Apply Kirchhoff's Current Law (KCL): Foundations — Hardware deep dive

This page builds every symbol, word, and picture that the KCL topic leans on — starting from "what is a charge" and ending at "why $\sum i_{k} = 0$ makes sense." If the parent note ever wrote a symbol without explaining it, we explain it here. Read top to bottom: each idea is a brick for the next.

1. Charge — the stuff that flows

Why the topic needs it. KCL is a rule about charge. Before we can say "charge in = charge out," we must agree that charge is a fixed, countable amount that particles carry around.

Figure — Apply Kirchhoff's Current Law (KCL)

The key fact we will lean on later: charge is never created or destroyed — this is Conservation of Charge. If 100 marbles enter a box and none stay inside, exactly 100 must leave.

2. Current $i$ — the rate charge flows

Charge sitting still is boring. What matters in a circuit is charge moving. That's current.

To write "amount per second" in math we need one new tool: the derivative.

3. Direction and sign — current is an arrow, not just a number

A count of marbles per second tells you how much, but not which way. Current has a direction.

4. Node and wire — where KCL is applied

5. The summation sign $\sum$ — "add them all up"

The parent note writes $\sum_{k} i_{k} = 0$ . That $\sum$ (Greek capital sigma) is just shorthand.

Putting the pieces together: $\sum_{k} i_{k} = i_{1} + i_{2} + i_{3} + \dots = 0$ with each $i_{k}$ carrying its sign (+ if leaving, − if entering).

6. Stored charge $Q_{node}$ and continuity — why the sum is zero

Now we can state the one equation that makes KCL true. Call $Q_{node}$ the charge currently sitting inside the node.

Prerequisite map

Equipment checklist

Recall Are you ready for KCL? Cover the answers.

What is electric charge $q$ , and what unit? ::: The "amount of electric stuff" a particle carries; measured in coulombs (C). State the core fact from conservation of charge. ::: Charge is never created or destroyed — what enters a region must leave or stay. What does current $i$ measure, in one phrase? ::: The amount of charge passing a point each second (amperes). Write current as a derivative and say why we use the derivative. ::: $i = \frac{d q}{d t}$ ; the derivative gives the instantaneous flow, not just an average over time. On a charge-vs-time graph, what is $\frac{d q}{d t}$ ? ::: The slope (steepness) of the curve at that instant. Why must a current carry a sign, not just a magnitude? ::: Because direction is physical — entering and leaving currents are opposites, so they must subtract, not add. Under "leaving = +", how is an entering current counted? ::: As negative. What is a node, and when are two points the same node? ::: A junction of two or more branches; all points joined by pure ideal wire are the same node. What does $\sum_{k} i_{k}$ tell you to do? ::: Add up every branch current at the node, each with its sign. Why does an ideal node force $\frac{d Q _{node}}{d t} = 0$ ? ::: It stores no charge, so its stored charge never changes, hence its rate of change is zero.

Connections

Apply Kirchhoff's Current Law (KCL) — the parent topic these foundations feed.
Conservation of Charge — the physical law underneath everything here.
Current and Current Density — defines $i = d q / d t$ .
Ideal Wires and Nodes — why same-wire points are one node.
Capacitor i-v Relationship — resolves the "capacitor stores charge" worry.
Kirchhoff's Voltage Law (KVL) — the sibling law for voltages around a loop.
Nodal Analysis — applies KCL systematically to whole circuits.