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Level 1 · The Water Rules

The Water
Rules.

The rules of flow.

The five ideas everything else stands on: push, flow, the pinch, the bucket, and the switch. Electronics explained with water — start here.

Fathohm Field Guide · Volume 1 of 3 · 2026
Fathohmfathohm.com
Level 1 · Contents

The rules
of flow.

Introduction
The five rules
Back matter
The Water Rulesi
Introduction

Electricity is water
you can't see.

Not literally. But the math is the same, the intuition is the same, and once you see the resemblance you'll never un-see it.

A wire is a pipe. A battery is a barrel of water held up high. A resistor is a pinch in the pipe. The thing we actually care about — the thing that does work at the far end — is the flow.

Anything water does, electricity does too. So throughout this book we'll look at the water and trust that the wire is doing the same thing, silently, at nearly the speed of light.

Water pressure  Voltage  V
Rate of flow  Current  I
A pinch in the pipe  Resistance  R
Opening the tap  Closing the circuit
Keep this table in your head. Every chapter is just a new thing you can do once water flows.
Introduction1
Chapter 01 · The Flow

Push, flow, and
the pinch.

Three quantities run all of electronics, and they're the three things you already understand about water.

Voltage is the push. Picture a barrel of water held up high. The taller the column, the harder it presses down on the pipe. A AA battery is a 1.5 V barrel; a car battery, a 12 V reservoir; a wall outlet, a 120 V tower. Voltage is potential — the push that could happen, whether or not anything is flowing.

Current is the flow. Open the tap and water moves; that rate of movement is current, measured in amps. In a wire it's the count of electrons passing a point each second — about 6.24 × 10¹⁸ of them per amp, per second. You don't set current with a knob. It falls out of how hard you push against how much you pinch.

Resistance is the pinch. Squeeze the pipe and less gets through, no matter how hard you push. Every wire has a little; every component has its own. A toaster's wire has a lot — and that's why it gets hot.

V = I × R
V — volts, the push  ·  I — amps, the flow  ·  R — ohms, the pinch
Read three ways: V = I·R  |  I = V/R  |  R = V/I. Same fact, different hat.
Worked example

A 9 V battery pushes through a 450 Ω resistor. How much flows?

I = V / R = 9 / 450 = 0.02 A = 20 mA — a typical, healthy current for a small LED.

More push, more flow. Tighter pinch, less flow. That's Ohm's Law, and it never stops being true.
The Flow2
Chapter 02 · The Branching

One path, or
many.

When the pipe splits or runs in a line, two simple rules tell you everything.

In series, parts sit one after another on a single path — like two pinches in the same pipe. The same flow goes through every one, and their resistances add up. More parts in a row, more total pinch, less current overall.

In parallel, the pipe splits into branches that rejoin. Each branch gets the full push, so adding branches gives the water more ways through — total resistance goes down, and total flow goes up. Two identical branches each carry half the current; the source sees double.

series: R = R₁ + R₂ + …
parallel (two): R = (R₁·R₂)/(R₁+R₂) — always smaller than either branch.
Worked example

Two 100 Ω resistors. In series: 200 Ω. In parallel: 50 Ω.

Same two parts — wiring decides whether you double the pinch or halve it.

Series shares the current. Parallel shares the voltage. Everything branches from those two facts.
The Branching3
Chapter 03 · Power & Heat

Where the energy
goes.

Push times flow is power. Power, in a resistor, comes out as heat — and that's not a bug.

Voltage is the push and current is the flow; multiply them and you get power — the rate energy is delivered, measured in watts. A waterwheel turns faster when there's both height and volume behind the water. Lots of push but a trickle of flow does little work; so does a flood with no pressure. You need both.

In a resistor, that power becomes heat. It's why a toaster glows, why a phone charger is warm, and why every part has a limit. Pick a resistor too small for the job and you'll find out — with a wisp of smoke.

P = V × I
P — watts. Also P = I²·R and P = V²/R (substitute Ohm's Law).
A ¼-watt resistor can shed about 0.25 W before it cooks.
Worked example

That 450 Ω resistor at 20 mA: P = I²·R = (0.02)² × 450 = 0.18 W.

Under ¼ watt — a standard resistor handles it with room to spare.

Watts = volts × amps. Heat isn't waste you forgot — it's where the energy always ends up.
Power & Heat4
Chapter 04 · The Bucket

A part that
stores it.

A capacitor is a bucket in the pipe. It fills with charge, holds it, and gives it back.

Put a small bucket across the flow and it fills up over time — fast at first, then slower as it nears the top. While it's filling, current flows into it; once full, the flow stops. Disconnect the push and the bucket pours its charge back out. That's a capacitor: it stores energy in an electric field and releases it on demand.

How long it takes to fill depends on the bucket's size and the pinch feeding it. Big bucket or tight pinch — slow fill. This fill-time is the secret behind every delay, every smooth power supply, and every blink.

τ = R × C
τ (tau) — the time constant, in seconds. After one τ the bucket is ~63% full; after five, essentially full.
C — capacitance, in farads (usually microfarads, µF).
Worked example

10 kΩ feeding a 100 µF bucket: τ = 10 000 × 0.0001 = 1 s.

So it's ~63% charged after 1 second, and full after about 5.

A resistor pinches flow. A capacitor stores it. Together they make time.
The Bucket5
Chapter 05 · The Switch

On, off, and
the birth of binary.

The simplest part is the most profound. A switch is a tap — and a tap that a machine can flip is a thought.

A switch opens or closes the pipe. Closed, water flows and the loop is complete; open, everything stops. There's no in-between that matters — it's all or nothing. That two-state, yes-or-no nature is exactly what we call binary: on or off, 1 or 0.

Stack switches in a row and the water only flows if all are closed — that's an AND. Put them side by side and flow gets through if any is closed — that's an OR. Those two patterns, repeated a billion times on a chip, are how a computer reasons.

AND = switches in series
OR = switches in parallel. Every logic gate is a clever arrangement of taps.
A switch is a one-bit decision. Computers are just a great many of them, deciding very fast.
The Switch6
Back matter

Glossary of
the rules.

The Level 1 vocabulary, each term tied back to the water.

Voltage V · volts
The push. Water pressure — how hard the source presses on the circuit, flowing or not.
Current I · amps
The flow. How fast charge moves past a point. It falls out of voltage and resistance; you don't set it directly.
Resistance R · ohms (Ω)
The pinch. How much the pipe narrows the flow. More ohms, less current for the same push.
Ohm's Law
V = I × R. The bedrock relation between push, flow, and pinch.
Series & parallel
One path or many. Series adds resistance and shares current; parallel lowers resistance and shares voltage.
Power P · watts
Push times flow, V × I. The rate energy is delivered — in a resistor, it leaves as heat.
Capacitor C · farads
A bucket. Stores charge in an electric field; fills and empties on the R·C clock. Resists voltage change.
Switch
A tap. Open or closed — the one-bit decision that, stacked, becomes logic.
Glossary7
Ω

You know the rules. Now meet the parts.

Level 2 — The Parts Catalog — turns these five rules into real components: the transistor, the diode, AC and DC, the inductor, and the 555 timer.

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