Kirchhoff's Laws

Overview

Kirchhoff’s laws are the conservation rules that make circuit analysis systematic. This lesson uses charge conservation at junctions and energy conservation around loops to derive resistance rules and solve simple d.c. circuit problems.

What You Need to Know

• Kirchhoff’s first law: the total current entering a junction equals the total current leaving it. This follows from conservation of charge.
• Kirchhoff’s second law: the sum of e.m.f.s around a closed loop equals the sum of potential differences around that loop. This follows from conservation of energy.
• In series, the same current passes through each resistor and potential differences add, giving R_total = R1 + R2 + ....
• In parallel, the potential difference across each branch is the same and currents add, giving 1 / R_total = 1 / R1 + 1 / R2 + ....
• Circuit problems often need a clear sign convention. Choose a loop direction and keep gains and drops consistent.
• A correct solution should include equations from the circuit, not just substituted numbers.

How to Work Through It

1. Begin with junction examples and write current equations using Kirchhoff’s first law.
2. Trace simple loops and identify energy gains across sources and energy transfers across components.
3. Derive the series and parallel resistance formulae from the two laws before using them directly.
4. Solve circuit questions by labelling currents, writing equations, and checking whether your final values are physically reasonable.

Check Your Understanding

• Why must current into a junction equal current out of the junction?
• What conservation principle supports Kirchhoff’s second law?
• Why is total resistance larger than any single resistor in a series combination?
• Why is total resistance smaller than the smallest branch resistance in a parallel combination?

Common Mistakes

• Adding parallel resistances directly instead of adding reciprocals.
• Forgetting that current is the same through components in series but p.d. is the same across branches in parallel.
• Mixing sign conventions halfway through a loop equation.
• Using memorised resistance formulae without showing how Kirchhoff’s laws apply to the circuit.

Next Steps

• Practise setting up equations before reaching for a calculator.
• Use the series and parallel ideas when analysing potential divider circuits next.