Close a dielectric-change energy ledger under two constraints

Define topology, recipient and initial conditions

A dielectric can polarize without conducting free charge across the gap. In an ideal linear material filling fixed parallel plates, capacitance increases from C0 to κC0. Before applying an energy formula, decide whether free plate charge or source voltage is held fixed. Both cannot generally remain fixed while capacitance changes.

Derive the quantitative response

For an isolated charged capacitor, Q stays at C0V0, voltage becomes V0/κ and stored energy becomes U0/κ. For a capacitor connected to a maintained V0 source, voltage stays fixed, charge becomes κQ0 and energy becomes κU0. The connected source does work V0(Qfinal−Qinitial)=2U0(κ−1), which exceeds the stored-energy increase U0(κ−1). That is not a conservation failure.

Check evidence and the limits of the model

Under the explicitly lossless quasistatic convention, ΔU=Wsource+Wexternal-mechanical. In either case the external agent can extract mechanical energy as the dielectric is drawn in, so its work on the capacitor system is negative. Actual friction, finite insertion speed, leakage and material loss would require additional terms. The calculations are a virtual thermodynamic-style system account, not instructions for inserting material into an energized or charged device.

Worked original case

Initially C0=2 µF and V0=6 V give Q0=12 µC and U0=36 µJ. For κ=3 isolated, voltage becomes 2 V and U=12 µJ, with external mechanical work −24 µJ. Connected, Q=36 µC and U=108 µJ; source work is 144 µJ and mechanical work −72 µJ.

Investigation

Compare isolated and connected constraints for κ=1,3,5. Retain source work and mechanical work separately instead of calling a stored-energy decrease unexplained loss. Explain why κ=1 is an exact no-change boundary.

Changed-case practice

Sources and evidence boundary

Original numerical cases and prose checked 30 October 2026 against the relevant primary explanation. All records are synthetic and all diagrams mathematical. No practical wiring, battery testing, capacitor handling, mains measurement or equipment safety rating is requested. Public feedback does not verify independent mastery; retain first predictions, contrasting recipes and criterion-based critique in the separate course evidence workspace.

The molecular polarization distinction is also checked against University Physics, dielectric model. The quasistatic source/mechanical ledger here is an original algebraic energy-balance derivation, not a copied apparatus exercise.