Derivation of Kirchhoff’s Second Law

We can resolve simple electrical circuits by applying Ohm’s Law, that is,

V = IR

For the circuits that are not too complicated, we resolve it by using the principle of conservation of energy or the definition of Potential differences/Voltage between two points in a circuit.

We calculate the resistance and potential difference using Ohm’s Law in simple circuits. But in actual real-time practice, we come across complicated circuits that include many resistances and several sources of EMFs. We cannot easily determine the effective resistance and the EMF from Ohm’s Law in such cases.

Gustav Robert Kirchhoff gave two rules known after him as the Kirchoff’s rules to understand circuits better and resolve them efficiently. These rules embody the principle of conservation of charges and energy.

Kirchhoff’s Circuit Laws

Before we state these rules, we shall define the following terms, which are fundamental to circuit theory and are necessary in this connection for the convenient understanding of electrical circuits..

1. Junction: A junction in a circuit is a point where three or more conductors are electrically connected. A junction is also called a node or a branch point.

2. Loop: A loop is a closed path in an electrical circuit. It is also called a mesh.

3. Node: Node is a point on the circuit of the joint of two or more circuital elements. (A node may or may not be a junction, but a junction will be a node.)

Kirchhoff’s 1st law/ Current Rule /The Junction Theorem:

The algebraic sum of all the currents flowing through any junction is nil or zero.

That is,

∑I = 0

Also, the sum of currents entering any junction (∑I)in must equal the sum of electric currents leaving that junction (∑I)out.

Sign Convention: 

The currents flowing towards a junction are +ve. The currents flowing away from the junction are negative, -ve.

Kirchhoff’s 2nd Rule/Voltage Rule/Loop Theorem:

The algebraic sum of the potential difference encountered in going around the closed loop is zero.

That is, ∑V = 0

This law is called the Voltage rule or the Loop theorem.

• Sign Convention: 

a. When we travel through a source in the direction – to +, the emf is considered positive. When we travel from + to -, the emf is negative.  

b. When we travel through a resistor in the same direction as the assumed current, the Voltage is negative and vice versa.

Kirchhoff’s Second Rule/2nd Law Calculation/Derivation

When applying this rule, one starts from a point on the loop and goes along it to reach the starting or initial point again. This movement can be either clockwise or anticlockwise. 

Here, during the journey through the loop, any drop in the potential encountered is taken as positive, and any rise in the potential is negative. The net summation of all these potential differences will lead to zero. 

In the figure, we show a loop ABCDEFA of a circuit. As we start from A  and go along the loop clockwise to reach the initial point A, we get the following potential differences:

Va – Vb = i1R1

Vb – Vc = i2R2

Vc – Vd = -E1

Vd – Ve =i3R3

Ve – Vf = – i4R4

Vb – Va = E2

Adding all these,

i1R1 +  i2 R2 -E1 +  i3R3  –  i4R4 + E2 = 0.

The Loop Law follows directly because electrostatic force is a conservative force in nature and the work it does in any closed path is zero.

Application of Circuit Laws

• Kirchhoff’s laws are fundamental to the circuit theory and are used to calculate the values of current, potential difference (Voltage) and internal resistance in the DC ( Direct Current) Circuit.

• We can find the application of Kirchhoff’s law helpful in determining the value of unknown resistors in the circuit by using the principle of Wheatstone Bridge and metre bridge.

Limitations of Kirchhoff’s Laws

• The Principle of Kirchhoff’s law, that is, KCL( Kirchhoff’s Current law) and KVL(Kirchhoff’s Voltage Law), is inconsistent with Alternative Current (AC) circuits of higher frequencies.

• Kirchhoff’s Voltage Law remains no longer consistent in a variable magnetic field because a changing magnetic field implies a changing electric field (from Faraday’s Law), which for a closed-loop is non-conservative. Hence, the linear integral, that of the electric field, is not nil, behaving inconsistently with KVL or Kirchhoff’s Voltage Law.

Conclusion

These fundamental rules to the circuit theory enable us to measure different electrical quantities using a Wheatstone bridge, metre bridge or a potentiometer.

• Kirchhoff’s Rules

1. ∑I = 0 (Junction Rule)

It follows the Principle of conservation of charge.

2. ∑V = 0 (Loop Rule)

It follows the Principle of Conservation of Energy.