#### Questions

State the two Kirchhoff’s rules used in electric networks. How are there rules justified?

State Kirchhoff's rules. Explain briefly how these rules are justified.

#### Solution 1

Kirchhoff’s first rule:

In any electrical network, the algebraic sum of currents meeting at a junction is always zero.

∑I=0

In the junction below, let I_{1}, I_{2}, I_{3}, I_{4} and I_{5} be the current in the conductors with directions as shown in the figure below. I_{5} and I_{3} are the currents which enter and currents I_{1}, I_{2} and I_{4} leave.

According to the Kirchhoff’s law, we have

(–I_{1}) + (−I_{2}) + I_{3} + (−I_{4}) + I_{5} = 0 Or I_{1} + I_{2} + I_{4} = I_{3} + I_{5}

Thus, at any junction of several circuit elements, the sum of currents entering the junction must equal the sum of currents leaving it. This is a consequence of charge conservation and the assumption that currents are steady, i.e. no charge piles up at the junction.

Kirchhoff’s second rule: The algebraic sum of changes in potential around any closed loop involving resistors and cells in the loop is zero. **or**

The algebraic sum of the e.m.f. in any loop of a circuit is equal to the algebraic sum of the products of currents and resistances in it.

Mathematically, the loop rule may be expressed as ∑E = ΣIR.

#### Solution 2

**Kirchhoff’s First Law − Junction Rule**

In an electrical circuit, the algebraic sum of the currents meeting at a junction is always zero.

*I*_{1}, *I*_{2} *I*_{3}, and *I*_{4} are the currents flowing through the respective wires.

Convention:

The current flowing towards the junction is taken as positive.

The current flowing away from the junction is taken as negative.

*I*_{3} + (− *I*_{1}) + (− *I*_{2}) + (− *I*_{4}) = 0

This law is based on the law of conservation of charge.

**Kirchhoff’s Second Law − Loop Rule**

In a closed loop, the algebraic sum of the *emf*s is equal to the algebraic sum of the products of the resistances and the currents flowing through them.

For the closed loop BACB:

*E*_{1} − *E*_{2} = *I*_{1}*R*_{1} + *I*_{2}*R*_{2} − *I*_{3}*R*_{3}

For the closed loop CADC:

*E*_{2} = *I*_{3}*R*_{3} + *I*_{4}*R*_{4} + *I*_{5}*R*_{5}

This law is based on the law of conservation of energy