓ}$ across the capacitor, the source current is

\frac{V _{s} - V _{c}}{R _{s}} = \frac{d ( C V _{c} )}{d t} + \frac{V _{c}}{R _{ℓ}} .

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_{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^{2} R_{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 $\int i^{2} R_{s} d t$ . 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} = \frac{1}{2} V_{c}^{2} D, 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 $D ≃ [C (q + h) - C (q - h)] / (2 h)$ . 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

σ_{D, C}^{2} = \frac{σ _{C +}^{2} + σ _{C -}^{2}}{( 2 h ) ^{2}} .

Shared reference and cable corrections can correlate the two readings. Include the covariance term $- 2 cov (C_{+}, C_{-}) / (2 h)^{2}$ when it is known; common offsets can cancel in a difference, whereas unrecognized drift does not. Coordinate calibration adds approximately $(D σ_{h} / h)^{2}$ 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 $h = 0.1000 \pm 0.0001 mm$ , the two capacitance readings differ by $0.360 pF$ , and each bridge reading has standard uncertainty $3 fF$ . The centered gradient is $D = 0.360 pF /0.2000 mm = 1.80 pF/mm$ . The bridge contribution is $3^{2} + 3^{2} fF / (0.2000 mm) = 0.0212 pF/mm$ and the coordinate contribution is $0.0018 pF/mm$ , combining to $0.0213 pF/mm$ , or $1.18%$ .

At $V_{c} = 100.0 \pm 0.2 V$ , the predicted force is $F_{q} = \frac{1}{2} V_{c}^{2} D = 9.00 μ N$ , with relative standard uncertainty

(\frac{σ _{F}}{F _{q}})^{2} = (2 \frac{σ _{V}}{V _{c}})^{2} + (\frac{σ _{D}}{D})^{2} = (1.25%)^{2},

or about $0.11 μ N$ . 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.