Kirchhoff's Loop Rule: Study Guide (Physics C: E&M 11.6)

Course note. This guide is for the calculus-based Physics C: Electricity and Magnetism course, Topic 11.6. The loop rule is one of the two Kirchhoff rules; the junction rule follows in Topic 11.7, and together they let you analyse any DC circuit in this course. Batteries, wires and meters are ideal unless a question says otherwise.

Energy and potential round a circuit

In Unit 9 you met the link between electric potential and potential energy: when a charge q moves through a potential difference ΔV, its electric potential energy changes by

ΔU_E = qΔV

Follow a small positive charge round a circuit and watch its potential.

• Through a battery from the − terminal to the + terminal, the chemical processes in the battery do work on the charge. Its potential rises by ℰ, so its electric potential energy rises by qℰ.
• Through a resistor in the direction of the current, the charge moves from high to low potential. Its potential falls by IR, and the energy qIR is transferred to the resistor (as thermal energy, or light in a bulb).
• Along an ideal wire, there is no resistance, so no potential difference is needed to keep the charge moving. The potential stays the same.

A real battery is an ideal emf in series with an internal resistance r (Topic 11.5), so crossing it from − to + gives a rise of ℰ followed by a fall of Ir.

The loop rule

Electric potential has a single value at each point in a steady circuit. So if a charge leaves a point, goes round any closed loop and arrives back at the same point, its potential is exactly what it was at the start. Its net change in electric potential energy is zero: the energy it gained in sources equals the energy it gave up in the other elements. This is conservation of energy, and it gives Kirchhoff’s loop rule:

ΣΔV = 0 round any closed loop

The sum includes every element in the loop: batteries, resistors, capacitors and any other element.

Background link to Unit 9. For static charges, ΔV = −∫E·dℓ, and the electrostatic field is conservative, so the integral round any closed path is zero: ∮E·dℓ = 0. The loop rule is that statement applied to a circuit. In Unit 13 you will see what changes when magnetic flux through the loop varies.

Writing a loop equation

1. Draw and label the circuit. Mark a direction for the current in each element; if you do not know it, guess.
2. Choose a loop and a direction to walk round it (clockwise is common).
3. Add a term for each element as you meet it, using one sign convention:

Element and direction you walk	Change in potential, ΔV
Resistor, walking with the assumed current	−IR
Resistor, walking against the assumed current	+IR
Battery, walking from − terminal to + terminal	+ℰ
Battery, walking from + terminal to − terminal	−ℰ
Ideal wire	0
Capacitor, walking from + plate to − plate	−Q/C

4. Set the sum to zero and solve.
5. Interpret the sign. If a current comes out negative, it flows opposite to your guess; its size is still correct.

The capacitor row is here for completeness. Circuits where capacitors charge and discharge are the subject of Topic 11.8.

For a single loop with one real battery of emf ℰ and internal resistance r, and external resistors R₁ and R₂, walking with the current gives ℰ − Ir − IR₁ − IR₂ = 0, so I = ℰ/(r + R₁ + R₂). This is the series result from Topic 11.5, now justified by energy conservation.

Graphs of potential against position

You can show the potentials round a loop on a graph. Put “position round the loop” on the horizontal axis and electric potential on the vertical axis, and choose one point as zero.

• A source of emf appears as a rise of ℰ, crossed from − to +.
• A resistor appears as a fall of IR in the direction of the current. For a uniform resistor the fall is a straight line, because equal lengths have equal resistance.
• An ideal wire appears as a flat (horizontal) section.
• The graph ends at the same potential it started at: that is the loop rule in picture form.

Figure 1 shows the circuit for Worked example 1, and Figure 2 shows its graph.

Worked example 1: a single loop, its potentials and its graph

Question. In Figure 1, ℰ = 9.0 V, r = 0.60 Ω, R₁ = 2.4 Ω and R₂ = 6.0 Ω. (a) Use the loop rule to find the current. (b) Taking V_A = 0, find the potentials at B, C and D, and sketch V against position. (c) Repeat (b) with V_C = 0. (d) Account for the energy of 2.0 C of charge going once round the loop.

(a) Walk clockwise from A, with the current. Through the battery from − to +: +ℰ. Through r, R₁ and R₂ with the current: −Ir, −IR₁, −IR₂.

ℰ − Ir − IR₁ − IR₂ = 0, so I = 9.0 ÷ (0.60 + 2.4 + 6.0) = 9.0 ÷ 9.0 = 1.0 A.

(b) Start at V_A = 0 and add each change.

1. Across the ideal emf: up 9.0 V. Across r: down (1.0)(0.60) = 0.60 V. So V_B = 8.4 V, which is also the terminal voltage.
2. Along the wire to R₁: no change. Across R₁: down 2.4 V. V_C = 6.0 V.
3. Across R₂: down 6.0 V. V_D = 0. The wire from D to A is flat, and we are back at 0, as the loop rule requires.

(c) With V_C = 0, subtract 6.0 V from every potential: V_A = V_D = −6.0 V, V_B = 2.4 V, V_C = 0. The graph has exactly the same shape, shifted down by 6.0 V. Every potential difference is unchanged, which shows that only differences in potential are physical.

(d) Using ΔU_E = qΔV with q = 2.0 C: the emf gives the charge 18 J. It transfers 1.2 J inside the battery (r), 4.8 J in R₁ and 12 J in R₂. Total: 18 − 1.2 − 4.8 − 12 = 0. Energy is conserved round the loop.

Worked example 2: the potential difference between two points

Question. An ideal 12 V battery has two branches connected across it. Branch 1 is R₁ = 2.0 Ω then R₂ = 4.0 Ω; branch 2 is R₃ = 6.0 Ω then R₄ = 3.0 Ω. R₁ and R₃ connect to the positive terminal. An ideal voltmeter is connected between M (between R₁ and R₂) and N (between R₃ and R₄), as in Figure 3. (a) Find the voltmeter reading and which point is at higher potential. (b) Check with a loop through the voltmeter. (c) Derive a symbolic expression for V_M − V_N and the condition for a zero reading.

(a) The voltmeter takes no current, so each branch is a simple series path. Take V = 0 at the negative terminal; the top wire is then at 12 V.

1. Loop through the battery and branch 1, walking with the current: 12 − I₁(2.0) − I₁(4.0) = 0, so I₁ = 2.0 A.
2. Loop through the battery and branch 2: 12 − I₂(6.0) − I₂(3.0) = 0, so I₂ = 1.33 A.
3. Walk from the top wire down to M: V_M = 12 − (2.0)(2.0) = 8.0 V. Down to N: V_N = 12 − (1.33)(6.0) = 4.0 V.
4. The voltmeter reads 4.0 V, with M at the higher potential.

(b) Walk the loop M → R₂ → bottom wire → R₄ → N → voltmeter → M. Down R₂ with its current: −(2.0)(4.0) = −8.0 V. Up R₄ against its current: +(1.33)(3.0) = +4.0 V. Across the voltmeter from N to M: +4.0 V. Sum: −8.0 + 4.0 + 4.0 = 0. ✓

(c) In general, I₁ = ℰ/(R₁ + R₂) and V_M = ℰR₂/(R₁ + R₂); similarly V_N = ℰR₄/(R₃ + R₄). So

V_M − V_N = ℰ[R₂/(R₁ + R₂) − R₄/(R₃ + R₄)] = ℰ(R₂R₃ − R₁R₄) / [(R₁ + R₂)(R₃ + R₄)]

The reading is zero when R₁/R₂ = R₃/R₄. Here R₁/R₂ = 0.5 but R₃/R₄ = 2, so the reading is not zero. Replacing R₄ with 12 Ω would make it zero.

Interpretation. You never needed the battery current (3.33 A, by the junction rule of Topic 11.7). Any path between two points gives their potential difference; the loop rule guarantees that every path gives the same answer.

Common misconceptions

• “Potential is used up as current flows.” Potential is a property of each point. Charge transfers energy as it moves to lower potential; the current is not used up.
• Mixing sign conventions within one loop. Pick one direction and apply the table every time.
• “A negative current means I made a mistake.” It means the current flows the other way.
• “The terminal voltage of a battery is always ℰ.” Crossing a real battery from − to + gives ℰ − Ir; for a battery being charged (current entering its + terminal) the terminal voltage is ℰ + Ir.
• Giving wires a potential drop. Ideal wires are flat sections on the graph.
• “The loop rule only works for single loops.” It holds for every closed loop, including loops through branches and meters.
• “The zero of potential is at the negative terminal.” That is a convenient choice, not a rule.
• Ending the graph at a different level from the start. A loop graph always closes.

Where this leads

Next, Topic 11.7 adds the junction rule (charge conservation), and together the two rules solve multi-loop circuits. Topic 11.8 uses the loop rule with a capacitor term to describe charging and discharging. If internal resistance or series and parallel feel unsure, return to Topic 11.5, Compound Direct Current Circuits. Practise now with the practice questions, then use the revision notes and the topic checklist. The next study guide is Topic 11.7, Kirchhoff’s Junction Rule.