Physics 9702 · Paper 2

Kirchhoff's Second Law

The first law counted charge at a junction. The second law counts energy around a loop.
Follow one coulomb of charge all the way round any loop: the energy it gains must equal the energy it gives out.

Energy round a loop

Around any closed loop, the sum of the e.m.f.s equals the sum of the p.d.s. That is .
The reason is conservation of energy. One volt is one joule per coulomb. A coulomb gains E joules in each source and loses V joules in each component. When it arrives back where it started, it must have the same energy as at the start, so the gains must equal the losses.

Your turn— tap to reveal the worked answer (9702/12/O/N/24 Q36)

A table pairs each of Kirchhoff's laws with a conservation law. Which row is correct?

Answer: first law with charge, second law with energy. This exact pairing question appears again and again. (9702/12/O/N/24 Q36)

Signs: which way each cell pushes

Pick one direction to walk round the loop. Then: an e.m.f. counts positive if it pushes the way you walk, negative if it pushes against you. Each $IR$ term counts positive if the current flows the way you walk.
Check the cell symbol to see which way it pushes: current is driven out of the long thin line (the positive terminal).

The two cells push opposite ways: walking round the loop, 6.0 V pushes with you and 2.0 V pushes against, so only 4.0 V is left to drive the current.

Worked example

Two cells pushing opposite ways in one loop

In the loop above, a 6.0 V cell and a 2.0 V cell are connected in opposite directions, with resistors of 10 Ω and 30 Ω. Find the current.

Sum of e.m.f.s, walking the way the 6.0 V cell pushes: $6.0 - 2.0 = 4.0$ V.
Sum of p.d.s: $10I + 30I = 40I$ .
Equate: $4.0 = 40I$ , so $I = 0.10$ A.

Answer

I = 0.10 \text{ A}

Common mistake

Adding both e.m.f.s no matter which way the cells point. Check the long thin line on each cell symbol first: cells facing opposite ways subtract. Here the driving voltage is 4.0 V, not 8.0 V.

Cells in series and in opposition

The second law also tells you what happens when cells are joined in series. Cells pointing the same way: the e.m.f.s add. One cell reversed: its e.m.f. subtracts.
Internal resistances always add in series, whichever way the cells point. Turning a cell round changes the direction it pushes, not its resistance.

Your turn— tap to reveal the worked answer (9702-style)

You have five 1.5 V cells in series. What total e.m.f. do you get if (a) all five point the same way, (b) three point one way and two the other, (c) four point one way and one the other?

Answer: (a) $5 \times 1.5 = 7.5$ V. (b) $(3-2) \times 1.5 = 1.5$ V. (c) $(4-1) \times 1.5 = 4.5$ V. (9702-style)

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