Kirchhoff's Laws for Current and Voltage

These mathematical rules explain how electric current and voltage flow

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. Kirchhoff's Current Law, also known as Kirchhoff's Junction Law, and Kirchhoff's First Law, define the way that electrical current is distributed when it crosses through a junction—a point where three or more conductors meet. Put another way, Kirchhoff's Laws state that the sum of all currents leaving a node in an electrical network always equals zero.

These laws are extremely useful in real life because they describe the relation of values of currents that flow through a junction point and voltages in an electrical circuit loop. They describe how electrical current flows in all of the billions of electric appliances and devices, as well as throughout homes and businesses, that are in use continually on Earth.

Kirchhoff's Laws: The Basics

Specifically, the laws state:

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 goes in must come out. Picture a well-known example of a junction: a junction box. These boxes are installed on most houses. They are the boxes that contain the wiring through which all electricity in the home must flow.

When performing calculations, the current flowing into and out of the junction typically has opposite signs. You can also state Kirchhoff's Current Law as follows:

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

You can further break down the two laws more specifically.

Kirchhoff's Current Law

In the picture, a junction of four conductors (wires) is shown. The currents v2 and v3 are flowing into the junction, while v1 and v4 flow out of it. In this example, Kirchhoff's Junction Rule yields the following equation:

v2 + v3 = v1 + v4

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. 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 (batteries, for example) or devices—lamps, televisions, and blenders—plugged into the circuit. 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 forcefield. The voltage represents the electrical energy in the system, so think of it 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 zero. If they 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. Choose a direction (clockwise or counterclockwise) 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.

Remember that when traveling around the circuit to apply Kirchhoff's Voltage Law, be sure you are always going in the same direction (clockwise or counterclockwise) 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 incorrect.

When crossing a resistor, the voltage change is determined by the formula:


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, so it is increasing.

Applying Kirchhoff's Voltage Law

The most basic applications for Kirchhoff's Laws relate to electrical circuits. You may remember from middle school physics that electricity in a circuit must flow in one continuous direction. If you flip off a light switch, for example, you are breaking the circuit, and hence turning off the light. Once you flip the switch again, you reengage the circuit, and the lights come back on.

Or, think of stringing lights on your house or Christmas tree. If just one light bulb blows out, the entire string of lights goes out. This is because the electricity, stopped by the broken light, has no place to go. It's the same as turning off the light switch and breaking the circuit. The other aspect of this with regard to Kirchhoff's Laws is that the sum of all electricity going into and flowing out of a junction must be zero. The electricity going into the junction (and flowing around the circuit) must equal zero because the electricity that goes in must also come out.

So, next time you're working on your junction box or observing an electrician doing so, stringing electric holiday lights, or turning on or off your TV or computer, remember that Kirchhoff first described how it all works, thus ushering in the age of electricity.

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Your Citation
Jones, Andrew Zimmerman. "Kirchhoff's Laws for Current and Voltage." ThoughtCo, Apr. 5, 2023, Jones, Andrew Zimmerman. (2023, April 5). Kirchhoff's Laws for Current and Voltage. Retrieved from Jones, Andrew Zimmerman. "Kirchhoff's Laws for Current and Voltage." ThoughtCo. (accessed May 28, 2023).