Foundations — Series vs parallel resistor combinations

Before you can read $V=IR$ with understanding, you need to know what each letter, each line, and each junction is. This page builds them one at a time, from the wire up. Nothing here is assumed — if the parent note leaned on it, we define it. (One symbol, $R_{eq}$, "equivalent resistance," shows up in the parent's parallel formula — we define it carefully in §8, so don't worry that you haven't met it yet.)

0. The picture everything sits on: charge flowing through a wire

Imagine a pipe full of water. Water molecules drift along; the more that pass a point each second, the "bigger the flow." Electricity is the same, but the thing flowing is electric charge — tiny particles inside the metal wire, each carrying a fixed amount of charge.

Look at the figure: charge drifts left-to-right through a wire. Two ideas are hiding in this one picture, and we peel them off next: how much flows (current) and how hard we push (voltage).

1. Current — the symbol $I$

Charge is measured in a unit called the coulomb (C). One electron carries a tiny fixed charge (the elementary charge), and it takes about $6.24\times 10^{18}$ electrons to make one coulomb. So a coulomb is really "a fixed quantity of charge," the way a litre is a fixed quantity of water.

The picture: watch the dashed line in figure s01 for one second. Add up how much charge (in coulombs) crosses it in that second — that total is $I$ in amperes. Fast, crowded flow → big $I$. A trickle → small $I$. The "per second" is essential: the same number of charges crossing twice as slowly is half the current.

Which way does $I$ point? We must pick a direction to call positive. Conventional current points the way positive charge would drift — from high voltage toward low. If real electrons (negative) actually drift the other way, that's fine; the bookkeeping still works. We just draw an arrow, label it $I$, and stick to it.

Why the topic needs it: the whole game is "same current or same voltage?" You cannot decide series vs parallel without a name for how much flows. In Kirchhoffs Current Law we will add currents that meet at a junction — so $I$ has to come first.

2. Voltage — the symbol $V$

The picture: in a water pipe, voltage is the height difference between two tanks. Water rolls downhill; more height drop → harder push → more flow.

In the figure, the tall tank on the left sits above the short tank on the right, and a pink resistor sits in the pipe joining them. That height gap is the voltage across the resistor. No gap (same height) → no push → no flow.

Polarity — which end is "+"? Voltage has a sign too. We mark the higher-pressure end with a + and the lower with a − (the tall tank is "+", the short tank is "−"). In figure s02 the current flows out of the "+" side of the tank, into the "+" side of the resistor and out its "−" side — i.e. current enters a resistor at its higher-voltage terminal. This "$I$ enters at $+$" convention is what makes $V=IR$ come out positive; we lean on it whenever we apply Kirchhoffs Voltage Law and walk a loop adding drops.

Why the topic needs it: Kirchhoffs Voltage Law adds up voltages around a loop. Series resistors "share the push out" between them; parallel resistors "feel the same push." You literally cannot state either rule without $V$ — and without agreeing on signs, the sum around a loop is meaningless.

3. Resistance — the symbol $R$, and what makes a resistor a component

The picture: a narrow, gravelly section of pipe. The narrower and rougher it is, the harder water pushes through — that's high resistance. A wide clean pipe is low resistance.

Wire vs resistor — the precise criterion. A plain connecting wire is treated as $R=0$: charge flows along it with no voltage drop, so both its ends are the same electrical point. A resistor is a component with a deliberate, non-zero $R$ placed between two points, so a voltage does appear across it when current flows. That is the exact test: if there's a voltage drop for the current passing, it's a resistor (a separate component); if there's no drop, it's just wire — part of a single node (§5). This criterion, not the word "narrow," is what tells you where one component ends and a node begins.

Why the symbol $\Omega$? It is just the agreed unit-name for ohms, the way "m" means metre. When you see $100\Omega$, read it "one hundred ohms."

Why the topic needs it: the entire parent page is about combining $R$ values. Everything else — $I$, $V$ — exists here to pin down what $R$ means.

4. The ratio that ties all three: Ohm's Law $V=IR$

Now we have three pictures — flow ($I$), push ($V$), narrowness ($R$). They are not independent; experiment shows they lock together in one relationship.

WHY multiply, not add? Because resistance is a ratio of push-to-flow. Double the push and (for a fixed narrowness) you get double the flow — a proportional, multiplicative link, not an additive one. That is exactly what a straight line through the origin captures.

The figure plots current $I$ against voltage $V$ for two resistors. Each is a straight line through the origin; the steeper line is the smaller resistor (more flow per volt). The slope is $1/R$. This single picture is the engine behind every derivation on the parent page — see Ohms Law.

5. Nodes and paths — what "series" and "parallel" actually see

Symbols are not enough; the parent page talks about arrangement. Two words carry it.

Look at the figure. Left: two resistors end-to-end, one path — the arrow of current has no fork, so both see the identical $I$ (series). Right: two resistors between the same top and bottom nodes — the current splits, but both endpoints are shared, so both feel the identical $V$ (parallel).

Why the topic needs this: "same current → voltages add" gives the series formula; "same voltage → currents add" gives the parallel formula. Misreading a branch as a straight path is the single most common error — see the parent's steel-manned mistakes.

6. Conductance — the symbol $G$ and the "$1/R$" you keep seeing

The parallel formula on the parent page is full of reciprocals like $\frac{1}{R_1}$. There's a name for $1/R$.

The picture: if resistance is a narrow gravelly pipe, conductance is "how wide-open" the pipe is. Two wide-open pipes side by side (parallel) let even more through — so conductances simply add in parallel, which is why the parallel law adds $1/R$ terms and then you invert at the end. See Conductance.

7. The summation symbol $\Sigma$ and "$\dots$"

The parent writes $R_{eq}=R_1+R_2+\cdots +R_n$ and $\frac{1}{R_{eq}}=\sum \frac{1}{R_i}$.

Why the topic needs it: the rules work for any number of resistors, so we need a way to say "add them all" without writing forever.

8. Equivalent resistance — the symbol $R_{eq}$

This is the symbol the parent page's very first formula uses, so let's earn it.

The picture: hide a tangle of resistors inside a box with two wires poking out. From outside you can only measure the push $V$ you apply and the flow $I$ you get. $R_{eq}=V/I$ is whatever resistance would give that same $V$-to-$I$ ratio. The parent page's job is to compute this number for series and parallel tangles.

Now the parent's opening formula $\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}$ reads cleanly: it's the equivalent resistance of a parallel pair, written via conductances (§6) and then inverted.

How the foundations feed the topic

Read this map as a build order, bottom of each arrow first: charge gives us current and voltage; those three with $R$ give Ohm's Law; Ohm's Law plus the node idea produces the "same current" / "same voltage" split; and those, together with conductance and summation, let us compute $R_{eq}$. Each box below is a section you just read — the diagram is your checklist that nothing was used before it was built.

Equipment checklist

Cover the right side and answer aloud — if any stumps you, reread that section before the parent page.

Connections

• Ohms Law — the $V=IR$ ratio built here in full.
• Kirchhoffs Voltage Law & Kirchhoffs Current Law — the "voltages add / currents add" rules that need $V$, $I$ and the sign conventions above.
• Conductance — the $G=1/R$ view that makes parallel intuitive.
• Parent: Series vs parallel — where all these symbols get combined.
• Equivalent Resistance and Network Reduction — collapsing many $R$'s into one $R_{eq}$.
• Power Dissipation in Resistors — needs $I$, $V$, $R$ together once currents are known.