Kirchhoff's two rules extend Ohm's law so that complicated multi-loop circuits can be solved.
Step 1: First rule at a node.
Consider any point where several wires meet. The first rule (junction/point rule) says the currents there must balance: \(I_1 + I_2 = I_3 + I_4 + \dots\), i.e. everything arriving must depart. Written as one equation with a sign convention (inward \(+\), outward \(-\)):
\[\sum_{k} I_k = 0\]
Step 2: Why the node rule holds.
A junction is just a point and can store no charge. If more charge arrived than left, charge would pile up there and grow without limit, which never happens in steady current. So the rule is simply charge conservation written at a point.
Step 3: Second rule around a mesh.
Pick a closed path and walk all the way around it. The second rule (mesh/loop rule) says every gain in potential (across a cell) is exactly cancelled by the losses (across resistors):
\[\sum \varepsilon = \sum I R\]
Add EMFs with a \(+\) sign when you enter the cell at \(-\) and leave at \(+\), and subtract each \(IR\) term when you move along the current.
Step 4: Why the mesh rule holds.
Electric potential is single-valued, so a test charge taken once around the loop comes back to the same potential. The total work done on it per unit charge over the closed path is therefore zero, which is energy conservation.
Step 5: Using the rules.
Assign a current to each branch, write one node equation per independent junction and one loop equation per independent mesh, then solve the simultaneous equations.
\[\boxed{\text{Node rule} \to \text{charge conserved}, \qquad \text{Loop rule} \to \text{energy conserved}}\]