# Kirchhoff’s Laws

Kirchhoff gave two laws to solve complex circuits which can not be solved by using Ohm’s law. Simplification of such complex circuits is impossible by series and parallel combinations.

In 1845, Gustav Kirchhoff, a German physicist, developed two rules or laws that deal with the conservation of current and energy within electrical circuits.

These two rules are commonly known as: *Kirchhoff’s Circuit Laws. *

- The first law is known as
**Kirchhoff’s Current Law, (KCL),**which deals with the current flowing around a closed circuit. - The second law is knows as
**Kirchhoff’s Voltage Law, (KVL),**which deals with the voltage sources present in a closed circuit.

#### Kirchhoff’s Current Law (KCL)

This law relates to the current at the junctions of an electric circuit and can be stated as :

*“The algebraic sum of the currents meeting at a junction in an electrical circuit is zero.”*

In an algebraic sum the sign of the quantity is taken into consideration. For example, Let’s consider five conductors carrying currents I_{1 },I_{2},I_{3},I_{4 }and I_{5 }and meeting at point O. This is shown in the fig.1 below.

Fig.1

If we take the sign of currents flowing towards the point O as positive, then currents flowing away from point O will be assigned negative sign,

Thus, applying Kirchhoff’s current law (KCL) to the junction O, we can get :

(I_{1}) + (I_{2}) +)+(I_{3})+(-I_{4})+(- I_{5}) = 0

Or I_{1} + I_{2}++I_{3= }= I_{4}+I_{5}

i.e. Sum of incoming currents = Sum of outgoing currents

Hence, Kirchhoff’s Current Law can also be stated as :

In an electric circuit, the sum of currents flowing towards any junction is equal to the sum of currents flowing away from that junction. Kirchhoff’s current law is also knowns as *Junction rule*.

#### Kirchhoff’s Voltage Law (KVL)

This law relates to e.m.f.s(voltage sources) and voltage drops in a closed circuit or loop and may be stated as :

*“In any closed electrical circuit or mesh, the algebraic sum of all the electromotive forces (e.m.f.s) and voltage drops is resistors is equal to zero.”*

i.e. In any closed circuit or mesh,

Algebraic sum of e.m.f.s + Algebraic sum of voltage drops = 0

To understand Kirchhoff’s Voltage law, let’s consider a closed loop ABCDA as shown in Fig.2 below.

Fig.2

If we start from point A in this closed circuit and go back to this point A after going around the circuit, then there will be no increase or decrease in potential.

It is important to maintain the same direction either clockwise or anti-clockwise or the final voltage sum will not be equal to zero.

So here, the algebraic sum of the e.m.f.s of all the sources plus the algebraic sum of the voltage drops in the resistances must be equal to zero.

Kirchhoff’s Voltage Law is also based on the law of conservation of energy.

Kirchhoff’s Voltage law is also known as Loop Rule.

**Sign Convention :**

While applying Kirchhoff’s voltage law to a closed circuit, algebraic sums are considered. therefore, it is very important to assign proper signs to e.m.f.s and voltage drops.

The following convention may be followed:

A rise in potential should be considered positive and fall in potential should be considered negative.

As we go from A to B, in Fig.3 (i),we go from negative terminal of the cell to the positive terminal, so there will be a rise in potential.

Fig.3

Similarly, in fig.3(ii), as we go from A to B, there is also rise in potential.

In fig.4(i), as we go from C to D, we go from positive terminal of the cell to the negative terminal, so there is a fall in potential.

Fig.4

Similarly, In fig.4(ii), as we go from C to D, there will be fall in potential.