Capacitors
▶︎ Watch it animatedinteractive step-through · ~3 min · optional ⚙︎ Open the appletCapacitor Bench · name what the wiring pins before anything else, then take the area, gap and filling one at a time while a probe walks a gap whose field will not changeCapacitance is a ratio fixed by construction, $C = \kappa\varepsilon_0 A/d$, so charging a capacitor harder raises $Q$ and $V$ together and leaves $C$ alone. When the geometry is changed, the answer depends on what the circuit pins: a connected battery holds $V$ fixed so $Q = CV$ moves, and a disconnected one traps $Q$ so $V = Q/C$ moves. The field between the plates is uniform across the gap, $E = V/d$, so a charge released there accelerates uniformly. The stored energy is $\tfrac12 CV^2 = Q^2/2C$, quadratic in both quantities.
Five errors dominate. Reading $C = Q/V$ causally, so a fuller capacitor gets a larger capacitance and an empty one gets zero. Changing the geometry while holding fixed the quantity the wiring did not hold fixed, or carrying both $Q$ and $V$ forward from the before-state. Making the gap field fade with distance from a plate, as though it came from a point source. Getting the area and separation dependences backward, so a wider gap raises the capacitance. And scaling the stored energy in proportion to the voltage rather than to its square.
The work
3 ways in · any order
Lesson
Capacitors
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Fixes capacitance to the hardware, makes the first line of every problem the question of what the wiring pins, keeps the gap field uniform, and squares the stored energy.
Diagnostic
10-item topic check
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Ten items spanning the failure modes of this topic: making capacitance grow with charge, freezing the wrong quantity when the geometry changes, fading the field across the gap, inverting the area and separation dependences, and scaling stored energy linearly with voltage. Take it cold to find which one is yours, or after the lesson to confirm it is not.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.