Q.In the circuit shown, a battery of emf 15 V (internal resistance negligible) is connected through a $100\ \Omega$ resistor carrying current $I_1$, and a battery of emf 9 V is connected through another $100\ \Omega$ resistor carrying current $I_2$; both branches meet at a junction and share a third $100\ \Omega$ resistor $R_3$ carrying current $I_3 = I_1 + I_2$. Calculate the currents $I_1$, $I_2$ and $I_3$ in the circuit.

Why Ohm's law alone is not enough. Ohm's law, $V=IR$, is directly useful only for simple, single-loop circuits with a single emf source. Real circuits frequently contain multiple loops, multiple junctions, and sometimes multiple emf sources, for which Kirchhoff's two generalised rules provide a complete and systematic method of analysis.

Kirchhoff's first rule (current rule / junction rule). At any junction of a circuit, the algebraic sum of the currents is zero -- current entering the junction is taken as positive, current leaving as negative. This is a direct statement of the conservation of electric charge: whatever charge flows into a junction in a given time must also flow back out, since charge cannot accumulate or vanish at a point. For a junction with several currents $I_1,\ldots,I_5$ meeting it, the rule reads $I_1+I_2-I_3-I_4-I_5=0$, i.e. total current in equals total current out.

Kirchhoff's second rule (voltage rule / loop rule). In any closed loop of a circuit, the algebraic sum of the IR products (voltage drops across resistors) and emfs around that loop is zero. This follows from the conservation of energy: the total energy supplied by every emf source around the loop must equal the total energy delivered to every resistor in that loop. The sign convention is essential: an IR term is positive when the loop is traversed in the SAME direction as the assumed current, negative when traversed against it; an emf is positive when traversed from the cell's negative to positive terminal, negative the other way. The voltage rule may only be validly applied once every current in the circuit has reached a steady state (no longer changing in time). …