Capacitors

A capacitor stores $+Q$ and $-Q$ on two conductors, and $C = Q/V$ is fixed by geometry alone: $C = \varepsilon_0 A/d$ for parallel plates, with no $Q$ and no $V$ in it. The charge collects on the facing inner surfaces, the field is $\sigma/\varepsilon_0$ in the gap and zero outside, and no charge crosses the gap. Series capacitors share $Q$ and add reciprocally; parallel capacitors share $V$ and add directly, the mirror of the resistor rules. The stored energy is $Q^2/2C = \tfrac12 QV = \tfrac12 CV^2$, three forms of one quantity that stop agreeing about the DIRECTION of a change as soon as the capacitor is altered.

The errors cluster. Reading $C = Q/V$ as a recipe, so that more charge means more capacitance. Adding series capacitances directly, or handing series capacitors equal voltages instead of equal charge. Using the full gap field to compute the force on a plate, which double-counts, or using half of it for a charge placed in the gap. Painting charge on both faces of each plate. Running a steady current through the insulating gap. And the largest one: quoting an energy change from a formula whose variable is not the one the wiring holds fixed, which reverses the sign of the answer.