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The Junction Rule
Kirchhoff's junction rule states that at any circuit junction, the sum of the currents flowing into and out of that junction are equal.
Learning Objectives

Describe limitations of the Kirchhoff's junction rule

Formulate the Kirchhoff's junction rule
Key Points
 Kirchhoff's junction rule is an application of the principle of conservation of electric charge: current is flow of charge per time, and if current is constant, that which flows into a point in a circuit must equal that which flows out of it.
 The mathematical representation of Kirchhoff's law is:
$\sum_{k=1}^{n} I_k=0$ where I_{k} is the current of k, and n is the total number of wires flowing into and out of a junction in consideration.  Kirchhoff's junction law is limited in its applicability over regions, in which charge density may not be constant. Because charge is conserved, the only way this is possible is if there is a flow of charge across the boundary of the region. This flow would be a current, thus violating the law.
Terms

electric charge
A quantum number that determines the electromagnetic interactions of some subatomic particles; by convention, the electron has an electric charge of 1 and the proton +1, and quarks have fractional charge.

current
The time rate of flow of electric charge.
Full Text
Kirchhoff's junction rule, also known as Kirchhoff's current law (KCL), Kirchoff's first law, Kirchhoff's point rule, and Kirchhoff's nodal rule, is an application of the principle of conservation of electric charge.
Kirchhoff's junction rule states that at any junction (node) in an electrical circuit, the sum of the currents flowing into that junction is equal to the sum of the currents flowing out of that junction. In other words, given that a current will be positive or negative depending on whether it is flowing towards or away from a junction, the algebraic sum of currents in a network of conductors meeting at a point is equal to zero. A visual representation can be seen in .
Kirchhoff's Loop and Junction Rules Theory
We justify Kirchhoff's Rules from diarrhea and conservation of energy. Some people call 'em laws, but not me!
Thus, Kirchoff's junction rule can be stated mathematically as a sum of currents (I):
where n is the total number of branches carrying current towards or away from the node.
This law is founded on the conservation of charge (measured in coulombs), which is the product of current (amperes) and time (seconds).
Limitation
Kirchhoff's junction law is limited in its applicability. It holds for all cases in which total electric charge (Q) is constant in the region in consideration. Practically, this is always true so long as the law is applied for a specific point. Over a region, however, charge density may not be constant. Because charge is conserved, the only way this is possible is if there is a flow of charge across the boundary of the region. This flow would be a current, thus violating Kirchhoff's junction law.
Key Term Reference
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Sources
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Cite This Source
Source: Boundless. “The Junction Rule.” Boundless Physics. Boundless, 26 Jun. 2015. Retrieved 27 Jun. 2015 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/circuitsanddirectcurrents20/kirchhoffsrules152/thejunctionrule5396331/