The node law (Kirchhoff's current law)

Statement of the law

The node law follows from the conservation of electric charge: in steady state, charge cannot accumulate at a node. Therefore:

Sum of incoming currents = sum of outgoing currents

In other words, if we denote I1, I2, I3... the currents of the branches arriving at a node (incoming) and J1, J2... those leaving it (outgoing):

I1 + I2 + I3 + ... = J1 + J2 + ...

Numerical example

At node A, two currents come in: I1 = 2 A and I2 = 3 A. Only one current goes out: I3.

Node law: I1 + I2 = I3 -> I3 = 2 + 3 = 5 A.

Another form: algebraic convention

The node law can also be written by counting incoming currents as positive and outgoing currents as negative. The algebraic sum of all the currents at a node is then zero:

I1 + I2 - I3 = 0

Common pitfall

Be careful with the arrow direction chosen for each current: if you get the direction wrong when drawing the diagram, you get a negative result, which simply means the actual current flows in the direction opposite to the one assumed. This is not a calculation error; it is physical information to be interpreted.

Key takeaway

The node law applies to every node in the circuit, independently of the resistor values or the generator voltage: it is a purely topological law, linked to the conservation of charge.