Dielectrics

A dielectric polarizes rather than conducts: bound surface charge appears on its faces, and the field inside drops to $E_0/\kappa$, a finite reduction rather than the complete cancellation a conductor produces. Filling a capacitor's gap replaces $\varepsilon_0$ with $\kappa\varepsilon_0$, so $C \to \kappa C_0$, and everything downstream depends on the constraint: with the battery attached $V$ is fixed, so $Q$ and $U$ both rise by $\kappa$ while $E$ is unchanged; disconnected, $Q$ is fixed, so $V$, $E$, and $U$ all fall by $\kappa$. A slab that fills part of the area makes two capacitors in parallel; one that fills part of the gap thickness makes two in series. The fringing field polarizes the slab and pulls it into the gap, with the force obtainable as $-dU/dx$.

Four errors dominate. Memorizing one constraint column and quoting it in the other scenario, which reverses the direction of most of the changes. Treating the slab as metal, so the internal field is set to zero and the bound charge is allowed to flow. Handling a partial fill by averaging $\kappa$ over the capacitor instead of splitting it along the boundary the slab actually makes, or choosing series where the geometry says parallel. And declaring that a neutral slab feels no force from a charged capacitor, when the induced polarization is exactly what pulls it in.