Introduction
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.
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FAQs on Kirchhoff’s Second Law
1. What is Kirchhoff's Second Law, also known as the Voltage Law (KVL)?
Kirchhoff's Second Law (KVL), or the loop rule, states that the algebraic sum of all the changes in potential (voltage) around any closed circuit loop must be zero. This means that for any complete loop in a circuit, the sum of electromotive forces (EMFs) from sources like batteries is equal to the sum of the voltage drops across all the resistors and other components in that loop.
2. What is the mathematical expression for Kirchhoff's Second Law?
The mathematical representation for Kirchhoff's Second Law is ΣΔV = 0. This formula signifies that if you trace a path around any closed loop in a circuit, the sum of all the potential gains (like from a battery) and potential drops (like across a resistor) will equal zero. It's a fundamental equation used for circuit analysis.
3. On what fundamental principle is Kirchhoff's Second Law based?
Kirchhoff's Second Law is based on the Law of Conservation of Energy. It implies that the net change in energy of a charge after it moves around a complete closed loop is zero. The energy supplied by the sources (EMF) within the loop is completely dissipated or consumed by the other circuit elements, like resistors, ensuring no energy is lost or created within the loop system.
4. How do we apply the sign convention when using KVL in a circuit analysis?
To correctly apply KVL, a consistent sign convention is crucial. As per the NCERT syllabus for the 2025-26 session, the standard convention is as follows:
- Potential Drop across a Resistor: When traversing a loop in the same direction as the current through a resistor, the change in potential is negative (-IR).
- Potential Gain across a Resistor: When traversing a loop in the opposite direction to the current, the change in potential is positive (+IR).
- EMF of a Source (Gain): When moving from the negative to the positive terminal of a battery or EMF source, the potential is considered positive (+ε).
- EMF of a Source (Drop): When moving from the positive to the negative terminal, the potential is considered negative (-ε).
5. What are some practical examples where Kirchhoff's Second Law is applied?
Kirchhoff's Second Law is essential for analysing any circuit that cannot be simplified using basic series and parallel combinations. Its main applications include:
- Calculating the current in each branch of a complex electrical network.
- Determining the voltage across different components in a multi-loop circuit.
- Analysing circuits containing multiple batteries or voltage sources.
- Serving as the foundation for advanced circuit analysis techniques like mesh analysis.
6. What is the primary difference between Kirchhoff's First Law (KCL) and Second Law (KVL)?
The primary difference lies in the conservation principle each law represents:
- Kirchhoff's First Law (KCL or Junction Rule) is based on the conservation of charge. It states that the total current entering a junction must equal the total current leaving it. It applies to nodes or junctions in a circuit.
- Kirchhoff's Second Law (KVL or Loop Rule) is based on the conservation of energy. It states that the sum of all voltage drops and gains in a closed loop is zero. It applies to closed loops within a circuit.
7. What are the key limitations of Kirchhoff's Second Law?
Kirchhoff's laws are based on the lumped-element model and have limitations. KVL is not applicable under certain conditions:
- It is not valid for high-frequency AC circuits, where the time it takes for the electromagnetic field to propagate across the circuit becomes significant.
- It fails in the presence of a time-varying magnetic field within the loop, as this induces an electric field that is non-conservative, violating the basis of the law.
8. Why is KVL not applicable in circuits with a changing magnetic field?
KVL is based on the assumption that the electric field within a circuit is conservative, meaning the work done to move a charge around a closed loop is zero. However, according to Faraday's Law of Induction, a changing magnetic field creates a non-conservative induced electric field. This induced EMF means the potential difference between two points is no longer unique and depends on the path taken, which directly contradicts the fundamental principle of KVL.
9. How does KVL help in analysing complex circuits that cannot be simplified by series/parallel rules?
For complex circuits with multiple power sources and interconnecting branches, simple series/parallel rules are insufficient. KVL provides a systematic method to solve them. By identifying the independent loops in the circuit, we can apply KVL to each one, creating a set of simultaneous linear equations. Solving this system of equations, often along with equations from KCL, allows us to find the unknown currents and voltages throughout the entire circuit.
