Choose the boundary before predicting dielectric insertion

Separate bound polarization from mobile free charge

A dielectric has bound charges that shift or orient under an electric field rather than moving freely through it as carriers do in a conductor. For an ideal linear homogeneous material, kappa = epsilon/epsilon0 and full insertion changes C to kappa C0. The bound-charge polarization field opposes the original field of the free plate charges.

Hold free charge fixed

For an isolated capacitor, Q stays Q0 while C increases. Thus V = V0/kappa and E = E0/kappa for unchanged gap. Stored energy becomes U0/kappa. Polarization does not imply electrons crossed the dielectric from one plate to the other. The fixed-free-charge statement is essential to the field-reduction claim.

Hold voltage fixed instead

With an ideal source maintaining V0, the final free charge is kappa Q0 and stored energy is kappa U0. The source replenishes charge, so the net gap field remains V0/d despite the opposing polarization contribution. Treating V and Q as simultaneously fixed while changing C violates Q = CV.

Optional reversible mechanical energy audit

For controlled lossless insertion, delta U = Wsource + Wmechanical, where Wmechanical is external work on the selected field/material system. At fixed voltage Wsource = V0 delta Q = 2U0(kappa − 1), while delta U = U0(kappa − 1); therefore Wmechanical = −U0(kappa − 1). At fixed Q the source term is zero and external mechanical work equals delta U = U0(1/kappa − 1), which is negative for kappa greater than one. Negative external work means energy is extracted during reversible insertion; real friction, hysteresis and breakdown require additional terms.

Worked original case

Initially C0 = 2 nF and V0 = 4 V, so Q0 = 8 nC and U0 = 16 nJ. Inserting kappa = 2 gives C = 4 nF. Isolated: Q = 8 nC, V = 2 V and U = 8 nJ. Maintained voltage: Q = 16 nC, V = 4 V and U = 32 nJ; source work is 32 nJ and reversible external mechanical work −16 nJ.

Investigation

For each kappa, predict the isolated and maintained-voltage ledgers before switching the boundary mode. Retain free charge, field, source work and mechanical work separately. Explain which microscopic polarization mechanism is shared and which external boundary produces different energy outcomes.

Changed-case practice

Sources and boundaries

Original cases checked against the official framework on 30 October 2026, with OpenStax verification on 31 October 2026 against the current official framework, Unit 10 and OpenStax capacitance. These are virtual supplied models, not instructions to charge, ground, wire or handle real equipment. Public answers are practice; retained plans and self-reports do not establish independent mastery, actual laboratory participation or AP authorization.