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Kirchhoff’s Second Law

Last updated date: 23rd May 2024
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Gustav Kirchhoff, a physicist from Germany, researched and found two laws concerning the electrical circuits involving lumped electrical elements. In the year 1845, he pursued the concepts of Ohm's law and Maxwell law and defined Kirchhoff’s first law (KCL) and Kirchhoff’s second law (KVL). 

Kirchhoff’s current law or KCL is based on the law of conservation of charge. According to this, the input current to a node must be equal to the output current of the node. Further, the second law is discussed below in detail.

State Kirchhoff’s Second Law

The second law by Kirchhoff is alternatively known as Kirchhoff’s voltage law (KVL). According to KVL, the sum of potential differences across a closed circuit must be equal to zero. Or, the electromotive force acting upon the nodes in a closed loop must be equal to the sum of potential differences found across this closed-loop. 

Kirchhoff’s 2nd law also follows the law of conservation of energy, and this can be inferred from the following statements. 

In a closed-loop, the amount of charge gained is equal to the amount of energy it loses. This loss of energy is due to the resistors connected in this closed circuit. 

Also, the sum of voltage drops across the closed circuit should be zero. Mathematically, it can be represented as \[\sum V =0 \].

Limitation and Application of Kirchhoff’s Law

As per Kirchhoff, the law holds only in the absence of fluctuating magnetic fields in this circuit. So, it cannot be applied if there is a fluctuating magnetic field. Take a look at the applications of KVL.


Sign Convention for KVL 

Refer to the above image to find the signs of voltage when the direction of current in this loop is as shown. 

Kirchhoff’s Law Examples 

Let us understand Kirchhoff’s voltage law with an example. 

Take a closed-loop circuit or draw one as shown in the figure. 

Draw the current flow direction in the circuit, and it might not be the actual direction of the current flow. 

At points A and B, \[I_{3}\] becomes the sum of \[ I_{1} and I_{2} \]. So, we can write \[I_{3} = I_{1} + I_{2} \]. 

According to Kirchhoff’s second law, the sum of the potential drops in a closed circuit will be equal to the voltage. From this statement, we have 

In loop 1: \[I_{1} * R_{1} + I_{3} * R_{3} = 10\]. 

In loop 2: \[ I_{2} * R_{2} + I_{3}* R3 = 20 \]. 

In loop 3: \[ 10 * I_{1} – 20 * I_{2} = 10 – 20 \]. 

By putting the value of \[ R_{1}, R_{2}, and R_{3} \] in the above equations, we have 

In loop 1: \[ 10 I_{1}+ 40 I_{3} = 10, or I_{1} + 4 I_{3} = 1 \]. 

In loop 2: \[20 I_{2} + 40 I_{3} = 20, or I_{2} + 2 I_{3} = 1\]. 

In loop 3: \[ 2 I_{2} - I_{1} = 1\]. 

According to Kirchhoff’s 1st law, we have I3 = I1 + I2. Substituting this in all 3 equations, we get

In loop 1: \[I_{1} + 4 (I_{2} + I_{2}) = 1, or 5 I_{1} + I_{2} = 1\]…………………(1)

In loop 2: \[I_{2} + 2 (I_{1} + I_{2}) = 1, or 2I_{1} + 3I_{2} = 1\]……………….(2)

By equating equation 1 and 2, we have 

\[5 I_{1} + I_{2} = 2I_{1} + 3I_{2}, or 3 I_{1} = 2 I_{2}\]

Therefore, \[I_{1} = -1/3 I_{2}\]

By putting the value of \[I_{1}\] in loop 3 equation, we have 

\[I_{1}\] = -0.143 A. 

\[I_{2}\] = 0.429 A. 

\[I_{3}\] = 0.286 A. 

The above speculations and calculations prove that Kirchhoff’s voltage law holds true for these lumped electrical circuits. 

Subsequently, you will be able to have a better understanding of KVL by acquiring study material from our Vedantu app. You can download the application to start learning from the comfort of your home.

FAQs on Kirchhoff’s Second Law

1. Kirchhoff’s Second Law is Based on the Law of Conservation of Energy. Explain?

According to the law of conservation of energy, the electric energy gained within a closed electrical circuit is also lost within the loop. Also, the sum of potential differences in a closed circuit will be zero.

2. Explain the Sign Convention of Kirchhoff’s Second Law?

The electromotive force of a battery is considered to be positive when current flows from the negative terminal towards the positive terminal. If the current traverses across the resistor, then its potential difference is deemed to be negative.

3. What are Some Examples of Kirchhoff’s Voltage law?

Single loop circuits, complex electrical circuits, and charging circuits are examples of Kirchhoff’s law. You can see applications of this law in such closed-loop circuits.

4. What is Kirchoff’s Second Law or voltage law?

The Voltage Law of Gustav Kirchhoff is the second of his essential laws that we can apply to circuit analysis. His voltage law claims that the algebraic sum of all the voltages combined around any closed loop in a circuit is zero for a closed-loop series path. This is due to the fact that a circuit loop has a closed conducting line, which means that no energy is lost.

To put it another way, the algebraic sum of ALL potential differences around the loop must equal zero, as follows: V = 0. It is worth noting that the phrase "algebraic sum" is used in reference to accounting for the polarities and signs of the sources and also the voltage losses around the loop.

The Conservation of Energy is a notion proposed by Kirchhoff that states that traveling through a closed-loop, or circuit, will return you to where you started in the circuit. Therefore, the same starting potential with no voltage loss. Hence, any voltage drops encountered along the way must be equal to any voltage sources encountered.

When applying Kirchhoff's voltage law to a given circuit element, we must pay close attention to the algebraic signs, (+ and -), of the voltage drops across elements and the emf's of sources, or our calculations will be incorrect.

5. What is a single circuit element?

For the sake of simplicity, we will assume that the current, I, flows in the same direction as the positive charge flow, which is known as conventional current flow.

The current flow through the resistor here is from point A to point B, or from a positive to a negative terminal. As we are moving in the same direction as current, there will be a drop in potential across the resistive element, resulting in a -IR voltage drop. 

If the current is flowing in the other direction from point B to point A. The potential across the resistive element would increase as we moved on from a - to a + potential, which results in a +I*R voltage drop.

As a result, in order to accurately apply Kirchhoff's voltage law to a circuit, we must first grasp the polarity, and as we can see, the sign of the voltage drop across the resistive element is determined by the direction of the current flowing through it. Generally, moving in the same direction as current across an element will lose potential, while moving in the direction of an emf source will gain potential.

Current flow around a closed circuit can be clockwise or anticlockwise, and either direction can be chosen. The conclusion will still be correct and valid if the chosen direction differs from the real direction of the current flow, but the algebraic answer will have a minus sign.

6. How are real circuits lumped with real elements?

At low frequencies, the lumped element approximation for a circuit is accurate. Leaked fluxes and fluctuating charge densities in conductors become prominent at higher frequencies. To some extent, parasitic components can still be used to simulate such circuits. If the frequencies are too high, it may be preferable to directly replicate the fields using finite element modeling or other techniques.

It is crucial to grasp the difference between actual circuit elements and ideal lumped elements when modeling circuits so that both laws can be applied. A wire, for example, is not an ideal conductor. Wires can inductively and capacitively join with each other. They have a finite propagation time, unlike a perfect conductor. To imitate capacitive coupling, parasitic capacitances spread across the conductors can be used, while parasitic (mutual) inductances can be used to model inductive coupling. Wires also contain some self-inductance, which necessitates the use of decoupling capacitors.

7. What are the limitations of Kirchoff’s circuit laws?

Kirchhoff's circuit rules are derived from the lumped-element model, and both are conditional on the model's applicability to the circuit in the issue. The laws do not apply when the model is not relevant.

The current law makes an assumption that the net charge in any junction, wire or lumped component remains constant. This will unlikely be the case when the electric field in between components of the circuit is non-negligible, such as when two wires are capacitively connected. This takes place in high-frequency AC circuits, where the lumped element model does not work.

In a transmission line, for example, the charge density in the conductor will be constantly oscillating.

The voltage law, on the other hand, is based on the notion that time-varying magnetic fields affect only specific components such as inductors. In actuality, an inductor's induced electric field is not contained, but the leaking fields are often insignificant.

8. What are Kirchoff’s circuit loops?

Kirchhoff's voltage law (KVL) asserts that the algebraic total of all voltage drops as you move around a closed circuit from a fixed point and return to the same place, taking polarity into account, is always zero.

Kirchhoff's second law is also known as the law of conservation of voltage, and it is very useful when dealing with series circuits since series circuits can also operate as voltage dividers, and the voltage divider circuit is a common application of many series circuits. The Vedantu app and website have free study materials.