Kirchhoff's Laws and Their Application to Multi-loop Circuits

Many practical networks -- containing several cells and several resistors wired together, with more than one closed loop -- cannot be reduced to a single equivalent resistance by the series-parallel rules of Section 3.8 alone, because no two resistors in the network share BOTH of their end-points exclusively with each other. Kirchhoff's two rules, both direct consequences of principles already established elsewhere in physics, provide a completely general method for solving any such network.

Kirchhoff's first rule (junction rule / current rule). At any junction (node) in a circuit, where three or more wires meet, the algebraic sum of all the currents meeting at that junction is zero -- equivalently, the sum of currents flowing INTO the junction equals the sum of currents flowing OUT of it:

This is simply a statement of the conservation of electric charge: charge cannot accumulate indefinitely at a junction (in a steady, direct-current circuit), so whatever charge flows in per unit time must flow out per unit time as well.

Kirchhoff's second rule (loop rule / voltage rule). Around any closed loop in a circuit, the algebraic sum of all the potential differences (EMFs of any cells, and $IR$ drops across any resistors) taken in order around the loop is zero:

This is a statement of the conservation of energy: since electric potential is a well-defined, single-valued property of each point in the circuit, the net change in potential on returning to the same starting point, having gone all the way around a closed loop, must be exactly zero.

Sign convention and method of solution. To apply the rules: (i) mark an assumed direction for the current in each independent branch of the circuit (if the assumed direction is wrong, the algebra will simply return a negative value for that current, indicating its true direction is the opposite one); (ii) at each junction, write the junction-rule equation relating the branch currents meeting there; (iii) for each independent loop, traverse it in a chosen sense (say, clockwise) and write the loop-rule equation, taking an EMF as positive if the loop is traversed from the cell's negative to its positive terminal (a rise in potential) and an $IR$ term as positive if the loop is traversed in the SAME direction as the assumed current through that resistor (a fall in potential); (iv) solve the resulting set of simultaneous equations for the unknown branch currents. …