Kirchhoff's laws solve circuits that series and parallel rules cannot. Drag the five sliders below — the two EMFs and three resistances of a two-loop circuit — and watch all three branch currents solve live, with each arrow flipping direction the instant its current changes sign.
This simulator drives the article's two-loop circuit and re-solves it on every drag. The five sliders each feed one part of the network: ε1 and ε2 set how hard each battery pushes, while R1, R2 and the shared R3 set how much each branch resists. Move any one and the branch currents respond first, before the power panel updates — so you can trace exactly which input drove the change.
Two readouts do the teaching. The three currents always obey the junction rule, I1 + I2 = I3, no matter what you drag — the sum is enforced, never coincidental. And the power check always balances: the power delivered by the sources equals the power dissipated in the resistors, because that is energy conservation, the loop rule written in watts. Under the hood the sim solves the same closed form the calculator uses — it computes det = R1·R2 + R1·R3 + R2·R3, then I1 = (ε1·(R2+R3) - ε2·R3) / det and its partner for I2 — so the numbers on screen are the exact solution, not an approximation.
The reason this page exists is the misconception no static diagram can fix. Push ε2 down while ε1 stays high and watch I2 fall through zero and go negative — its arrow flips right there on the canvas. A negative current is not an error; it is the answer telling you the real direction of flow. Source 2, too weak to push against source 1, is now being charged, and the sim shows it as a reversed arrow and a “charging” status. Put exact numbers to any configuration with the Kirchhoff's Law calculator, read the full derivation in Kirchhoff's Laws (Current & Voltage), revisit the V = IR that each loop equation is built from, or open another model from the full simulation collection.
A negative sign means the current flows opposite to the direction drawn for that branch — the arrow flips on screen to show it. The magnitude is still exactly right: a reading of -1 A is a real 1 A flowing the other way. This is the simulator's whole point. It usually happens to I2 when source 2 is too weak to push against source 1, so current is forced backwards through it and it is being charged rather than delivering.
They set the two EMFs and the three resistances of the article's two-loop circuit: ε1 and R1 in the first source branch, ε2 and R2 in the second, and R3 in the branch the two loops share. Every time you move one, the simulator re-solves the junction and loop equations and updates all three branch currents at once.
Because energy is conserved — it is Kirchhoff's voltage law written in terms of power. The power delivered by the sources always equals the power dissipated in the resistors, no matter how you set the sliders. When a battery shows a negative current it is absorbing energy (being charged), so it counts as a load rather than a source, and the totals still match.
No — the sliders clamp at 0.5 ohm. A true zero-resistance branch is an ideal short circuit, which is not a lumped resistor and would make the network equations singular, leaving the current undefined. Keeping every resistance above zero guarantees the solver always has a well-defined answer.
No, the simulator is DC only: fixed EMFs and pure resistances. The same two rules — the junction rule and the loop rule — do extend to AC, but there you work with phasors and impedance instead of plain voltages and resistances, because capacitors and inductors introduce phase shifts. Master the DC case here first; the structure carries over.