10.2 Kirchhoff’s laws

At a circuit junction, total current entering equals total current leaving: ΣI_in=ΣI_out.

Choose current directions before solving; a negative answer means the true direction is opposite to the assumed arrow.

If 2.0 A and 0.5 A enter a node and one branch carries 1.5 A out, the remaining branch carries 1.0 A out.

Current is not “used up” at a junction; charge conservation determines the branch relation.

Around any closed circuit loop, the algebraic sum of e.m.f.s and potential changes is zero: ΣV=0.

Choose a loop direction and keep rises and drops signed consistently. Include internal resistance where the loop contains a real source.

A 12 V source feeding 4 V and 8 V drops satisfies 12−4−8=0.

A loop equation is not a claim that every component has the same voltage; it balances signed energy transfers per charge.

The combined resistance of a network is the single resistance that gives the same total current for the same applied voltage.

Reduce simple series or parallel sections step by step, preserving which elements share current or p.d. before applying the next rule.

Two 6 Ω resistors in parallel have equivalent resistance 3 Ω, then adding a 2 Ω resistor in series gives 5 Ω total.

Equivalent resistance is not the arithmetic sum for every network; parallel branches always give a value below the smallest branch resistance.

For series resistors, the same current flows through each and total resistance is the sum of individual resistances.

Use V_total=IR_total and note that p.d. divides in proportion to resistance when current is common.

4 Ω and 6 Ω in series give 10 Ω; at 2 A the voltage drops are 8 V and 12 V.

Series components do not share equal voltage unless their resistances happen to be equal.

Combine series and parallel sections where possible, then apply junction current conservation and loop energy conservation to the remaining circuit.

Label branch currents and polarities before writing equations; a negative solved current only reverses the assumed direction.

Reduce two parallel branches first, then use the source loop equation to find total current and branch voltages.

Equivalent resistance simplifies terminal behaviour but does not erase the branch currents or internal voltage distribution.

For parallel resistors, each branch has the same potential difference and currents add, giving 1/R_total=Σ(1/R_i).

Use reciprocal values carefully and check that the equivalent resistance is below the smallest branch resistance.

Two 6 Ω resistors in parallel give 1/R=1/6+1/6, so R_total=3 Ω.

Parallel branches do not carry equal current unless their resistances are equal; they share voltage, not necessarily current.

Assign unknown currents, use ΣI=0 at junctions and ΣV=0 around loops, then solve the simultaneous equations.

Choose current directions and loop orientations consistently; include source internal resistance and resistor drops where relevant.

A two-loop circuit can be solved by one junction equation plus one loop equation per independent loop.

A negative current is not a failed circuit—it means the chosen arrow was opposite to the physical current.