Reconcile field pressure with source and separating work
For vacuum parallel plates in the no-fringe approximation, field energy density is ε0E²/2 and attractive pressure has the same numerical value in SI. The plates have area 0.01 m², initial gap 0.001 m and initial voltage 10 V. We change the gap slowly while declaring either fixed initial charge or maintained voltage; these are different mechanical experiments.
Learning outcome: Compare final gap ratios 0.5,1,1.5,2 under both constraints. Retain field, force, stored-energy change and source/external work separately. Verify the exact no-change state and explain why the sign of stored-energy change alone cannot establish force direction under maintained voltage.
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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Let U0=ε0A(10 V)²/(2d0). Doubling the gap at fixed initial charge doubles stored energy, leaves attraction fixed and requires external work +U0. Doubling it at maintained voltage halves stored energy, returns U0 to the source and requires external work +U0/2. Attraction in the latter case is one quarter of its initial value.
Open the experimentPredict
When the gap doubles at maintained voltage, how does ideal attraction change?
Experiment
Compare final gap ratios 0.5,1,1.5,2 under both constraints. Retain field, force, stored-energy change and source/external work separately. Verify the exact no-change state and explain why the sign of stored-energy change alone cannot establish force direction under maintained voltage.
Reconcile field stress with fixed-charge or fixed-voltage work
Worked teaching model · synthetic values · Ideal vacuum parallel plates area .01 m², initial gap .001 m and initial voltage10 V. Quasistatic change to the selected gap, with either fixed initial free charge or maintained10 V. No fringe, elastic, pull-in, breakdown or radiative model. External work is positive for increasing gap, negative for compression, and zero for unchanged gap; source work depends on the constraint. No capacitor handling.
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Electrostatic attraction under the selected constraint
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: field-energy stress (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Electrostatic attraction under the selected constraint
field-energy stress
| gap ratio to initial | attractive force (N) |
|---|---|
| 0.5 | 1.76839e-5 |
| 0.51875 | 1.64286e-5 |
| 0.5375 | 1.53024e-5 |
| 0.55625 | 1.42882e-5 |
| 0.575 | 1.33716e-5 |
| 0.59375 | 1.25404e-5 |
| 0.6125 | 1.17843e-5 |
| 0.63125 | 1.10947e-5 |
| 0.65 | 1.04638e-5 |
| 0.66875 | 9.88530e-6 |
| 0.6875 | 9.35346e-6 |
| 0.70625 | 8.86341e-6 |
| 0.725 | 8.41088e-6 |
| 0.74375 | 7.99215e-6 |
| 0.7625 | 7.60393e-6 |
| 0.78125 | 7.24332e-6 |
| 0.8 | 6.90777e-6 |
| 0.81875 | 6.59500e-6 |
| 0.8375 | 6.30301e-6 |
| 0.85625 | 6.02999e-6 |
| 0.875 | 5.77433e-6 |
| 0.89375 | 5.53459e-6 |
| 0.9125 | 5.30948e-6 |
| 0.93125 | 5.09783e-6 |
| 0.95 | 4.89858e-6 |
| 0.96875 | 4.71079e-6 |
| 0.9875 | 4.53360e-6 |
| 1.00625 | 4.36622e-6 |
| 1.025 | 4.20794e-6 |
| 1.04375 | 4.05812e-6 |
| 1.0625 | 3.91615e-6 |
| 1.08125 | 3.78151e-6 |
| 1.1 | 3.65369e-6 |
| 1.11875 | 3.53225e-6 |
| 1.1375 | 3.41676e-6 |
| 1.15625 | 3.30685e-6 |
| 1.175 | 3.20215e-6 |
| 1.19375 | 3.10235e-6 |
| 1.2125 | 3.00714e-6 |
| 1.23125 | 2.91625e-6 |
| 1.25 | 2.82942e-6 |
| 1.26875 | 2.74641e-6 |
| 1.2875 | 2.66700e-6 |
| 1.30625 | 2.59099e-6 |
| 1.325 | 2.51817e-6 |
| 1.34375 | 2.44839e-6 |
| 1.3625 | 2.38147e-6 |
| 1.38125 | 2.31725e-6 |
| 1.4 | 2.25560e-6 |
| 1.41875 | 2.19637e-6 |
| 1.4375 | 2.13945e-6 |
| 1.45625 | 2.08471e-6 |
| 1.475 | 2.03205e-6 |
| 1.49375 | 1.98135e-6 |
| 1.5125 | 1.93253e-6 |
| 1.53125 | 1.88550e-6 |
| 1.55 | 1.84015e-6 |
| 1.56875 | 1.79643e-6 |
| 1.5875 | 1.75424e-6 |
| 1.60625 | 1.71353e-6 |
| 1.625 | 1.67421e-6 |
| 1.64375 | 1.63624e-6 |
| 1.6625 | 1.59954e-6 |
| 1.68125 | 1.56406e-6 |
| 1.7 | 1.52975e-6 |
| 1.71875 | 1.49655e-6 |
| 1.7375 | 1.46443e-6 |
| 1.75625 | 1.43333e-6 |
| 1.775 | 1.40320e-6 |
| 1.79375 | 1.37402e-6 |
| 1.8125 | 1.34574e-6 |
| 1.83125 | 1.31832e-6 |
| 1.85 | 1.29174e-6 |
| 1.86875 | 1.26595e-6 |
| 1.8875 | 1.24092e-6 |
| 1.90625 | 1.21663e-6 |
| 1.925 | 1.19304e-6 |
| 1.94375 | 1.17014e-6 |
| 1.9625 | 1.14788e-6 |
| 1.98125 | 1.12626e-6 |
| 2 | 1.10524e-6 |
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.
| Quantity | Value | Unit |
|---|---|---|
| Final gap | 0.0015 | m |
| Final capacitance | 5.89463e-11 | F |
| Final voltage | 10 | V |
| Final free plate charge | 5.89463e-10 | C |
| Final gap field | 6666.667 | N/C |
| Attractive force magnitude | 1.96488e-6 | N |
| Field pressure | 0.0001964876 | Pa |
| Stored-energy change | -1.47366e-9 | J |
| Source work during gap change | -2.94731e-9 | J |
| External work during gap change | 1.47366e-9 | J |
| Energy residual | 0 | J |
A fixed-charge gap change keeps the ideal field and attraction constant; fixed voltage gives a different force and source-energy account.
Vacuum energy density is epsilon0 E²/2, and the attractive pressure has the same numerical value in SI. At fixed Q, E=Q/(epsilon0 A) is constant and external work increases stored energy linearly with gap. At fixed V, C falls as1/d; the source work V DeltaQ is negative during separation, while external work is positive. Compression reverses both work signs; unchanged gap gives zero work. In both cases DeltaU=Wsource+Wexternal. Do not infer elastic stability, pull-in or breakdown from this rigid-plate energy model.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
A fixed-charge gap change keeps the ideal field and attraction constant; fixed voltage gives a different force and source-energy account.
Vacuum energy density is epsilon0 E²/2, and the attractive pressure has the same numerical value in SI. At fixed Q, E=Q/(epsilon0 A) is constant and external work increases stored energy linearly with gap. At fixed V, C falls as1/d; the source work V DeltaQ is negative during separation, while external work is positive. Compression reverses both work signs; unchanged gap gives zero work. In both cases DeltaU=Wsource+Wexternal. Do not infer elastic stability, pull-in or breakdown from this rigid-plate energy model.
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