Kirchhoff's Second Law

Kirchhoff's Second Law

Kirchcoff’s Law or circuit Laws is comprised of two equality mathematical equations that deal with resistance, current and voltage in the lumped elemental model of electrical circuits. The laws are fundamental to circuit theory. They quantify how current flows and voltages vary through a loop in a circuit. Gustav Robert Kirchhoff, a German physicist contributed to the fundamental understanding of electrical circuits.

What are Kirchhoff’s Laws?

There are two laws as follows:

  • Kirchhoff’s first law, also known as the Kirchhoff’s current law (KCL) states that the current flowing into a node must be equal to the current flowing out of the node. This is a consequence of charge conservation.
  • Kirchhoff’s second law, also known as the Kirchhoff’s voltage law (KVL) states that the sum of all voltages around a closed loop in any circuit must be equal to zero. This again is a consequence of charge conservation and also conservation of energy.

Here in this short piece of article, we will be discussing Kirchhoff’s second law.

Kirchhoff’s Voltage Law

Kirchhoff’s Second Law or the voltage law states that

The net electromotive force around a closed circuit loop is equal to the sum of potential drops around the loop

It is termed as Kirchhoff’s Loop Rule which is an outcome of an electrostatic field which is conservative.
Hence,

  • If a charge moves around a closed loop in a circuit, it must gain as much energy as it loses.
  • The above can be summarised as ” the gain in energy by the charge = corresponding losses in energy through resistances
  • Mathematically, the total voltage in a closed loop of a circuit is expressed as \(\sum V=0\).

The below figure illustrates that the total voltage around a closed loop must be zero.

Kirchhoff's Law

This law manages the voltage drops at different branches in an electrical circuit. Consider one point on a closed loop in an electrical circuit. If somebody goes to another point in a similar ring, he or she will find that the potential at that second aspect might be not quite the same as the first point.

On the off chance that he or she keeps on setting off to some unique point on the loop and he or she may locate some extraordinary potential in that new area. If he or she goes on further along that closed loop, eventually he or she achieves the underlying point from where the voyage was begun.

That implies, he or she returns to a similar potential point in the wake of the intersection through various voltage levels. It can be then again said that the gain in electrical energy by the charge is equal to corresponding losses in energy through resistances.

Solving circuit using Kirchhoff’s Second Law

Circuit Diagram

  • The first and foremost step is to draw a closed loop to a circuit. Once done with it draw the direction of the flow of current.
  • Defining our sign convention is very important
    Sign Convention
  • Using Kirchhoff’s first law, at B and A we get,

    \(I_{1}+I_{2}=I_{3}\)

  • By making use of above convention and Kirchoff’s Second Law
  • From Loop 1 we have:

    \(10=R_{1}*I_{1}+R_{3}*I_{3}\)

    \(=10I_{1}+40I_{3}\)

    \(1=I_{1}+4I_{3}\)

    From Loop 2 we have :

    \(20=R_{2}*I_{2}+R_{3}*I_{3}\)

    \(20I_{2}+40I_{3}\)

    \(1=I_{2}+2I_{3}\)

    From Loop 3 we have :

    \(10-20=10I_{1}-20I_{2}\)

    \(1=-I_{1}+2I_{2}\)

  • By making use of Kirchhoff’s First law \(I_{1}+I_{2}=I_{3}\)

    Equation reduces as follows (from Loop 1 ) :

    \(1=5I_{1}+4I_{2}\)

    Equation reduces as follows ( from Loop 2 ) :

    \(1=2I_{1}+3I_{2}\)

  • This results in the following Equation:

    \(I_{1}=-\frac{1}{3}I_{2}\)

  • From last three equations we get,

    \(1=\frac{1}{3}I_{2}+2I_{2}\)

    \(I_{2}=0.429A\)

    \(I_{1}=0.143A\)

    \(I_{3}=0.286A\)

Advantages and Limitations of Kirchhoffs Law

The advantages of the laws are:

  • It makes the calculation of unknown voltages and currents easy
  • The analysis and simplification of complex closed-loop circuits becomes manageable

Kirchhoff’s laws work under the assumption that there are no fluctuating magnetic fields in the closed loop. Electric fields and emf could be induced which results in the breaking of Kirchhoff’s rule under the influence of the varying magnetic field.

Practise This Question

Two charged particles A and B repel each other by a force kr2, where k is a constant and r is the separation between them. The particle A is clamped to a fixed point in the lab and the particle B which has a mass m, is released from rest with an initial separation r0  from A. Find the change in the potential energy of the two particle system as the separation increases to a large value. What will be the speed of the particle B in this situation?

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