Circuit connections impose conservation rules on currents and voltages. Kirchhoff's laws express charge conservation at nodes and energy consistency around closed loops.

Kirchhoff's current law

At a node, charge does not accumulate in the steady lumped-circuit model. Therefore the algebraic sum of currents is zero:

$$\sum I=0.$$

Equivalently, total current entering a node equals total current leaving it.

Kirchhoff's voltage law

Around any closed loop,

$$\sum \Delta V=0.$$

After returning to the starting point, the electric potential must have returned to its original value.

Sign conventions

A current direction can be chosen arbitrarily before solving. If the calculated current is negative, the actual current flows opposite to the assumed direction.

Likewise, voltage rises and drops must be assigned consistently while traversing a loop.

From topology to equations

Kirchhoff's laws describe the network structure; component equations such as $V=IR$ describe how individual elements behave. Together they turn a circuit diagram into simultaneous algebraic equations.