Kirchhoff's Law: Find Unknown Resistance R in the Circuit. | Class 12 CBSE | Application Based |

Using Kirchhoff’s Rule determine the value of unknown resistance R into the circuit so that no current flows through 4Ω resistance. Also find the potential difference between A and D.

Kirchhoff's Laws are fundamental principles in electrical engineering and physics that are used to analyze electrical circuits. They were first described in 1845 by German physicist Gustav Kirchhoff. These two laws are essential for understanding how current and voltage behave in complex circuits.

There are two main laws:

Kirchhoff's Current Law (KCL) - Also known as Kirchhoff's First Law or the Junction Rule.

Statement: The algebraic sum of currents entering any junction (or node) in an electrical circuit is equal to the algebraic sum of currents leaving that junction.
In simpler terms: Charge is conserved. Current is the flow of charge, so at any point where wires connect (a junction), no charge can accumulate or be lost. Whatever current flows into the junction must flow out of it.

Mathematical representation: ΣI=0 (where current entering is positive and current leaving is negative, or vice versa, consistently applied).
Analogy: Imagine water flowing through pipes. If several pipes meet at a junction, the total amount of water flowing into that junction must equal the total amount of water flowing out of it.

Kirchhoff's Voltage Law (KVL) - Also known as Kirchhoff's Second Law or the Loop Rule.

Statement: The algebraic sum of the potential differences (voltages) around any closed loop in an electrical circuit must be zero.
In simpler terms: Energy is conserved. As you trace a path around a closed loop in a circuit, any energy gained (e.g., from a battery) must be equal to the energy lost (e.g., across resistors) by the time you return to your starting point. There's no net gain or loss of potential energy for a charge completing a loop.
Mathematical representation: ΣV=0 (where potential rises are positive and potential drops are negative, or vice versa, consistently applied).
Analogy: Imagine a roller coaster. If you start at a certain height, go through various dips and climbs, by the time you return to your starting point, your net change in height (potential energy) must be zero.
Key Applications:

Circuit Analysis: Kirchhoff's Laws are used extensively to determine unknown currents and voltages in complex circuits that cannot be easily simplified using just series and parallel combinations.
Network Analysis: They form the basis for systematic methods of circuit analysis like nodal analysis (primarily using KCL) and mesh analysis (primarily using KVL).
Design and Troubleshooting: Electrical engineers use these laws to design functional circuits and to troubleshoot problems by understanding how current and voltage should behave.
Limitations:

While incredibly powerful for most common circuits, Kirchhoff's Laws are based on the "lumped element model" and are approximations of Maxwell's equations. They are accurate for DC circuits and for AC circuits at frequencies where the wavelengths of electromagnetic radiation are very large compared to the circuit's physical dimensions. At very high frequencies or in circuits with significant electromagnetic fields (like long transmission lines), more advanced analysis methods based on Maxwell's equations are required because charge densities might not be constant in time within elements, and electric and magnetic fields can "leak" outside the circuit.

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