Visual walkthrough — Define electric charge, electron, proton, and the coulomb

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Step 1 — One electron, one indivisible packet

WHAT. We start with the smallest thing there is to count: a single electron. It carries a fixed amount of negative charge. We give that amount a name, the letter $e$ — the elementary charge. Think of $e$ as the size of one marble, nothing more.

WHY. Before we can talk about "how much charge", we need a unit lump to measure in. Nature is kind here: charge does not come in smooth syrup, it comes in identical lumps. So a single electron is our ruler.

WHAT IT LOOKS LIKE. Look at the single coral marble in the figure. Its label $-e$ says: this one lump carries one unit of negative charge. There is no smaller piece — you cannot slice the marble.

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Step 2 — Charge only comes in whole marbles (quantisation)

WHAT. You can have $1$ electron, $2$ electrons, $3$ electrons… but never $1.5$. Charge is quantised — it exists only in whole-number multiples of $e$.

WHY. This is an experimental fact (Millikan's oil-drop experiment). It is what lets us count instead of measure. Counting is easier than measuring, and it makes our formula exact.

WHAT IT LOOKS LIKE. In the figure the allowed charges sit on a ladder: rungs at $e$, $2e$, $3e$, $4e$. The greyed-out gap between rungs (the dashed marble at $1.5e$) is forbidden — there is nowhere for it to stand.

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Step 3 — Adding marbles: charge is additive

WHAT. Put $N$ marbles together. Because charge simply adds up, the total is $Q=\underset{N\text{ marbles}}{\underbrace{e+e+\cdots +e}}.$

Let us read this equation piece by piece:

• $Q$ — the total charge of the whole pile (what we want).
• each $e$ — one marble's charge (from Step 1).
• $N$ — how many marbles are in the pile (a plain counting number: $1,2,3,\dots$).

WHY. Charges do not shrink or grow when you bring them together; two lumps side by side just carry both lumps' worth. "Total = sum of parts" is exactly what makes charge countable.

WHAT IT LOOKS LIKE. The figure stacks marbles in a row and a running total climbs beside them: $1$ marble $\to e$, $2$ marbles $\to 2e$, $3$ marbles $\to 3e$. The total rises in equal steps of $e$.

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Step 4 — Repeated addition IS multiplication: $Q=Ne$

WHAT. Adding the same lump $N$ times is just multiplying. So the long sum collapses into one clean line: $\boxed{Q=Ne}$

Term by term, right where each symbol lives:

• $Q$ on the left — the answer, total charge in coulombs.
• $N$ — the count of elementary charges (electrons or protons). Pure number, no unit.
• $e$ — the charge per marble, $1.602\times 10^{-19}\text{C}$. This is what carries the unit.

WHY. We prefer $N\times e$ over writing out $e+e+\dots$ because $N$ can be astronomically large (billions of billions). Multiplication lets us handle huge piles in one stroke — you would never write out $10^{18}$ plus signs.

WHAT IT LOOKS LIKE. The figure shows the tall column of many marbles on the left collapsing into a single multiplication box on the right: "$N$ copies of $e$" $=$ "$N\times e$". Same pile, shorter sentence.

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Step 5 — Reading it backwards: this is what "coulomb" MEANS

WHAT. Now flip the formula. Ask: how many marbles make one whole coulomb? Set $Q=1\text{C}$ and solve for $N$: $N=\frac{Q}{e}=\frac{1}{1.602\times 10^{-19}}\approx 6.242\times 10^{18}.$

Reading the division:

• $Q=1\text{C}$ — the target pile size (one coulomb).
• $e$ in the denominator — how big each marble is. Dividing "total wanted" by "size of one piece" answers "how many pieces?".
• $N\approx 6.242\times 10^{18}$ — the count that comes out.

WHY divide, not something else? "How many small things fit in a big thing?" is always a division: metres in a kilometre, cups in a bucket, marbles in a coulomb. Division is the how-many-fit tool.

WHY does this matter? This number is not a lucky coincidence — it is the definition of the coulomb read in reverse. The coulomb was set up to be this pile size.

WHAT IT LOOKS LIKE. The figure shows a giant bucket labelled $1\text{C}$ being filled with a swarm of tiny marbles; a counter on the bucket reads $6.242\times 10^{18}$. One marble is drawn beside the bucket at true (invisibly tiny) relative scale to show how absurdly many it takes.

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Step 6 — The sign: positive, negative, and exactly zero

WHAT. Our formula gives a size. The sign comes from which marble you counted:

• Count electrons → total is negative: $Q=-Ne$.
• Count protons → total is positive: $Q=+Ne$.
• Equal numbers of each → they cancel to zero: $Q=+Ne-Ne=0$ (a neutral atom).

WHY. There are only two kinds of charge (+ and −), so there is only one extra thing to track: direction on a number line. Zero is not a special third case — it is just the point where the two counts balance.

WHAT IT LOOKS LIKE. Three scenes in the figure:

1. a lavender pile of protons → arrow points right (positive),
2. a coral pile of electrons → arrow points left (negative),
3. equal piles facing off → the arrows cancel, needle rests at $0$.

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Step 7 — The degenerate cases: $N=0$ and huge $N$

WHAT.

• If $N=0$ (no spare marbles), then $Q=0\times e=0$. No charge. This is the balanced, neutral case — nothing to move.
• If $N$ is astronomically large, $Q$ can be many coulombs. There is no largest charge; you just keep stacking marbles.

WHY show this. A formula you trust must survive its extremes. At $N=0$ it correctly says "nothing"; at giant $N$ it keeps working without breaking. No hidden surprise waits at the edges.

WHAT IT LOOKS LIKE. The figure plots $Q=Ne$ as a straight line through the origin: at $N=0$ it sits exactly on zero; it climbs forever with no ceiling. The line is straight precisely because every marble is identical.

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Worked example, seen on the ladder

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The one-picture summary

WHAT. One figure carries the whole story left-to-right: one marble $e$ → stack $N$ of them → multiply $Q=Ne$ → fill the $1\text{C}$ bucket ($N=1/e$). That single arrow of pictures is the derivation.

Recall Feynman retelling — the walkthrough in plain words

Charge comes in tiny identical marbles; one marble's worth is called $e$ (Step 1). You can never have half a marble — only whole ones (Step 2). Pile up $N$ marbles and the charge just adds up (Step 3), and adding the same thing $N$ times is the same as multiplying, so $Q=Ne$ (Step 4). Turn that around and ask how many marbles fill one big bucket called a coulomb: divide the bucket by one marble's size, and out pops $6.24$ billion-billion (Step 5). Whether the total counts as plus or minus just depends on which marble you piled — protons push the needle right, electrons push it left, equal piles rest at zero (Step 6). With no marbles you get zero charge, and you can always stack more for a bigger charge — the formula never breaks (Step 7). That's the entire idea: count marbles, multiply by marble-size, read the sign.

Recall Quick self-test

• Why can we multiply instead of adding one $e$ at a time? ::: Because every marble is identical, so repeated addition of $e$ collapses to $N\times e$.
• What does dividing $1$ by $e$ answer? ::: How many elementary charges fit in one coulomb — i.e. the definition of the coulomb.
• Where does the minus sign in $Q=-Ne$ come from? ::: From choosing to count electrons rather than protons; it is direction on the +/− number line, not part of the size.
• What is $Q$ when $N=0$? ::: Exactly zero — the neutral case.

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Connections

• Define electric charge, electron, proton, and the coulomb (index 1.1.1) — the parent topic this walkthrough unpacks.
• Electric Current and the Ampere — moving these coulombs per second.
• Voltage and Potential Difference — the push that moves the marbles.
• Conductors and Insulators — whether the marbles are free to slide.
• Coulomb's Law — the force between two of these charges.
• Atomic Structure — where the marbles live.