AP® Physics 2: Algebra-Based review sheet from Aim for Five (aimforfive.com/physics-2/units/11/11-6)
Unit 11 · Topic 11.6
11.6 Kirchhoff’s Loop Rule
Kirchhoff's loop rule says the potential differences around any closed loop add to zero. It's conservation of energy for charge: a charge that goes all the way around a loop ends at the same potential it started at. With it you can find unknown voltages, currents and resistances.
Key terms
- Kirchhoff's loop rule
- conservation of energy
- potential drop
- emf
- closed loop
Energy around a loop
When a charge q moves through a potential difference ΔV, its electric potential energy changes by ΔU = qΔV. A battery raises the charge's potential energy; resistors and bulbs lower it, turning that energy into thermal energy and light.
Potential is like height on a hiking trail. Walk any loop and return to your starting point, and your total change in height is zero, however many hills you climbed and descended along the way. Kirchhoff's loop rule says the same for potential:
It works because of conservation of energy. If a charge could come back to its starting point with extra energy, you could circle it forever and get free energy.
Signs when you walk a loop
Choose a direction to walk around the loop and add up the changes in potential.
- Crossing a battery from − to +: potential rises by ε. From + to −: it drops by ε.
- Crossing a resistor in the same direction as the current: potential drops by IR (current flows from high to low potential through a resistor).
- Crossing a resistor against the current: potential rises by IR.
- Crossing an ideal wire: no change.
Graphs of potential around a loop
A graph of potential against position around a loop shows the rule at work. Start at the battery's negative terminal and call it 0 V. Crossing the battery, the graph jumps up by ε. Along ideal wires it stays flat. Through each resistor it slopes down by that resistor's IR. When you arrive back at the start, you're at 0 V again.
Bigger resistors in series make bigger drops, since they share the same current. Elements in parallel always have the same potential difference, because their ends connect to the same two points; the loop rule applied to the loop containing both branches proves it.
Using the rule
For a single-loop circuit, the loop rule gives ε − IR₁ − IR₂ − ... = 0, which is the same as I = . Its real power shows in multi-loop circuits, where you write one loop equation for each independent loop and combine them with the junction rule (11.7).
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Potential at each point
A 12 V battery is connected in series with a 2.0 Ω resistor and a 4.0 Ω resistor. Calling the negative terminal 0 V, find the potential at each point around the loop, and describe the potential–position graph.
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- Step 1: Current: I = 12/(2.0 + 4.0) = 2.0 A.
- Step 2: Leaving the positive terminal: 12 V.
- Step 3: After the 2.0 Ω resistor: drop of (2.0)(2.0) = 4.0 V, so 8.0 V.
- Step 4: After the 4.0 Ω resistor: drop of (2.0)(4.0) = 8.0 V, so 0 V, back to the start. Check: +12 − 4.0 − 8.0 = 0.
- Step 5: Graph: a jump from 0 to 12 V at the battery, flat along wires, a 4.0 V slope down through the first resistor, flat, then an 8.0 V slope down through the second.
Answer: 12 V after the battery, 8.0 V between the resistors, 0 V after the second resistor
- Example 2Calculator allowed
Two batteries in one loop
A loop contains a 9.0 V battery, a 2.0 Ω resistor, a 3.0 V battery and a 4.0 Ω resistor, all in series. The 3.0 V battery is turned so that it opposes the 9.0 V battery. Find the current and check with the loop rule.
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- Step 1: The batteries oppose each other, so the net emf is 9.0 − 3.0 = 6.0 V, and the current flows in the direction the 9.0 V battery pushes.
- Step 2: I = 6.0 V / (2.0 + 4.0) Ω = 1.0 A.
- Step 3: Loop rule, walking with the current: +9.0 − (1.0)(2.0) − 3.0 − (1.0)(4.0) = 0. ✓ (The 3.0 V battery is crossed from + to −, so it's a drop.)
Answer: 1.0 A, in the direction set by the 9.0 V battery
Common mistakes
- Getting the sign wrong through a resistor. Going with the current is a drop (−IR); going against it is a rise.
- Assuming each element in a series loop gets the full battery voltage. The drops share it, in proportion to resistance.
- Forgetting that parallel branches must have equal potential differences, which comes straight from the loop rule.
On the exam
- Expect to sketch a graph of potential against position around a loop, with flat sections for wires and drops for resistors.
- When justifying, name the loop rule and conservation of energy explicitly; “the potential differences around the loop add to zero” earns the reasoning point.
Connected topics
Videos
Check yourself
4 questions on 11.6 Kirchhoff’s Loop Rule. Pick an answer to see if you got it, and why.
A 12 V ideal battery is connected in series with a 2.0 Ω resistor and a 4.0 Ω resistor. What is the potential difference across the 4.0 Ω resistor?
Kirchhoff's loop rule says that the potential differences around any closed loop add to zero. Which conservation law is this rule based on?
A battery is connected in series with resistors R and 2R. A student graphs the electric potential against position, going once around the loop starting from the battery's negative terminal. Which describes the graph?
A single loop contains a 9.0 V battery, a 3.0 V battery and a 3.0 Ω resistor. The batteries are connected so that they push current in opposite directions around the loop. What is the current in the resistor?
0 of 4 answered