Kirchhoff’s circuit laws — 2D schematic lab

Switch to 3D
Controls
Live readouts
Numerical checks
What to try
  1. Start with "One source, divider": E2 = 0, so R3 acts as a plain load to ground and the node voltage follows the parallel-resistor formula.
  2. Choose "Balanced": two equal EMFs with equal resistances make V_B equal to the EMF and the current through R2 vanish.
  3. Reverse E2 (negative): the current in the E2 branch flips and the dots reverse direction.
  4. Sweep R2 in the graph: the power in R2 peaks when R2 equals R1 in parallel with R3 (maximum power transfer).
  5. Watch the KCL and KVL residuals in the checks panel: they stay at numerical zero for any values.
The science

Kirchhoff’s two laws (1845) follow from conservation of charge and of energy in a lumped circuit.

KCL: ∑ Iin = 0 at every node  ⇒  I1 + I2 + I3 = 0 at node B KVL: ∑ ΔV = 0 around every closed loop  ⇒  E1 − I1R1 + I2R2 = 0 VB = (E1/R1 + E2/R3) / (1/R1 + 1/R2 + 1/R3)   (Millman’s theorem) Maximum power in R2 when R2 = Rth = R1R3 / (R1 + R3)

The solution is computed twice: by the node-voltage method and by an independent mesh-current Gaussian elimination, and the checks compare them, the KCL and KVL residuals, the power balance between sources and resistors, and a brute-force scan for the maximum-power resistance.

Simplifications: ideal voltage sources, ideal wires and linear resistors (no temperature dependence), steady DC conditions. Drifting dots are illustrative; real electron drift speeds are far slower.