Principle

Junction and loop rules

junction \sum I = 0, loop \sum Δ V = 0

Statement

Charge does not pile up at a junction; going right round a loop must return you to the same potential.

Symbols and units

$I$	currents meeting at a junction	A
$Δ V$	potential change across each element	V

When it applies

Lumped circuits: no charge piling up, no changing magnetic flux through the loop.

Don't use when

Assumes a lumped circuit with no charge accumulating at a junction and no changing magnetic flux through a loop — high-frequency or radiating circuits need a fuller electromagnetic treatment.

Derivation

Charge is conserved and does not accumulate at a point, so what flows in must flow out:

in \sum I = out \sum I

The electric force is conservative, so the potential change around any closed path is zero:

\oint E \cdot d s = 0 ⟹ loop \sum Δ V = 0

Sign rules when you traverse the loop — with the current through a resistor, against it, and through a source from − to + or the reverse:

- I R, + I R, + ε, - ε

Write (junctions − 1) junction equations plus enough loop equations to cover every branch, then solve the linear system.

Worked examples

Two sources, three branches

Junction rule:

I_{3} = I_{1} + I_{2}

Two independent loops:

ε_{1} - R_{1} I_{1} - R_{3} I_{3} = 0, ε_{2} - R_{2} I_{2} - R_{3} I_{3} = 0

Three unknowns, three equations, solved as an ordinary linear system.

Watch out for

Signs are where it goes wrong, not the algebra. Guess current directions freely — a negative answer simply means the arrow points the other way.

Connects to

Terminal voltage Combining resistors Charging