Kirchhoff's Laws for Current and Voltage

The sum of all the voltages around a loop is equal to zero. v1 + v2 + v3 - v4 = 0
The sum of all the voltages around a loop is equal to zero. v1 + v2 + v3 - v4 = 0. Kwinkunks/Wikimedia Commons/CC BY 3.0

In 1845, German physicist Gustav Kirchhoff first described two laws that became central to electrical engineering. The laws were generalized from the work of Georg Ohm, such as Ohm's Law. The laws can also be derived from Maxwell’s equations, but were developed prior to the work of James Clerk Maxwell.

The following descriptions of Kirchhoff's Laws assume a constant electrical current. For time-varying current, or alternating current, the laws must be applied in a more precise method.

Kirchhoff's Current Law

Kirchhoff's Current Law, also known as Kirchhoff's Junction Law and Kirchhoff's First Law, defines the way that electrical current is distributed when it crosses through a junction - a point where three or more conductors meet. Specifically, the law states that:

The algebraic sum of current into any junction is zero.

Since current is the flow of electrons through a conductor, it cannot build up at a junction, meaning that current is conserved: what comes in must come out. When performing calculations, current flowing into and out of the junction typically have opposite signs. This allows Kirchhoff's Current Law to be restated as:

The sum of current into a junction equals the sum of current out of the junction.

Kirchhoff's Current Law in Action

In the picture, a junction of four conductors (i.e. wires) is shown. The currents i2 and i3 are flowing into the junction, while i1 and i4 flow out of it.

In this example, Kirchhoff's Junction Rule yields the following equation:

i 2 + i 3 = i 1 + i 4

Kirchhoff's Voltage Law

Kirchhoff's Voltage Law describes the distribution of electrical voltage within a loop, or closed conducting path, of an electrical circuit. Specifically, Kirchhoff's Voltage Law states that:

The algebraic sum of the voltage (potential) differences in any loop must equal zero.

The voltage differences include those associated with electromagnetic fields (emfs) and resistive elements, such as resistors, power sources (i.e. batteries) or devices (i.e. lamps, televisions, blenders, etc.) plugged into the circuit. In other words, you picture this as the voltage rising and falling as you proceed around any of the individual loops in the circuit.

Kirchhoff's Voltage Law comes about because the electrostatic field within an electric circuit is a conservative force field. In fact, the voltage represents the electrical energy in the system, so it can be thought of as a specific case of conservation of energy. As you go around a loop, when you arrive at the starting point has the same potential as it did when you began, so any increases and decreases along the loop have to cancel out for a total change of 0. If it didn't, then the potential at the start/end point would have two different values.

Positive and Negative Signs in Kirchhoff's Voltage Law

Using the Voltage Rule requires some sign conventions, which aren't necessarily as clear as those in the Current Rule. You choose a direction (clockwise or counter-clockwise) to go along the loop.

When traveling from positive to negative (+ to -) in an emf (power source) the voltage drops, so the value is negative. When going from negative to positive (- to +) the voltage goes up, so the value is positive.

Reminder: When traveling around the circuit to apply Kirchhoff's Voltage Law, be sure you are always going in the same direction (clockwise or counter-clockwise) to determine whether a given element represents an increase or decrease in the voltage. If you begin jumping around, moving in different directions, your equation will be correct.

When crossing a resistor, the voltage change is determined by the formula I*R, where I is the value of the current and R is the resistance of the resistor. Crossing in the same direction as the current means the voltage goes down, so its value is negative.

When crossing a resistor in the direction opposite the current, the voltage value is positive (the voltage is increasing). You can see an example of this in our article "Applying Kirchhoff's Voltage Law."

Also Known As

Kirchoff's Laws, Kirchoff's Rules