Kirchhoff's loop rule says that the sum of potential differences around any single closed circuit loop equals zero. In AP Physics 2, this is a conservation of energy rule: a charge may gain electric potential energy across a battery and lose electric potential energy across circuit elements, but after a full loop it returns to the same electric potential.

Use the loop rule when a circuit question asks you to relate voltage rises and voltage drops in one closed path. Your equation should track signs consistently and end with $\sum \Delta V = 0$ .

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Energy Changes in Electrical Circuits

When charges move through circuit elements, they experience changes in electric potential energy. This relationship is expressed mathematically as:

$\Delta U_{E} = q \Delta V$

This equation tells us that when a charge (q) moves through a potential difference (ΔV), it undergoes a change in electrical potential energy (ΔUE). This fundamental relationship helps us understand energy transformations in circuits.

In a battery, charges gain electric potential energy as they move from the negative to positive terminal
In a resistor, charges lose electric potential energy, which is converted to thermal energy
The total energy must be conserved throughout the entire circuit

Conservation of Energy in Circuits

Kirchhoff's Loop Rule is a direct application of the conservation of energy principle in electrical circuits.

Energy cannot be created or destroyed within a closed loop of a circuit, only transformed from one form to another. When charges complete a full loop around a circuit, they must return to their starting point with the same energy they began with - otherwise, energy would be created or destroyed, violating a fundamental law of physics.

Charges may gain energy from batteries (voltage sources)
Charges may lose energy through resistors and other components
The total energy change must sum to zero for a complete loop

The Loop Rule Equation

Kirchhoff's Loop Rule is mathematically expressed as:

$\sum \Delta V = 0$

This equation states that the algebraic sum of all potential differences (voltage drops and rises) around any closed loop in a circuit must equal zero.

When applying this rule:

Voltage rises (like those across batteries) are considered positive
Voltage drops (like those across resistors) are considered negative
The direction you choose to trace around the loop is arbitrary, but must be consistent

For example, if a charge gains 12V going through a battery, it must lose exactly 12V going through the rest of the circuit components in that loop.

Electric Potential Graphs

Electric potential can be visualized graphically as a function of position within a circuit loop. These graphs provide an intuitive way to understand voltage changes throughout a circuit.

The vertical axis represents electric potential (V)
The horizontal axis represents position in the circuit
Steep slopes indicate large voltage changes (like across resistors)
Vertical jumps indicate ideal batteries or voltage sources

In these graphs, you can clearly see that when you trace a complete loop, the potential returns to its starting value, visually confirming Kirchhoff's Loop Rule.

Practice Problem 1: Applying Kirchhoff's Loop Rule

A circuit contains a 12V battery and three resistors with resistances of 2Ω, 3Ω, and 5Ω connected in series. Using Kirchhoff's Loop Rule, determine the current flowing through the circuit.

Solution

To solve this problem, we'll apply Kirchhoff's Loop Rule: $\sum \Delta V = 0$

Step 1: Identify all voltage changes in the loop.

Battery: +12V (voltage rise)
2Ω resistor: -I × 2Ω (voltage drop)
3Ω resistor: -I × 3Ω (voltage drop)
5Ω resistor: -I × 5Ω (voltage drop)

Step 2: Apply Kirchhoff's Loop Rule. $+12V - I(2Ω) - I(3Ω) - I(5Ω) = 0$

Step 3: Solve for the current I. $12V = I(2Ω + 3Ω + 5Ω)$ $12V = I(10Ω)$ $I = 12V/10Ω = 1.2A$

Therefore, the current flowing through the circuit is 1.2 amperes.

Practice Problem 2: Electric Potential Graph Analysis

A circuit consists of a 9V battery and two resistors with resistances of 3Ω and 6Ω in series. Draw a qualitative electric potential graph for this circuit and explain how it demonstrates Kirchhoff's Loop Rule.

Solution

To create an electric potential graph for this circuit:

Step 1: Calculate the current in the circuit. $I = 9V/(3Ω + 6Ω) = 9V/9Ω = 1A$

Step 2: Calculate the voltage drops across each resistor.

Voltage drop across 3Ω resistor: $V_1 = I × R_1 = 1A × 3Ω = 3V$
Voltage drop across 6Ω resistor: $V_2 = I × R_2 = 1A × 6Ω = 6V$

Step 3: Create the electric potential graph.

Starting at an arbitrary point (0V)
At the battery: Vertical jump up by 9V (to 9V)
Across 3Ω resistor: Sloped line down by 3V (to 6V)
Across 6Ω resistor: Steeper sloped line down by 6V (back to 0V)

This graph demonstrates Kirchhoff's Loop Rule because the total change in potential around the complete loop is zero. The potential rises by 9V at the battery and then falls by a total of 9V across the two resistors, returning to its starting value and confirming that $\sum \Delta V = 0$ .

Vocabulary

The following words are mentioned explicitly in the College Board Course and Exam Description for this topic.

Term	Definition
closed loop	A complete path in a circuit that starts and ends at the same point, allowing current to flow continuously.
conservation of energy	The principle that the total energy in an isolated system remains constant, with energy transforming between different forms but not being created or destroyed.
electric potential	A scalar quantity that represents the electric potential energy per unit charge at a point in space, measured in volts.
electric potential difference	The difference in electric potential energy per unit charge between two points in a circuit, measured in volts; also called voltage.
Kirchhoff's loop rule	A principle stating that the sum of potential differences across all circuit elements in a single closed loop must equal zero, based on conservation of energy.

Frequently Asked Questions

What is Kirchhoff's loop rule?

Kirchhoff's loop rule says the algebraic sum of all potential differences around a closed circuit loop equals zero. In AP Physics 2, it is a direct application of conservation of energy.

Why does the sum of voltage changes around a loop equal zero?

A charge that completes a full circuit loop returns to its starting electric potential. Any potential rise from a battery must be balanced by potential drops across circuit elements, so the total change is zero.

How do you write a loop rule equation?

Choose a direction around the loop, assign signs consistently for voltage rises and drops, write each potential difference, and set the sum equal to zero.

Is Kirchhoff's loop rule the same as Kirchhoff's voltage law?

Yes. Kirchhoff's voltage law is another name for the loop rule: the total potential difference around a closed loop is zero.

How do batteries and resistors appear in loop rule equations?

An ideal battery is usually a voltage rise when crossed from negative to positive terminal. A resistor is a voltage drop in the direction of current, often written as -IR.

How does an electric potential graph show the loop rule?

An electric potential graph rises across sources and falls across circuit elements. After one complete loop, the graph returns to the original potential, showing that the net change is zero.