Kirchhoff’s Loop Rule

Kirchhoff's Loop Rule — also called Kirchhoff's Voltage Law (KVL) — states that the sum of all voltage changes around any closed loop in a circuit is exactly zero. This is not an arbitrary rule invented for convenience; it is a direct consequence of the law of conservation of energy. If you carry a unit of positive charge around a complete loop and return to your starting point, the charge must have gained precisely as much energy as it lost. No energy can appear from nowhere or vanish without trace. Think of it like a hiking trail that forms a closed loop around a mountain. If you start at base camp, climb hills, descend valleys, and return to base camp, your net change in altitude is zero — regardless of the path taken. In a circuit, voltage plays the role of altitude. A battery is like climbing a hill (gaining voltage), whilst a resistor is like descending into a valley (losing voltage). By the time you complete the loop, the gains and losses must perfectly cancel. Kirchhoff's Loop Rule is one of two laws formulated by Gustav Kirchhoff in 1845. Together with his Junction Rule (conservation of charge), these laws provide a systematic method for analysing circuits that are far too complex to handle with simple series and parallel rules alone. Mastering the Loop Rule unlocks the ability to solve multi-loop circuits precisely.

Key takeaways
  • Choose a closed loop
  • Assign current directions
  • Apply the sign convention for EMF sources
  • Apply the sign convention for resistors
  • Sum all terms and set equal to zero

Choose a closed loop

Identify a closed path within the circuit — one that starts and ends at the same node. In a multi-loop circuit there may be several valid loops (called meshes). Label each loop clearly and decide on a direction of traversal (clockwise or anticlockwise). Your choice of direction does not affect the final answer; it only determines the sign convention you apply.

Assign current directions

Label the current in each branch with a symbol (I₁, I₂, etc.) and an assumed direction using an arrow. If your assumed direction turns out to be wrong, the mathematics will return a negative value for that current, telling you the real current flows the other way. This is perfectly fine — the algebra corrects any wrong assumption automatically.

Apply the sign convention for EMF sources

As you traverse the loop, if you pass through a battery from its negative terminal to its positive terminal (i.e. in the direction of EMF), record its EMF as positive (+ε). If you pass through it from positive to negative terminal, record it as negative (−ε). This rule reflects whether the source is acting as an energy supplier or opposing the traversal direction.

Apply the sign convention for resistors

For each resistor, if you traverse it in the same direction as your assumed current, the voltage drop is negative (−IR), because the charge is losing energy. If you traverse it against the assumed current direction, the voltage change is positive (+IR). This mirrors the physical reality: current flows from high potential to low potential through a resistor.

Sum all terms and set equal to zero

Add together every EMF term and every voltage-drop term encountered on your traversal. Set the total equal to zero. This gives you one equation per loop. With multiple loops, you generate a system of simultaneous equations. Solve these equations using substitution or elimination to find all unknown currents and voltages in the circuit.

The Key Equation

∑k=1nVk=0\sum_{k=1}^{n} V_k = 0
  • \sum — Summation (—): Indicates that all voltage terms around the complete closed loop are added together.
  • V_k — Voltage change at element k (Volt (V)): The voltage rise or drop across each individual component (battery or resistor) encountered during the traversal. For a resistor: V = IR. For a battery: V = ±ε.
  • n — Number of elements in the loop (—): The total count of components (EMF sources and resistors) encountered as you travel once around the closed loop.

Worked Example

Problem. A single closed loop contains a 12 V battery (internal resistance r = 1 Ω) and two external resistors R₁ = 3 Ω and R₂ = 2 Ω connected in series. Find the current flowing in the loop.

I=εr+R1+R2=121+3+2=2 AI = \frac{\varepsilon}{r + R_1 + R_2} = \frac{12}{1+3+2} = 2 \text{ A}

Answer. The current in the loop is 2 A.

Figure
Diagram
Kirchhoff's Loop Rule — Single Loop Circuit+−ε = 12 VR₁ = 3 ΩR₂ = 2 Ωr = 1 ΩI →↓ I← I−IR₁−IR₂+ε−Ir+ε − Ir − IR₁ − IR₂ = 0
Origins · 1 minGustav Robert Kirchhoff and how this was discovered
GR
Gustav Robert Kirchhoff
1824–1887 · German

Kirchhoff formulated his two circuit laws in 1845 whilst still a university student at Königsberg, aged just 21. His Voltage Law (Loop Rule) and Current Law (Junction Rule) provided the first rigorous, general framework for analysing electrical networks of any complexity. These laws remain indispensable in electrical engineering and physics to this day. He later co-discovered caesium and rubidium and made foundational contributions to spectroscopy and the theory of blackbody radiation. Kirchhoff derived his circuit laws as part of his doctoral dissertation — meaning he published one of the most practically important results in all of electrical engineering before he had even completed his degree.

  • Kirchhoff's Junction Rule — The Junction Rule (conservation of charge at nodes) is Kirchhoff's other law and is applied alongside the Loop Rule to solve multi-loop circuit problems completely.
  • Ohm's Law — Ohm's Law (V = IR) provides the voltage-drop expression for each resistor that is substituted directly into the Loop Rule equation.
  • Electromotive Force and Internal Resistance — Understanding EMF and internal resistance is essential for correctly assigning voltage rises and drops when applying the Loop Rule to real batteries.
  • Series and Parallel Circuits — The Loop Rule is a generalisation of the voltage rules for series circuits and underpins why voltages add in series — making this a natural conceptual predecessor.