Kirchhoff’s Law - Equation, Example, FAQs

Kirchhoff's circuit rules are two equalities that deal with the current and potential difference in the lumped element model of electrical circuits (often known as voltage).In 1845, a German scientist named Gustav Kirchhoff was the first to describe them. This broadened Georg Ohm's work and came before James Clerk Maxwell's. Kirchhoff's rules, often known as Kirchhoff’s laws, are widely utilized in electrical engineering. These rules apply in both time and frequency domains and serve as the foundation for network analysis.
Kirchhoff’s current law (KCL LAW) and Kirchhoff’s voltage law (KVL LAW) were defined in 1845 after he pursued the notions of Ohm's law and Maxwell law.
Kirchhoff’s current law, or KCL LAW, is based on the principle of charge conservation. The input current to a node must be equal to the node's output current, according to this rule.

State and explain Kirchhoff’s law.

Gustav Kirchhoff, a German scientist, found the two sets of laws that would help us comprehend the notion of current and energy conservation in a particular electrical circuit in 1845. Kirchhoff’s laws of electrical circuits are the name given to these two rules. Kirchhoff's rules of electrical circuits are useful for assessing and determining the electrical resistance and impedance of any complex alternating current (AC) circuit. To state kirchhoff’s law, we must also be familiar with the directions of current flow.

Kirchhoff’s laws describe how current flows in a circuit and how voltage varies around a loop.

Kirchhoff’s current law is applicable to both alternating current and direct current circuits. It is inapplicable to magnetic fields that change over time.

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Kirchhoff’s current law (KCL LAW):

Kirchhoff’s current law: - The current coming into a node (or a junction) must match the current flowing out of it, according to Kirchhoff's first law. This is due to charge conservation.

The conventional technique to explain Kirchhoff’s current law is to write Kirchhoff’s equation in which the sum of all currents entering the junction equals the sum of currents exiting the junction. Example:

The current that enters a junction equals the current that leaves that junction.

i₁ + i₄ = i₂ + i₃

This can be generalized to n wires joined together at a node as,

Kirchhoff’s voltage law (KVL LAW):

Kirchhoff’s voltage law(KVL definition): - The sum of all voltages across components that supply electrical energy (such as cells or generators) in any entire loop inside a circuit must match the sum of all voltages across all other components in the same loop, according to Kirchhoff’s voltage law (2nd Law). This law is the result of both charge and energy conservation.

When there are multiple junctions in a circuit, we must be careful to apply this law to only one loop at a time. In practice, this means selecting only one option at each crossroads. In any given circuit,

The total of all voltages around a loop equals zero.

v₁ + v₂ + v₃ + v₄= 0

Kirchhoff’s voltage law can be generalized.

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Advantages of Kirchhoff’s law

Because of the numerous benefits of using Kirchhoff's rules, they are an important element of the fundamentals of circuit theory. For beginners, calculating unknown voltage and the current becomes much simpler. Numerous complex circuits are closed in a structure Kirchhoff’s current law, on the other hand, makes the analysis and calculation of these complex circuits straightforward and comprehensible where circuit analysis is typically difficult. There are numerous other benefits, but these are the most important.

Limitations of Kirchhoff’s law

Both Kirchhoff’s laws contain a constraint in that they work under the premise that the closed-loop has no fluctuating magnetic field. In the presence of a fluctuating magnetic field, electric fields and emf can be induced, causing Kirchhoff’s loop rule to fail.

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Solved Example Based on Kirchhoff's Law

Example 1: Dimension of spectral emissive power is

1) $\left[M T^{-2}\right]$
2) $\left[M L^2 T^{-3}\right]$
3) $\left[M L^{-1} T^{-3}\right]$
4) $\left[M T^{-3}\right]$

Solution:

As we have learnt,

Spectral Emissive Power -

$\begin{aligned}
& e_\lambda=\frac{\text { Energy }}{\text { Area } \times \text { times } \times \text { wavelength }} \\
& \text { unit }=\frac{J}{m^2 \cdot \text { s.A }} \\
& =\frac{\left[M L^2 T^{-2}\right]}{\left[L^2\right][T][L]}=\left[M L^{-1} T^{-3}\right]
\end{aligned}$

Hence, the answer is option (3).

Example 2: Which of the following relation is correct for total emissive power

1) $e=\int_0^{\infty} e_\lambda^2 d \lambda$
2) $e=\int_0^{\lambda_0} e_\lambda d \lambda$
3) $e=\int_0^{\infty} e_\lambda d \lambda$
4) $e=\int_0^{\lambda_0} e_\lambda^2 d \lambda$

Solution:

Total Emissive power is defined as the total amount of thermal energy emitted per unit time, per unit area of the body for all possible wavelengths.

$e=\int_0^{\infty} e_\lambda d \lambda$

Hence, the answer is option (3).

Example 3: For a perfectly black body emissivity is:

1) zero
2) 1
3) $\infty$
4) None of these

Solution:

The emissivity of a body at a given temperature is defined as the ratio of the total emissive power of the body (e) to the total emissive power of a perfectly black body.

$\varepsilon=\frac{e}{E}$

$\varepsilon=1 \text { - for a perfectly black body }$

Hence, the answer is option (2).

Example 4: The emissivity of a perfectly reflecting body is :

1) zero
2) 1
3) $\infty$
4) None of these

Solution:

As we have know,

$\text { Emissivity }=\varepsilon=\frac{e}{E}$

$\varepsilon=0 \text { - for perfectly reflecting body }$

Hence, the answer is option (1).

Summary
Kirchhoff's Law for emissive and absorptive power states that the emissivity, the effectiveness of a surface in emitting energy as thermal radiation for a body in thermal equilibrium, is the same as its absorptivity, the fraction of incident radiation that gets absorbed by the body. That is to say, a good absorber of radiation must be a good emitter. The law is thus key to the understanding of the thermal behaviour of materials and is a basis for many other sciences, such as thermodynamics, material science, and more. Other real-life applications include radiative cooling system design and efficient solar panels; it has numerous space applications, principally thermal radiation analysis.

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