Kirchhoff's Circuit Laws Explained

Ohm's law tells you what a single resistor does. It says nothing about what happens when three resistors and two batteries are wired into a loop that feeds back on itself. For that you need two statements that Gustav Kirchhoff published in 1845, while he was still a student: the junction rule and the loop rule.

Neither is a new law of nature. The junction rule is conservation of charge, written for a wire. The loop rule is conservation of energy, written for a charge that goes all the way around and comes back to where it started. Everything else in DC circuit analysis — series resistance, parallel resistance, potential dividers, internal resistance — falls out of those two.

Before either law makes sense you need three words, and the first one is the one people get wrong.

Charge does not pile up in a wire. Whatever flows into a junction in one second must flow out of it in the same second, so:

It is often more convenient to count everything as leaving the node, letting currents that actually flow inwards come out negative. Then the law is simply:

Electric potential is a property of a point in the circuit. If you walk from a point all the way around a loop and arrive back at that same point, you must be back at the same potential — there is only one value there. So the rises and drops you accumulated on the way round have to cancel exactly:

Physically: a cell of EMF $E$ does $qE$ joules of work pushing charge $q$ through itself, and every resistance the charge then passes through takes some of that energy back out as heat. Round a full loop, the energy given equals the energy taken. KVL is conservation of energy per unit charge.

Almost every mistake in Kirchhoff problems is a sign, not a formula. Two habits make it reliable.

First: guess the current directions and commit. You do not need to guess correctly. Mark an arrow on every branch, do the algebra, and if a current comes out negative that simply means the real flow is the other way. A negative answer is information, not an error.

Second: pick a direction to walk each loop and apply the same two rules at every component you meet.

This is the circuit the simulation opens on, with its default values. Two cells and three resistors are wired between the same pair of nodes, $A$ at the top and $B$ at the bottom.

Take $B$ as the zero of potential and let $V_A$ be the unknown. Each branch current follows from Ohm's law applied across the whole branch — the potential you start at, minus the potential you end at, plus whatever the cell adds, all over the resistance:

Now apply the junction rule at $A$. Two branches feed it and one drains it:

That is one equation in one unknown. Collecting terms gives $0.6746\,V_A = 4.3810$, so:

Check it with the loop rule. Walking loop ① up the left branch and back down the middle:

The rules you were taught first are what the two laws reduce to in simple topologies. Switch the simulation between the Series and Parallel presets and watch the readout.

Series. A chain of resistors has no junctions between them, so KCL leaves no choice: the current is identical in every branch. KVL round the single loop then gives $E = I(r + R_1 + R_2 + R_3)$, which is the same as one resistor of value:

Parallel. All three resistors sit between the same two nodes, so they all have the same potential difference $V$ across them. KCL at the top node says the branch currents add to the total:

An ideal cell would hold its terminals at $E$ volts no matter how much current you drew. A real one cannot: its own chemistry has resistance $r$, and the current has to get through that too. What the outside world actually sees is the terminal voltage:

This is why headlights dim when you crank an engine, and why a battery reads a healthy voltage with nothing attached and sags the moment it has to do work. Put the simulation on the single-loop preset, drag the voltmeter leads onto the two nodes, and lower $R_1$: as the current climbs, the reading falls away from $E$.

Each cell delivers $P = EI$ into the circuit, and each resistance turns $P = I^2R$ of it into heat. Because KVL is conservation of energy, multiplying the loop equation by the current proves these must be equal overall:

The simulation shows both totals side by side, and they agree to the last decimal in every circuit — that is a genuine check on the solution, not decoration. In the default two-loop network it comes to 13.84 W each way, and note that E₂'s contribution to the left-hand side is negative: it is absorbing energy, not supplying it.

Guessing loops by hand does not scale. The simulation uses nodal analysis, which turns the junction rule into a matrix equation and solves it by Gaussian elimination — the same method any circuit simulator uses.

Every branch is treated as an EMF in series with a resistance, so its current is a function of the potentials at its two ends. Substituting that into $\sum I = 0$ at each node gives one linear equation per unknown potential:

// Assemble the conductance system G·v = i, one row per unsolved node.
for (const branch of branches) {
  const g = 1 / branch.R;
  for (const [id, s] of [[branch.from, 1], [branch.to, -1]]) {
    const row = unknowns.get(id);
    if (row === undefined) continue;       // the grounded node has no row
    G[row][row] += g;
    const col = unknowns.get(s === 1 ? branch.to : branch.from);
    if (col !== undefined) G[row][col] -= g;
    rhs[row] -= s * branch.emf * g;
  }
}

// Solve for the node potentials, then read every branch current back off them.
const V = gaussian(G, rhs);
for (const branch of branches) {
  branch.I = (V[branch.from] - V[branch.to] + branch.emf) / branch.R;
}

Unlike the rest of the site's simulations, there is nothing to integrate here — no positions, no velocities, no timestep. A resistive circuit settles instantly, so its state is the exact solution of a linear system, recomputed from scratch every frame. The only thing that advances with time is the animation of the charge carriers.