Conservation of Energy and Charge in Circuits

Kirchhoff's two laws are conservation of charge and conservation of energy, written for circuits.

Kirchhoff's first law — the sum of the currents entering a junction equals the sum of the currents leaving it.

Σ I(in) = Σ I(out)

This is conservation of charge: charge does not accumulate at a junction or vanish there, so whatever arrives must leave.

Example: currents of 0.80 A and 0.50 A enter a junction and 0.30 A leaves along one wire. The remaining wire must carry 1.30 − 0.30 = 1.0 A.

Kirchhoff's second law — around any closed loop, the sum of the e.m.f.s equals the sum of the potential differences.

Σ ε = Σ IR

This is conservation of energy: a unit charge travelling once round a loop returns to where it started, so all the energy given to it by sources must be transferred away by components.

Resistors in series — the same current passes through each, and the p.d.s add:

R = R₁ + R₂ + R₃ …

Resistors in parallel — the same p.d. is across each, and the currents add:

1/R = 1/R₁ + 1/R₂ + 1/R₃ …

A useful check — the total resistance of a parallel combination is always less than the smallest individual resistance, because adding another branch gives the current an extra route. If a parallel calculation gives an answer larger than one of the resistors, it is wrong, and the usual cause is forgetting to invert at the end.

Example: two 6.0 Ω resistors in parallel give 1/R = 1/6 + 1/6 = 1/3, so R = 3.0 Ω — half of one of them, and less than either, as it must be.

Current and p.d. in each arrangement — in series the current is the same everywhere and the p.d. divides in proportion to the resistances. In parallel the p.d. is the same across each branch and the current divides, with the largest current through the smallest resistance.