Foundations — Series and parallel capacitors — derivations

This page builds every symbol the parent note leans on, from absolute zero. Nothing below assumes you have seen a capacitor before. Read top to bottom — each idea is a brick for the next.

1. Charge — the symbol $Q$

The picture: imagine a metal plate. If we scrape a certain number of extra electrons onto it, the plate carries negative charge; if we remove electrons, it carries positive charge. $Q$ is just the score-keeper for "how much extra".

Why the topic needs it: the whole point of a capacitor is to store $Q$. Every derivation begins with "how much charge sits here?" See Charge conservation for the rule that charge is never created or destroyed — we will use that in the series proof.

2. Voltage — the symbol $V$

The picture: think of two water tanks at different heights. Water wants to flow from high to low. The height difference is the "pressure" = voltage. A battery is a pump that keeps one side permanently higher.

Why the topic needs it: the parent note's key sentence "parallel = same $V$, series = same $Q$" is entirely about voltage. We must know exactly what "same voltage" means before that sentence makes sense.

3. A capacitor and its plates

The picture: two flat sheets facing each other with a thin gap. One goes positive, the other negative by exactly the same amount — the gap does not let charge cross, so the plates stay balanced at $+Q$ and $-Q$.

Often the gap is filled with an insulating material to boost storage — see Dielectrics in capacitors.

4. The master rule — $Q=CV$

The picture: capacitance is the width of a bucket. A wide bucket (big $C$) holds a lot of water (charge $Q$) while the water level (voltage $V$) barely rises. A narrow bucket fills to a high level with only a little water.

5. Wires, nodes, and "same voltage"

The picture: a wire is a perfectly flat, level lake. Drop a leaf anywhere on it — same water height everywhere. Two plates wired to the same node are forced to the same electrical "height".

Why the topic needs it: the parent's parallel proof opens with "$V_1=V_2=V_3=V$". That line is only legal because of equipotential wires and nodes.

6. Two conservation laws we will lean on

The picture: a road fork. Every car entering the junction must leave by some branch — none vanish. In the parallel proof this splits the battery's charge among branches: $Q=Q_1+Q_2+Q_3$.

The picture: a hiking loop. However many ups and downs, when you get back to camp your net height change is zero. In the series proof this gives $V=V_1+V_2+V_3$.

7. Equivalent capacitance $C_{eq}$

Why the topic needs it: it is literally the answer both derivations are hunting for. Everything else is a means to compute this one number.

8. Reciprocals — the symbol $1/C$

Prerequisite map

Equipment checklist

Cover the right side; answer, then reveal.