Kirchhoff's second law: energy around a loop

By the end of this lesson

By the end you can state Kirchhoff's second law, explain why it follows from conservation of energy, and use it to find an e.m.f. or a p.d. in a circuit loop.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$R$	resistance
$V_1$	p.d. across the first component
$V_R$	p.d. across the resistor

Two kinds of energy transfer per unit charge

Before the law itself, you need the two quantities it compares. Both are energies divided by charge, so both are measured in volts: $1\ \text{V} = 1\ \text{J C}^{-1}$ .

The electromotive force (e.m.f.) of a source is the energy transferred per unit charge in driving charge round a complete circuit. A 12 V cell gives each coulomb 12 J of electrical energy.
The potential difference (p.d.) across a component is the energy transferred from electrical energy to other forms per unit charge passing through it. An 8.0 V p.d. across a resistor means 8.0 J of electrical energy becomes thermal energy for every coulomb.

The full treatment of e.m.f., terminal p.d. and internal resistance belongs to the practical-circuits lessons (topic 10.1). Here you only need e.m.f. as "energy given per unit charge" and p.d. as "energy taken per unit charge".

Why it works

Imagine walking a circular route that brings you back to your starting point. However many hills you climb and descend, your net change in height is zero, because you finished where you began.

Charge going round a loop is like that, except the "height" is energy per unit charge. It gains energy in the source and gives it up in the components, so when it returns to its starting point it has gained nothing overall.

Where the analogy breaks: the charge does not "remember" the journey, and energy is not stored as height. The point is simply that the energy gains and losses must cancel.

Definition

Kirchhoff's second law

The sum of the electromotive forces around any closed loop of a circuit is equal to the sum of the potential differences around that loop.

Why it follows from conservation of energy

Conservation of energy says energy cannot be created or destroyed, only transferred between forms.

Take one coulomb of charge all the way round a loop. The source gives it some electrical energy. The components take electrical energy away and transfer it to the surroundings. The charge comes back to where it started, so it cannot have a net gain.

Therefore the energy given per unit charge equals the energy taken per unit charge: the total e.m.f. equals the total p.d. That is the loop rule.

Using signs without tears

In calculations, choose a direction to go round the loop and then be consistent.

Crossing a source from its negative terminal to its positive terminal adds e.m.f.
Crossing a resistor in the direction of the current takes energy away, so it contributes a p.d. $IR$ .
Crossing a resistor against the current would add energy back, which is why direction matters.

In practice, most AS questions have one source and several resistors in a single loop, so the arithmetic is just "e.m.f. = sum of the p.d.s".

Worked example: Checking the loop rule in a two-resistor circuit

A cell of e.m.f. 12 V is connected to a 4.0 Ω resistor and a 2.0 Ω resistor in series. Show that the loop rule holds.

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