Gustav Kirchhoff formulated two laws in 1845 that let any resistor
network be solved exactly. This circuit board has a voltage source and
a series resistor R1 feeding a junction where the current
splits between two parallel branches, R2 and R3,
before rejoining and returning to the source.
I1 = I2 + I3.EMF = I1·R1 + I2·R2 and EMF = I1·R1 + I3·R3.R3 from the circuit — watch I1 drop and all current reroute through R2.Kirchhoff's laws are a direct consequence of charge conservation (KCL) and energy conservation in an electrostatic-like field (KVL) — the same principles used to analyse everything from breadboard prototypes to continent-spanning power grids.
A 3D circuit board with a voltage source, a series resistor and two parallel resistor branches meeting at a junction — the sim solves the network live and shows currents and voltages that always satisfy KCL and KVL.
Current I1 leaving the source splits at the junction into I2 and I3 through the parallel resistors, and always recombines exactly (KCL). The voltage drops around each loop always sum to the source EMF (KVL).
Adjust the source EMF and the three resistor values, or open the branch B switch to remove R3 entirely. Watch the glowing particles speed up, slow down, or stop as the currents recalculate in real time.
Kirchhoff's laws, published in 1845, are special cases of the conservation of charge and energy — the same equations engineers still use today to design everything from phone chargers to national power grids.