Introduction

In Class XI you studied why materials conduct electricity in the first place -- the origin of electrical conductivity, especially in metals -- and you learned how to collapse a chain of resistors wired purely in series, or purely in parallel, into one single equivalent resistance. Those series-parallel rules are enough for simple circuits, but many real circuits contain several resistors and cells connected in a more tangled way, with multiple loops and multiple junctions, that cannot be reduced to one loop just by repeatedly applying the series and parallel formulas.

Such complex networks are instead analysed using two rules formulated by the German physicist Gustav Robert Kirchhoff (1824-1887). Kirchhoff's rules do not replace Ohm's law; they extend it to any network, however many branches and loops it has, by combining Ohm's law with two very general conservation principles -- conservation of electric charge and conservation of energy. This chapter opens by stating and using these two rules (Section 9.2), and then shows how the same systematic circuit-analysis idea underlies three practical instruments built from resistor networks: the Wheatstone bridge and its everyday laboratory form, the metre bridge, both used to measure an unknown resistance very precisely (Section 9.3); the potentiometer, used to measure potential difference and emf without drawing any current from the source being measured (Section 9.4); and the moving-coil galvanometer, the sensitive current-detecting device that, with small circuit modifications, becomes either a working ammeter or a working voltmeter (Section 9.5). The chapter closes with a related but conceptually separate topic, the thermoelectric (Seebeck) effect, in which a temperature difference alone -- with no chemical cell at all -- can drive a current around a loop of two dissimilar metals (Section 9.6).