Foundations — Energy stored in capacitor U = ½CV²

This page builds every symbol the parent note uses, from absolute zero, in an order where each one leans only on the ones before it. If a word or squiggle on the parent page confused you, it is defined here.

0 · Charge $q$ and $Q$ — the stuff being moved

• Lowercase $q$ = the charge sitting on a plate right now, mid-way through filling (a running total).
• Uppercase $Q$ = the final charge when we stop filling.

Why the topic needs it: the entire question is "how much work to build up charge $Q$?", so we must be able to talk about the charge at every intermediate moment $q$, not just the end.

Look at the beads in the figure: the pile height is $q$. When we stop adding beads, the height is $Q$.

1 · Voltage $V$ (and instantaneous $v$) — the "hill height"

• Lowercase $v$ = the hill height right now, while charging.
• Uppercase $V$ = the final hill height.

Why two symbols again? Because the whole trick of the derivation is that the hill grows as you fill it. When the plate is empty the hill is flat ($v=0$, free to add charge); when full the hill is at its tallest ($v=V$).

2 · Capacitance $C$ — how "roomy" the bucket is

Picture two ways to think of it:

• A wide bucket (big $C$) holds lots of charge for only a little rise in level — voltage stays low.
• A narrow test-tube (small $C$) fills its level up fast — a little charge, big voltage.

Why the topic needs it: $C$ is the fixed constant linking charge and voltage. Because $v=q/C$ at every moment, capacitance is the number that turns "how much charge is on there" into "how tall the hill is now".

3 · The proportionality $Q=CV$ — a straight line

Because $C$ is constant, plotting charge $q$ against voltage $v$ gives a straight line through the origin with slope $C$.

See Capacitance and Q = CV for this relation in full.

4 · "$d$" of something ($dq$, $dW$) — a tiny sliver

Why the topic needs it: the hill height keeps changing, so we cannot use one fixed voltage for the whole job. We chop the job into slivers so tiny that during each sliver the voltage $v$ is effectively constant. Then we can safely say $dW=vdq$.

5 · The integral $\int$ — "add up all the slivers"

Read the whole line in plain words:

"Add up (work per sliver = current hill height $\times$ sliver of charge), from empty to full."

Because the hill grows as a straight ramp, this sum equals the area of the triangle in the figure of §3 — which is why the answer $U=\frac{Q^2}{2C}$ carries a $2$ in the denominator (triangle = half of rectangle).

6 · ${\epsilon }_0$, $A$, $d$, $E$ — the parallel-plate cast

These appear when the parent note turns energy into energy density. Meet them once:

For a parallel plate capacitor these combine into $C=\frac{{\epsilon }_{0}A}{d}$ — bigger plates or smaller gaps make a roomier bucket. Full story: Parallel plate capacitor C = ε₀A/d.

Why the topic needs them: they let us rewrite the stored energy as $u=\frac{1}{2}{\epsilon }_0{E}^2$ and reveal that energy lives in the field, not on the metal. See Energy density of electric field.

7 · $u$ vs $U$ — total energy vs energy per volume

Why the topic needs both: $U$ answers "how much energy total?"; $u$ answers "how is that energy spread through space?". Same letter, different case, different meaning — keep them straight.

The prerequisite map

Every arrow says "you need the left box before the right box makes sense". Follow them top-down and you arrive, fully equipped, at the parent topic: the topic note.

Related building blocks: Dielectrics and capacitance, Work done by a battery and Joule heating, Capacitors in series and parallel.

Equipment checklist

Test yourself — cover the right side and answer out loud.