Kirchhoff's Laws of Electric Circuits

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Introduction

While studying the electric circuit it’s mandatory to learn the laws of electricity associated with them. Kirchhoff’s law of electric circuits is assumed to be the heart of electric circuits and circuit analysis. With the help of Kirchoff’s law, we can find the values of electrical components such as resistor, inductors, etc... Kirchoff has given two individual laws corresponding to the calculation of current and voltage in a given circuit. These are known as Kirchhoff’s current law and Kirchhoff’s voltage law. Here we will define Kirchhoff's law and understand the concepts.


What are Kirchhoff’s laws? 

In 1845, Gustav Kirchhoff a german physicist discovered the two sets of law that will help us to understand the concept of conservation of current and energy in a given electrical circuit. These two laws are commonly known as Kirchhoff’s laws of electrical circuits. Kirchhoff’s laws of the electrical circuits are helpful in analyzing and calculating the electrical resistance, impedance of any complex AC circuits. To state Kirchoff’s law we must be well versed with the directions of current flow as well. Now let us have a look at Kirchoffs law in detail. Below we will state Kirchhoff’s law.


Kirchhoff’s Current Law

Gustav Kirchhoff was the creature of law. Kirchhoff’s current law is also known as Kirchhoff’s first law of electric circuits. It says that the algebraic sum of all the currents in any given circuit will be equal to zero. In other words, it states that the total current flowing into a node or junction in an electric circuit must be equal to the total current flowing out. Kirchhoff’s current law is considered to be a consequence of charge conservation.

We know that the motion of charge carriers results in the current. The flow of current occurs when charge carriers move throughout the electric circuits. We know that in physics charge is a conserved quantity. So in the context of the electrical circuit, it is satisfied by checking whether the amount of current flowing inside is equal to the amount of current flowing out.

The standard way of expressing Kirchhoff’s current law is by writing the equation for the sum of all the current entering is being equal to the sum of currents leaving the junction. Let us consider an example as shown below:

  1. In the given figure below there are two arms from which the current is entering the node and from the third, the current is leaving the circuit. Now, according to Kirchhoff’s current law, the algebraic sum of currents entering the node must be equal to the algebraic sum currents leaving the node. Mathematically we write,

     ⇒ I1 + I2 = I3


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  1. In the second example given below, we can see all the currents in the wire are entering the node, then according to Kirchhoff’s current law we write,

     ⇒ I1 + I2 + I3 = 0


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This can be generalized for n wires connected at a node together as,

\[\Rightarrow \sum_{n=1}^{k}I_{n}=0\]

It is important to understand that sign convention plays an important role in describing the current law. If we are writing a positive current, that means the current considered is flowing in the indicated direction. Always current flows from positive of the battery to negative of the battery. If it is moving in the opposite direction we will consider that as a negative current.


Kirchhoff’s Voltage Law

  • Kirchhoff’s voltage law states that the algebraic sum of all the voltages in a given circuit will be equal to zero. It is also known as the Loop law in general.

  • As the charge carriers in an electrical circuit pass through the components present in it, they will either gain electrical energy or lose depending upon the component. The component refers to either a cell, resistor, etc…

  • The work is done on the electrical charges or by the charges due to the electrical forces inside the components. 

  • The total work done on charge carriers by electrical forces in supply components (Such as cell) must be equal to the work done by the charge carriers in the rest of the components (like the resistors)  in a given circuit. This will result in that, the sum of all the potential differences across the components involved in the given circuit must be zero.

When a circuit contains a number of junctions, we need to be careful to choose just one loop each time and apply this law. In effect, this means choosing only one option at each junction. Kirchhoff’s voltage law can be generalized in any given circuit,

\[\Rightarrow \sum_{n=1}^{k}V_{k}=0\]


Applications of Kirchhoff’s Law

Kirchhoff’s laws are widely used in analog electronics for solving complex electrical circuits. Kirchhoff’s rules are used to analyze any AC electrical circuit by modifying them for those circuits with electromotive forces, resistors, capacitors, and more. Technically speaking, however, these rules are only useful for characterizing those circuits that cannot be simplified by combining elements in series and parallel. 

Series and parallel combinations of electrical circuits are typically much easier to perform than applying either of Kirchhoff’s rules, but Kirchhoff’s rules are widely applicable and should be used to solve problems involving complex AC circuits that cannot be simplified by combining circuit components in series or parallel.


Examples

1. Calculate the Potential Difference Between the 5Ω and 7Ω in the Given Circuit.


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Sol:

For solving the questions from electronics it’s mandatory to observe and note down the sign convention of the currents in the circuit. Let us redraw the circuit with the direction of the current flow.

We know that direction of current is always from positive terminal to negative terminal, therefore we mark the direction of current as shown in the figure below.


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We are asked to calculate the potential difference across the 5Ω and 7Ω for calculating the potential difference across given resistors we will use ohm’s law. Before starting with the potential difference calculation, let us first determine the current in the given circuit.

We have two circuit laws, but we have to choose one among them depending upon the convenience. For the given circuit it will be useful to apply loop law or Kirchhoff’s voltage law. 

So, according to Kirchhoff’s voltage law we have, the algebraic sum of voltages across the components and the supply must be equal to zero. Let i be the current flowing through the given circuit. Then, we get,

⇒ 10 - 5i - 7i = 0

Simplifying the above expression we get,

⇒ 12i = 10

⇒ i = 10/12 = 0.833 = 833mA

Therefore, the current flowing through the given circuit is 833mA.

Now, let us find the potential difference across 5Ω and 7Ω . According to Ohm’s law w have,

V = IR

Where,

V - The potential difference across the resistor R

 I - The current flowing through a closed-loop


For a 7Ω resistor, the potential difference is given by:

⇒ V = iR = (833 x 10-3) (7) volts

⇒ V = 5.831 volts

Therefore the Potential Difference Across the 7Ω Resistor is 5.831 volts.

For a 5Ω resistor, the potential difference is given by:

⇒ V = iR = (833 x 10-3) (5) volts

⇒ V= 4.165 volts

Therefore the Potential Difference Across the 5Ω Resistor is 4.165 volts.

FAQs (Frequently Asked Questions)

1. State and Explain Kirchhoff’s Law? Define Kirchhoff's Law.

Ans: Kirchoff has given two individual laws corresponding to the calculation of current and voltage in a given circuit. These are known as Kirchhoff’s current law and Kirchhoff’s voltage law. Therefore the basic laws for analyzing an electric circuit are as follows:

Kirchhoff’s current law states that the total current flowing into a node or junction in an electric circuit must be equal to the total current flowing out. It is also known as the junction law.

Kirchhoff’s Voltage Law states that the algebraic sum of all the voltages in a given circuit will be equal to zero. It is also known as the Loop law in general.

2. Define Kirchhoff’s First Law.

Ans: Kirchhoff’s 1st law states that the algebraic sum of all the currents in any given circuit will be equal to zero.

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