Topic 11.6: Kirchhoff's Loop Rule

What Topic 11.6 requires

Topic 11.6 has one learning objective, four essential knowledge statements, no sub-statements, and no boundary statement on its CED page. There is very little to learn. Almost all of the difficulty is in execution.

• 11.6.A Describe a circuit or elements of a circuit by applying Kirchhoff's loop rule.

The four statements underneath it:

• 11.6.A.1 "Energy changes in simple electrical circuits may be represented in terms of charges moving through electric potential differences within circuit elements." Relevant equation: $\Delta U_E = q \Delta V$.
• 11.6.A.2 "Kirchhoff's loop rule is a consequence of the conservation of energy."
• 11.6.A.3 "Kirchhoff's loop rule states that the sum of potential differences across all circuit elements in a single closed loop must equal zero." Relevant equation: $\sum \Delta V = 0$.
• 11.6.A.4 "The values of electric potential at points in a circuit can be represented by a graph of electric potential as a function of position within a loop."

The suggested skills say what the exam does with those four sentences: 1.C, create qualitative sketches of graphs that represent features of a model or the behavior of a physical system; 2.A, derive a symbolic expression from known quantities by selecting and following a logical mathematical pathway; 2.C, compare physical quantities between two or more scenarios or at different times and locations in a single scenario; and 3.B, apply an appropriate law, definition, theoretical relationship, or model to make a claim.

1.C is the one that catches people out. This topic has a graph in it, and it is 11.6.A.4's graph of potential against position, not a current-voltage graph. Unit 11 is weighted at 15 to 18 percent of the AP Physics 2 exam and the CED allots it about 12 to 20 class periods.

Why the sum has to be zero

Statement 11.6.A.2 gives the reason in one line: the loop rule is a consequence of the conservation of energy. Statement 11.6.A.1 supplies the link between energy and potential:

Take a charge $q$ around a closed loop and bring it back to the point it started from. Electric potential is a property of the point, not of the trip, so the charge is back at the same potential and its electric potential energy is back to its starting value. Over the round trip $\Delta U_E = 0$. Divide by $q$ and you have $\sum \Delta V = 0$.

Every gain has to be paid for. The battery lifts charge to a higher potential; the resistors drop it back down. The rule is a bookkeeping identity: over one lap, whatever the source puts in, the rest of the circuit takes out, exactly.

The argument has the same shape as the one behind gravitational potential energy in AP Physics 1. Walk a closed path on a hillside and you return to the height you left, whatever route you took. The loop rule is conservation of energy written per unit charge, and the electric potential defined in Topic 10.5 is the electrical analogue of height.

Two words in 11.6.A.3 are worth pausing on. Single: the rule applies to one loop at a time, and a circuit with several loops gives you one equation for each loop you choose to walk. Closed: the path must return to its starting point. Apply it to half a loop and it guarantees nothing.

The sign convention, declared once

Every worked example and every statement below this point uses the convention in this table, and none of them changes it partway through. Write it at the top of your scratch paper on the day and do not improvise.

1. Draw the circuit and mark a direction for the current in every branch. Guess. It does not have to be right.
2. Choose a direction to walk the loop, clockwise or counterclockwise. This choice is free too.
3. Walk the loop, writing down the change in potential at each element you cross, using these four cases and no others.
4. Set the total to zero and solve.

Two short reasons fix the two halves of that table, so you can rebuild it rather than memorise it.

For a resistor, charge flows from high potential to low, so walking with the current is walking downhill and the change in potential is negative. Walk against the current and you are climbing, so it is positive. Only the relation between your walk and the current matters, never which way the resistor is drawn.

For a battery, the sign has nothing to do with the current at all. It is set by which terminal you enter and which you leave. A battery being charged by a stronger source is still crossed from plus to minus for $-\mathcal{E}$, even though current is being pushed backwards through it.

A battery with internal resistance is two entries in that table, not one: cross the ideal source for $\pm\mathcal{E}$, then cross the internal resistor $r$ for $\mp Ir$. Doing exactly that is where the terminal-voltage result $\Delta V_{\text{terminal}} = \mathcal{E} - Ir$ from Topic 11.5 comes from. Note that Topic 11.2's boundary statement fixes the other half of the notation for you: unless otherwise specified, all circuit schematic diagrams are drawn using conventional current, so a current arrow on an exam diagram is always the direction positive charge would move.

The signs are the whole difficulty

After 11.6.A.2 there is no physics left in this topic. Everything that goes wrong is bookkeeping, and it goes wrong in four specific ways.

Confusing the two directions. The walking direction and the current direction are different things, and either can point either way. Only their relative orientation decides the sign at a resistor. Mark the current arrows on the diagram, then draw a separate curved arrow for the loop you are walking, so you can see the relation instead of holding it in your head.

Reading a battery's sign off the current. For a battery it is the terminals that matter, never the current. A plus-to-minus crossing contributes $-\mathcal{E}$ whether current is entering that terminal or leaving it.

Panicking at a negative answer. If the algebra returns $I = -2.0$ A, the magnitude is 2.0 A and the current actually runs opposite to the arrow you drew. That is the answer, not an error to fix, and worked example 3 produces one on purpose. Do not go back and flip signs to make it positive: the sign is now carrying information the rest of your work depends on.

Changing convention halfway. Walking one loop clockwise and the next counterclockwise is legal, because each loop is its own independent equation. Changing the direction of a current arrow partway through a single equation is not, and neither is writing $-IR$ in one term and $+IR$ in another for the same resistor on the same walk.

One free check catches nearly all of these. Once you have numbers, walk the loop again and add up the actual potential changes. They must total zero to the last digit. A sign error usually still produces a plausible-looking current, so the answer alone will not tell you.

Suggested skill 2.C turns up here as questions of the form "the switch is closed, what happens to the potential difference across $R_2$?" The loop rule answers those without any computation, because the terms have to keep summing to zero: if one drop grows, another must shrink, or the source term must grow to match.

Reading a graph of potential against position

Statement 11.6.A.4 is the only place in Unit 11 that asks for a graph of electric potential against position: "The values of electric potential at points in a circuit can be represented by a graph of electric potential as a function of position within a loop." Paired with suggested skill 1.C, that is an instruction to be able to sketch one.

Building one takes three steps. Pick a point in the loop, call it position zero, and assign it a potential of zero; the negative terminal of the battery is the usual choice, and any point works. Then walk the loop in one direction using the convention above. Then plot the running potential against distance travelled around the loop.

What the pieces look like:

• Ideal wire: flat. No resistance means no potential difference along it, so any two points joined only by wire sit at the same potential.
• Resistor crossed with the current: a step down of height $IR$. Drawn to scale along the resistor it is a ramp; on a schematic where the resistor is a point, it is a vertical step.
• Battery crossed from negative to positive: a step up of height $\mathcal{E}$.
• Back at the start: the trace must return to its starting value. That is $\sum \Delta V = 0$, drawn.

The graph is the loop rule made visible, and it doubles as a check on an answer: if your trace does not close, your signs do not balance.

Two readings the exam asks for. The largest step down identifies the element with the largest $IR$, which in a series loop means the largest resistance, because every element shares one current. The vertical gap between any two points is the potential difference a voltmeter connected across those two points would read, which is how a question about $V_A - V_B$ between two labelled points gets answered without computing anything else. Worked example 2 tabulates one of these traces.

What the equation sheet prints, and what it does not

The Electricity group of the AP Physics 2 equation sheet has 20 entries, and none of them is the loop rule. $\sum \Delta V = 0$ appears in the CED as a relevant equation under 11.6.A.3, printed on the topic page, not on the sheet you are handed in the exam room. You are expected to know it. The junction rule, $\sum I_{\text{in}} = \sum I_{\text{out}}$, is in the same position under 11.7.A.2. Neither Kirchhoff rule is printed.

What the sheet does give you is 11.6.A.1's energy statement and the tools you apply the rule with.

Ohm's law is printed with the delta on it, and that matters more here than anywhere else in the unit. $\Delta V$ is a difference between two points, and a loop equation is nothing but a list of such differences. Ask yourself which two points every single time.

You can count the Electricity group for yourself on the AP Physics 2 formula sheet rather than taking the number on trust.

Traps that cost marks

• Applying the rule to a path that is not closed. If the path does not return to its start, the sum of the potential differences is not zero: it is the potential difference between the two ends. That is a useful quantity and it is how you find $V_A - V_B$, but it is a different calculation and it needs a different sentence in your answer.
• Using the battery current for every resistor. $\sum \Delta V = 0$ is written in volts, and each resistor contributes $IR$ using the current in that resistor. In a single-loop circuit there is only one current. As soon as the circuit branches, there is more than one, and they need separate symbols.
• Treating the battery's label as the terminal potential difference. If the battery has internal resistance, the loop contains both $\mathcal{E}$ and $-Ir$, and the terminal potential difference is what is left over. Topic 11.5 sets that model up.
• Writing one equation for a two-loop circuit. Two independent loops need two independent loop equations, plus the junction rule to relate the branch currents. One equation with three unknowns does not solve, and no amount of algebra will rescue it.
• Inventing terms the question did not give you. Topic 11.5's boundary statement makes batteries, wires, and meters ideal unless a question states otherwise, so an $Ir$ term you were not given costs time and accuracy.
• Treating a negative current as a mistake. Covered above. It is information about direction, and it is often the point of the question.

Where the loop rule stops and the junction rule starts

The two Kirchhoff rules are a pair, and each one enforces a different conservation law.

Topic 11.7 owns the charge half of this and states it precisely: the total amount of charge entering a junction per unit time must equal the total amount exiting that junction per unit time. You will usually need both rules on the same problem, because the loop rule produces equations in the branch currents and the junction rule is what relates those currents to one another.

When do you need either? Only when the network will not reduce. If every group of resistors is cleanly in series or cleanly in parallel, the equivalent-resistance method in Topic 11.5 is faster, and the series and parallel guide runs that routine end to end. Reach for the loop rule when a circuit has two batteries in one loop, when a resistor bridges the middle of two branches so it has no series or parallel partner, or when the question asks for the potential difference between two named points rather than across one element.

The loop rule also survives into Topic 11.8. An RC circuit is analysed at the two instants the CED singles out: immediately after the switch closes, when an uncharged capacitor "acts like a wire" (11.8.B.2.i), and after a long time, when the capacitor branch carries zero current (11.8.B.2.iv). At each of those instants the circuit is an ordinary resistor network and the loop rule applies unchanged.