Kirchhoff’s Current and Voltage Laws

Kirchhoff’s current law (KCL) states that the sum of currents entering a junction equals the sum leaving it (conservation of charge), and Kirchhoff’s voltage law (KVL) states that the sum of the potential differences around any closed loop is zero (conservation of energy); together they allow any circuit to be analyzed. In brief: choose a direction for each current, apply KCL at the junctions and KVL around independent loops, using the sign convention that a potential rise (going through a battery from − to +) is positive and a drop (going through a resistor in the direction of the current) is −IR. Solve the resulting simultaneous equations. A negative answer for a current means that it flows opposite to the assumed direction.

Ohm’s law and the series-parallel rules are not enough for circuits with several sources or bridges. Kirchhoff’s laws handle them all. This guide states the laws and the sign convention, gives a step-by-step method, compares KCL and KVL in a table, works through a two-loop circuit with two batteries and corrects common mistakes.

What is the current law?

At any junction (node) in a circuit, charge cannot accumulate, so the total current into the junction equals the total current out: ΣI_in = ΣI_out (or ΣI = 0, with currents toward the junction positive and away negative). If a current of 5.0 A flows into a node, and currents of 2.0 A and 3.0 A leave it by two branches, the law is satisfied; if two branches carry 2.0 A out and the third unknown, it carries 3.0 A out. KCL applies to any closed surface in the circuit as well, and holds at all times for direct currents.

What is the voltage law?

The electric potential has a single value at each point, so going around a closed loop and returning to the starting point, the total change in potential is zero: ΣV = 0 around any closed loop. Equivalently, the sum of the emfs in a loop equals the sum of the potential drops (IR) across the resistors. The sign rules, going around a loop in a chosen direction:

• a battery crossed from the negative to the positive terminal: +ε; positive to negative: −ε;
• a resistor crossed in the same direction as the assumed current: −IR; opposite to the current: +IR.

What is the method?

1. Label each branch current with an assumed direction (the choice is arbitrary).
2. Apply KCL at the junctions (one fewer than the number of junctions gives independent equations).
3. Apply KVL to as many independent loops as needed so that the number of equations equals the number of unknowns.
4. Solve the simultaneous equations.
5. Interpret the signs, and check with a loop that you have not used, or with a power balance.

KCL and KVL compared

Worked example: a two-loop circuit

A circuit has two batteries and three resistors. Battery 1 (ε₁ = 12 V) is in series with R₁ = 2.0 Ω, battery 2 (ε₂ = 6.0 V) is in series with R₂ = 4.0 Ω, and both branches are connected across a shared resistor R₃ = 6.0 Ω. Let I₁ flow from battery 1 and I₂ from battery 2 toward the junction at the top of R₃, and I₃ through R₃.

1. KCL at the junction: I₃ = I₁ + I₂.
2. KVL, loop 1 (battery 1, R₁, R₃): 12 − 2.0 I₁ − 6.0 I₃ = 0.
3. KVL, loop 2 (battery 2, R₂, R₃): 6.0 − 4.0 I₂ − 6.0 I₃ = 0.
4. Substitute I₃: loop 1: 12 = 2.0I₁ + 6.0(I₁ + I₂) = 8.0I₁ + 6.0I₂. Loop 2: 6.0 = 4.0I₂ + 6.0(I₁ + I₂) = 6.0I₁ + 10I₂.
5. Solve: from the first, I₁ = 1.5 − 0.75I₂. Into the second: 6.0 = 6.0(1.5 − 0.75I₂) + 10I₂ = 9.0 + 5.5I₂, so I₂ = −3.0 ÷ 5.5 = −0.545 A. Then I₁ = 1.5 + 0.409 = 1.909 A, and I₃ = 1.909 − 0.545 = 1.364 A.
6. Interpretation: I₂ is negative, so the current in battery 2 actually flows into its positive terminal: battery 2 is being charged by battery 1.
7. Check with the outer loop (battery 1, R₁, R₂, battery 2), which was not used in the solution. Going around it, the voltage law gives ε₁ − I₁R₁ + I₂R₂ − ε₂ = 12 − 2.0 × 1.909 + 4.0 × (−0.545) − 6.0 = 12 − 3.82 − 2.18 − 6.0 = 0 ✓.
8. Power check: the power supplied by battery 1 is 12 × 1.909 = 22.9 W, and battery 2 absorbs 6.0 × 0.545 = 3.3 W. The resistors dissipate 2.0 × 1.909² + 4.0 × 0.545² + 6.0 × 1.364² = 7.29 + 1.19 + 11.16 = 19.6 W. Supplied minus absorbed: 22.9 − 3.3 = 19.6 W ✓.

Who was Kirchhoff?

Gustav Robert Kirchhoff (1824 to 1887) formulated the circuit laws in 1845 as a 21-year-old student at Königsberg. He also made major contributions to spectroscopy (with Robert Bunsen, 1859), and the law of thermal radiation. His laws generalize Ohm’s law to networks, and they are the basis of circuit simulation programs such as SPICE.

Which mistakes are common?

Mistake 1: Inconsistent signs in the loop equations

Decide on the direction around the loop and apply the same rules to every element.

Mistake 2: Thinking that a negative current is an error

It only means that the actual direction is opposite to the assumed one.

Mistake 3: Writing dependent equations

The outer loop of a two-loop circuit gives no new information. Use independent equations only.

Mistake 4: Forgetting the internal resistance of batteries

Real batteries have internal resistance, which adds a term in the loop equation.