# Kirchhoff’s laws

The algebraic sum of currents in any node of the electric circuit is equal to zero (that is, the number of charges leaving through this node must be equal to the number of incoming charges).

The sum of the voltages in any closed circuit of an electrical circuit is zero.

Gustav Kirchhoff’s career was in many ways typical of a 19th century German physicist. Germany, later than its western neighbors, approached the industrial revolution and therefore needed more advanced technologies that would contribute to the accelerated development of industry. As a result, scientists, especially natural scientists, were highly valued in Germany. In the year he graduated from the university, Kirchhoff married the professor’s daughter, “thus observing,” as one of his biographers writes, “two prerequisites for a successful academic career.” But even before that, at the age of twenty-one, he formulated the basic laws for calculating currents and voltages in electrical circuits, which now bear his name.

The middle of the 19th century was just the time of active research into the properties of electrical circuits, and the results of these studies quickly found practical applications. The basic rules for calculating simple circuits, such as Ohm’s law, were already well developed. The problem was that from wires and various elements of electrical circuits, it was technically already possible to make very complex and branched networks – but no one knew how to model them mathematically in order to calculate their properties. Kirchhoff managed to formulate rules that make it easy to analyze the most complex circuits, and Kirchhoff’s laws still remain an important working tool for specialists in the field of electronic engineering and electrical engineering.

Both Kirchhoff’s laws are formulated quite simply and have a clear physical interpretation. The first law states that if we consider any knot circuit (that is, the branch point where three or more wires converge), then the sum of the electric currents entering the circuit will be equal to the sum of the outgoing currents, which, generally speaking, is a consequence of the law of conservation of electric charge. For example, if you have a T-shaped node in the electrical circuit and electric currents are supplied to it through two wires, then the current will flow along the third wire in the direction from this node, and it will be equal to the sum of the two incoming currents. The physical meaning of this law is simple: if it were not fulfilled, an electric charge would continuously accumulate in the node, but this never happens.

The second law is no less simple. If we have a complex, branched chain, it can be mentally broken down into a series of simple closed circuits. The current in the circuit can be distributed in different ways along these circuits, and the most difficult thing is to determine which route the currents will flow in a complex circuit. In each of the circuits, electrons can either acquire additional energy (for example, from a battery), or lose it (for example, on a resistance or other element). Kirchhoff’s second law states that the net increase in electron energy in any closed circuit is zero. This law also has a simple physical interpretation. If this were not the case, each time passing through a closed circuit, the electrons would gain or lose energy, and the current would continuously increase or decrease. In the first case, it would be possible to obtain a perpetual motion machine, and this is forbidden by the first law of thermodynamics; in the second, any currents in electrical circuits would inevitably attenuate, but we do not observe this.

We observe the most common application of Kirchhoff’s laws in the so-called series and parallel circuits. IN serial circuit (a vivid example of such a circuit is a Christmas tree garland consisting of light bulbs connected in series) electrons from the power source pass through a series of wires sequentially through all the light bulbs, and on the resistance of each of them the voltage drops according to Ohm’s law.

IN parallel circuit the wires, on the other hand, are connected in such a way that an equal voltage is supplied to each element of the circuit from the power source, which means that each element of the circuit has its own current strength, depending on its resistance. An example of a parallel circuit is a ladder connection of lamps: voltage is applied to the busbars and the lamps are mounted on crossbars. The currents passing through each node of such a circuit are determined according to the first Kirchhoff’s law.

Gustav Robert KIRCHHOF
Gustav Robert Kirchhof, 1824–77

German physicist. Born in Königsberg (present-day Kaliningrad). He formulated the laws of calculating electrical circuits while still a student at the University of Königsberg. He continued his brilliant career at a number of German universities, the last of which was Berlin, where he was professor of theoretical physics from 1875 until his death. While working at the University of Breslau (on the territory of modern Poland), together with Robert Bunsen, he developed the basics of spectroscopy (cm. Discovery of Kirchhoff-Bunsen). In addition, he discovered another cycle of Kirchhoff’s laws describing heat absorption and radiation. Although Kirchhoff spent the second half of his life in a wheelchair as a result of an accident, all his contemporaries spoke of him as a pleasant person and a convinced optimist.