Kirchhoff’s circuital laws are a set of equations that govern the behaviour of current flowing in a circuit. This involves predicting the behaviour of current at junctions and how the voltage drop across the circuit is spread. Kirchhoff’s laws can be traced back to 1845 when German physicist Gustav Kirchhoff first came up with them.
In their most simple form, Kirchhoff’s laws can be written as a set of two laws, the first one being Kirchhoff’s current law, and the second one being Kirchhoff’s voltage law. Kirchhoff’s first law describes how current passes a junction in a circuit. A junction is a point in a circuit where two or more paths are available for the current to travel through. The total current entering the junction is equal in magnitude to the total current leaving the junction. The current may be entering the junction from multiple sources and also leaving the junction in multiple directions. This law is the result of the conservation of charge and non-accumulation of charge in a circuit.
The total charge that is flowing in the circuit must be the same at each instant. This is because, in classical mechanics, charge can neither be created nor destroyed. Also, the charge does not pile up anywhere in the circuit. This means that the total charge entering and leaving all cross-sectional areas of the conductor is the same at each instant. Thus, it naturally follows that to prevent accumulation of charge at a junction, while also maintaining the principle of the conservation of charge, the total current entering and leaving the junction must be the same. If that is not so, there will be the presence of excess charge at the junction, which is impossible.
Kirchhoff’s second law deal with the voltage drop across a circuit. It states that if we consider the voltage drop at each point of a circuit moving in either clockwise or counterclockwise direction, then the sum of all the voltage drop across the closed-loop must come out to be zero. This is the result of the conservation of electric potential. We will return to this topic later in the article.
Kirchhoff’s laws are widely used in electrical engineering to find out the current flowing through each arm of a complex circuit if the voltage drop at each point of the circuit is known, or to find out the voltage drop at a point in a circuit if the current flowing through the arms of the circuit are known. They are especially useful in circuit analysis and can be adequately applied in time and frequency domains.
Kirchhoff’s second law is also called the mesh rule because it is frequently applied to circuits and situations where a lot of junctions are present, which looks like a mesh. In simplest terms, Kirchhoff’s second law states that voltage drop across a closed loop in a circuit is zero.
The above statement can be understood by an example. Electric potential is defined as the amount of work that needs to be done to move a charge to a certain point in the presence of an electric field. Let us assume that a point in a circuit has a certain potential. If we move a charge that is resting at that point and move it around the circuit in either clockwise or counterclockwise direction, then the amount of work done in moving the charge around the circuit and ending at the same point is zero. This is because a point has a fixed potential and if the charge is brought back to it after moving it around the circuit then there is no change in the electric potential of the charge. If the charge was to have a different potential at the same point, then that would mean that a single point in the circuit has two different values of electric potential associated with it, which is absurd and impossible.
Mathematically, Kirchhoff’s second law can be stated as:
Here,
V_{K} is the voltage drop at each point of the circuit.
Kirchhoff’s second law applies to all the common circuits we encounter in everyday life. It is also true for imaginary circuits because electric potential behaves the same way regardless of the presence of physical circuits. It is also tremendously useful in situations where static electricity is involved.
Kirchhoff’s second law has its limitations. It is only true for lumped circuits. Lumped circuits are those where the physical properties, like resistance, that are spread across the entirety of the circuit are localised at a single point.