Foundations — Kirchhoff's current law (KCL), Kirchhoff's voltage law (KVL)

This page assumes you have seen nothing. Before you can read the parent note Kirchhoff's Laws, you need to genuinely feel what each squiggle means. We build them one at a time, each on top of the last.

1. Charge — the stuff that flows

The picture: imagine a bucket holding some water. The number of litres in the bucket is like the amount of charge $Q$. You can have more or less, but the litres never magically appear or vanish — they only move from one place to another.

Why the topic needs it: KCL is literally the statement "charge does not appear or vanish at a junction." You cannot understand that sentence until $Q$ means something concrete to you.

2. Current — how fast charge flows

$I=\frac{\Delta Q}{\Delta t}$

Read this out loud: "current equals the change in charge, $\Delta Q$, divided by the time it took, $\Delta t$." The symbol $\Delta$ (Greek letter delta) is shorthand for "the change in" — the after value minus the before value. So $\Delta Q$ = (charge now) − (charge a moment ago).

Why a rate and not just an amount? Because circuits are about flow. A wire doesn't care how much total charge has ever passed — it cares how fast charge streams through right now. That "per second" is exactly what makes $I$ a current (the word even comes from a river current).

3. The derivative — "instantaneous rate"

The parent note writes $\frac{dQ}{dt}$, not $\frac{\Delta Q}{\Delta t}$. Why the switch from $\Delta$ to $d$?

Why do we need this sharper tool? With $\Delta t$ being some visible chunk (say a whole second), if the flow is speeding up or slowing down you only get its average over that second. The letter $d$ answers a different, sharper question: "exactly how fast is charge changing at this precise moment?" For KCL's derivation we set this instant-by-instant rate to zero, so we need the instant version.

4. Node and Loop — the two shapes we do bookkeeping on

Why exactly these two? Because each conservation law lives on one of them:

• Charge conservation → applied at a node (nothing piles up there) → KCL.
• Energy conservation → applied around a loop (a round-trip nets to zero) → KVL.

Learn to spot nodes and loops in a diagram and half of circuit analysis is already done.

5. Electric potential and voltage — "electrical height"

The picture: think of a hilly landscape. Being high up = high potential. A battery is a pump that lifts charge up to a high point; a resistor is a slide that lets charge drop back down, releasing energy as it goes. See Electric Potential for the deep version.

Why "difference," not absolute height? Only differences in height make things move — the actual sea-level you measure from doesn't matter. That is why circuits care about $V_A-V_B$, never $V$ alone.

6. EMF ($\epsilon$) — the pump

Why a separate symbol from $V$? Because $\epsilon$ is a source of energy (a rise you can count on), while a resistor's $V$ is a drop where energy leaves. Giving the source its own letter keeps the "rise vs drop" bookkeeping clean in KVL.

7. Resistance $R$ and Ohm's Law — the slide's steepness

Read it: the bigger the current $I$ or the bigger the resistance $R$, the bigger the voltage drop $V$ you lose crossing it. This is the glue tying current to voltage — see Ohm's Law. Kirchhoff's laws give you equations; Ohm's law lets you turn the unknown voltages into unknown currents so you can actually solve them.

8. Signed sums and the symbol $\sum$

So $\sum _k{I}_k=0$ just says: "when you add up all the currents (with their signs), you get zero."

How it all feeds the topic

Equipment checklist

Connections

• Ohm's Law — supplies $V=IR$, the link between current and voltage.
• Conservation of Charge — why $Q$ never appears or vanishes; the root of KCL.
• Conservation of Energy — why a loop round-trip nets to zero; the root of KVL.
• Electric Potential — the "electrical height" idea in full.
• Series and Parallel Resistors — the first structures built from these foundations.
• Mesh and Nodal Analysis — systematic use of the symbols assembled here.
• Wheatstone Bridge — a circuit needing every tool on this page at once.