# 0.1 Kirchhoff's circuit laws

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Kirchhoff’s circuit laws are facilitating rules for analyzing electrical circuits. These rules are handy where circuits are more complex beyond the scope of series and parallel combination of resistances and where circuits involve intermixing of electrical sources and resistances (appliances or resistors). Kirchhoff’s laws are brilliant reflection of fundamental laws like conservation of charge and energy in the context of electrical circuits.

There are two Kirchhoff’s laws which are known by different names :

1: Kirchhoff’s current law (KCL) : It is also referred as Junction or point or Kirchhoff’s first rule.

2: Kirchhoff’s voltage law (KVL) : It is also referred as Loop or Mesh or Kirchhoff’s second rule.

## Kirchhoff’s current law (kcl)

No point in the circuit accumulates charge. This is the basic consideration here. Then, the principle of conservation of charge implies that the amount of current flowing towards a point should be equal to the amount of current flowing away from that point. In other words, net current at a point in the circuit is zero. We follow the convention whereby incoming current is treated as positive and outgoing current as negative. Mathematically,

$\sum I=0$

There is one exception to this law. A point on a capacitor plate is a point of accumulation of charge.

Problem : Consider the network of resistors as shown here :

Each resistor in the network has resistance R. The EMF of battery is E having internal resistance r. If I be the current that flows into the network at point A, then find current in each resistor.

Solution :

It would be very difficult to reduce this network and obtain effective or equivalent resistance using theorems on series and parallel combination. Here, we shall use the property of symmetric distribution of current at each node and apply KCL. The current is equally distributed to the branches AB, AD and AK due to symmetry of each branch meeting at A. We should be very careful about symmetry. The mere fact that resistors in each of three arms are equal is not sufficient. Consider branch AB. The end point B is connected to a network BCML, which in turn is connected to other networks. In this case, however, the branch like AK is also connected to exactly similar networks. Thus, we deduce that current is equally split in three parts at the node A. If I be the current entering the network at A, then applying KCL :

Current flowing away from A = Current flowing towards A

As currents are equal in three branches, each of them is equal to one-third of current entering the circuit at A :

${I}_{AB}={I}_{AD}={I}_{AK}=\frac{I}{3}$

Currents are split at other nodes like B, D and K symmetrically. Applying KCL at all these nodes, we have :

${I}_{BC}={I}_{BL}={I}_{DC}={I}_{DN}={I}_{KN}={I}_{KL}=\frac{I}{6}$

On the other hand, currents recombine at points C, L, N and M. Applying KCL at C,L and N, we have :

${I}_{LM}={I}_{CM}={I}_{NM}=\frac{I}{3}$

These three currents regroup at M and finally current I emerges from the network.

## Kirchoff’s voltage law (kvl)

This law is based on conservation of energy. Sum of potential difference (drop or gain) in a closed circuit is zero. It follows from the fact that if we start from a point and travel along the closed path to the same point, then the potential difference is zero. Recall that electrical work done in carrying electrical charge in a closed path is zero and hence potential difference is also zero :

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