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SCIENCE · INVESTIGATE AND EXPLAIN

Keep battery work in an energy-derived force

An energy derivative gives mechanical force only after the electrical constraint is stated. The supplied lateral-overlap approximation is C(x)=C0[1+3x/ell], with C0=100 pF and ell=0.1 m. It corresponds to relative permittivity 4 occupying part of a plate area at the same gap. Initial voltage at zero overlap is 10 V.

Learning outcome: Compare fixed charge and fixed voltage at overlap fractions 0,0.5,1. Retain source work, stored-energy change and external work before inferring force. Explain why the same geometric Cprime does not make force independent of the electrical constraint.

Starting recap

Review charge conservation, conductor equipotential conditions, definite field integrals and energy derivatives under a stated electrical constraint. Convert capacitance and charge units before interpreting small energies.

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Try

At half overlap, C=250 pF. With maintained 10 V, source work from zero overlap is 15 nJ, stored-energy increase is 7.5 nJ and external work is −7.5 nJ; pull is 150 nN. With fixed initial charge 1 nC, voltage instead becomes 4 V and the pull is 24 nN.

Open the experiment

Predict

At maintained voltage, why is stored-energy change alone insufficient to infer mechanical work?

Experiment

Compare fixed charge and fixed voltage at overlap fractions 0,0.5,1. Retain source work, stored-energy change and external work before inferring force. Explain why the same geometric Cprime does not make force independent of the electrical constraint.

Differentiate capacitance while retaining source work

Worked teaching model · synthetic values · Lateral dielectric overlap of ideal plates length .1 m. C(x)=100 pF[1+3x/.1], corresponding to relative permittivity4 and equal gap. Initial voltage10 V at zero overlap. Slow lossless motion. Bulk-capacitance approximation estimates energy/force; actual edge fields transmit the force and are not resolved. No insertion procedure.

Reset keeps your prediction, captures and practice history.

Energy-derived force under the selected constraint

force toward greater overlap (N)overlap fraction011.50000e-70

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: fixed-charge or fixed-voltage energy derivative (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Energy-derived force under the selected constraint

fixed-charge or fixed-voltage energy derivative

overlap fractionforce toward greater overlap (N)
01.50000e-7
0.01251.50000e-7
0.0251.50000e-7
0.03751.50000e-7
0.051.50000e-7
0.06251.50000e-7
0.0751.50000e-7
0.08751.50000e-7
0.11.50000e-7
0.11251.50000e-7
0.1251.50000e-7
0.13751.50000e-7
0.151.50000e-7
0.16251.50000e-7
0.1751.50000e-7
0.18751.50000e-7
0.21.50000e-7
0.21251.50000e-7
0.2251.50000e-7
0.23751.50000e-7
0.251.50000e-7
0.26251.50000e-7
0.2751.50000e-7
0.28751.50000e-7
0.31.50000e-7
0.31251.50000e-7
0.3251.50000e-7
0.33751.50000e-7
0.351.50000e-7
0.36251.50000e-7
0.3751.50000e-7
0.38751.50000e-7
0.41.50000e-7
0.41251.50000e-7
0.4251.50000e-7
0.43751.50000e-7
0.451.50000e-7
0.46251.50000e-7
0.4751.50000e-7
0.48751.50000e-7
0.51.50000e-7
0.51251.50000e-7
0.5251.50000e-7
0.53751.50000e-7
0.551.50000e-7
0.56251.50000e-7
0.5751.50000e-7
0.58751.50000e-7
0.61.50000e-7
0.61251.50000e-7
0.6251.50000e-7
0.63751.50000e-7
0.651.50000e-7
0.66251.50000e-7
0.6751.50000e-7
0.68751.50000e-7
0.71.50000e-7
0.71251.50000e-7
0.7251.50000e-7
0.73751.50000e-7
0.751.50000e-7
0.76251.50000e-7
0.7751.50000e-7
0.78751.50000e-7
0.81.50000e-7
0.81251.50000e-7
0.8251.50000e-7
0.83751.50000e-7
0.851.50000e-7
0.86251.50000e-7
0.8751.50000e-7
0.88751.50000e-7
0.91.50000e-7
0.91251.50000e-7
0.9251.50000e-7
0.93751.50000e-7
0.951.50000e-7
0.96251.50000e-7
0.9751.50000e-7
0.98751.50000e-7
11.50000e-7

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
QuantityValueUnit
Capacitance2.50000e-10F
Voltage10V
Free plate charge2.50000e-9C
Stored energy1.25000e-8J
Source work from zero overlap1.50000e-8J
External quasistatic work from zero overlap-7.50000e-9J
Pull toward increased overlap1.50000e-7N
Energy residual0J

At fixed voltage the source contributes work, so differentiating stored energy alone gives the wrong mechanical sign.

The parallel-area approximation gives C=C0[1+(kappa−1)x/ell]. At fixed charge, F=−d(Q²/2C)/dx=Q² Cprime/(2C²). At maintained voltage, source work is V² dC while stored energy rises by V²dC/2, leaving positive mechanical output Fdx=V²dC/2. The external agent work is negative during insertion. Actual edge fields transmit this force; the bulk approximation does not resolve them or authorize handling.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

At fixed voltage the source contributes work, so differentiating stored energy alone gives the wrong mechanical sign.

The parallel-area approximation gives C=C0[1+(kappa−1)x/ell]. At fixed charge, F=−d(Q²/2C)/dx=Q² Cprime/(2C²). At maintained voltage, source work is V² dC while stored energy rises by V²dC/2, leaving positive mechanical output Fdx=V²dC/2. The external agent work is negative during insertion. Actual edge fields transmit this force; the bulk approximation does not resolve them or authorize handling.

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

State conductor and source constraints

An energy derivative gives mechanical force only after the electrical constraint is stated. The supplied lateral-overlap approximation is C(x)=C0[1+3x/ell], with C0=100 pF and ell=0.1 m. It corresponds to relative permittivity 4 occupying part of a plate area at the same gap. Initial voltage at zero overlap is 10 V.

Derive charge, voltage and energy

At fixed charge, U=Q²/(2C), so the pull toward increasing overlap is −dU/dx=Q²Cprime/(2C²). At maintained voltage, stored energy instead increases by V²dC/2, but the source does work VdQ=V²dC. The remaining half is mechanical output, giving the same instantaneous form F=V²Cprime/2 under the appropriate voltage.

Check evidence and excluded mechanisms

Thus differentiating stored energy alone at fixed voltage would give the wrong mechanical sign. The external agent does negative work while slowly resisting insertion, and ΔU=Wsource+Wexternal. This bulk-capacitance approximation estimates force through energy; actual edge fields transmit that force and are not resolved here. No charged-capacitor insertion or handling is requested.

Worked original case

At half overlap, C=250 pF. With maintained 10 V, source work from zero overlap is 15 nJ, stored-energy increase is 7.5 nJ and external work is −7.5 nJ; pull is 150 nN. With fixed initial charge 1 nC, voltage instead becomes 4 V and the pull is 24 nN.

Investigation

Compare fixed charge and fixed voltage at overlap fractions 0,0.5,1. Retain source work, stored-energy change and external work before inferring force. Explain why the same geometric Cprime does not make force independent of the electrical constraint.

Changed-case practice

Try the changed case before feedback

At full overlap and fixed initial charge 1 nC, find the voltage in volts.

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original derivations and synthetic numerical packets checked 30 October 2026 against the primary explanation. No charged-capacitor handling, high-voltage apparatus, material safety limit or practical certification is requested. Preserve first predictions, independent critique, revision and newly assigned work separately from public practice feedback.

Apply

At maintained voltage, why is stored-energy change alone insufficient to infer mechanical work?

Public practice checks a specific claim. It does not establish independent mastery.

Which geometry, calibration or source assumption could fail while an exact arithmetic fit remains possible?

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