Kirchhoff's Current Law (KCL)

Algebraic sum of currents at any junction is zero. Equivalently: sum of currents entering a junction equals sum leaving. Statement of charge conservation.

Statement

At any junction in a circuit, the algebraic sum of currents is zero:

$$\boxed{\sum_{\text{junction}} I = 0}$$

With sign convention: currents entering the junction are positive, currents leaving are negative (or vice versa — consistently).

Basis: charge conservation

In steady state, charge does not accumulate at any junction. The rate of charge arriving must equal the rate leaving:

$$\sum I_{in} = \sum I_{out}$$

If charge were to accumulate, the growing electric field would repel incoming charges and attract outgoing ones until the balance was restored — steady state enforces KCL automatically.

Application

For a junction with $n$ branches: write one KCL equation per junction. For a circuit with $j$ junctions, only $(j-1)$ equations are independent (the last is a linear combination of the others).

Example: T-junction

Currents $I_1$ into junction, $I_2$ and $I_3$ out:

$$I_1 - I_2 - I_3 = 0 \implies I_1 = I_2 + I_3$$