Kirchhoff’s Current Law (KCL) is a foundational principle in electrical circuit theory. It is essential for analyzing electric circuits and forms the basis for techniques such as nodal analysis. This article explores KCL in detail, including its definition, mathematical formula, derivation and examples with solutions.
What is Kirchhoff’s Current Law?
Kirchhoff’s Current Law states that:
“The total current entering a junction (node) in an electrical circuit is equal to the total current leaving the junction.”
In simpler terms, it means that electric charge is conserved at any junction point in a circuit—no charge is lost or created.
Mathematical Form of KCL
Let multiple branches connect at a node. If currents entering the node are considered positive and those leaving are negative (or vice versa), then the algebraic sum of all currents at the node is:
\[
\sum I = 0
\]
Physical Interpretation
KCL is based on the law of conservation of electric charge, which states that charge can neither be created nor destroyed. In a circuit, this means that all the current that flows into a junction must also flow out of it.
\[
\sum I_{\text{in}} = \sum I_{\text{out}}
\]
we will use sign convention
1. If current entering node then it is + current
2. if current is leaving node then it is - current
using
\[
\sum I = 0
\]
\[I_1+I_3+I_4 -I_2-I_5=0\]
after simplifying
\[I_1+I_3+I_4 =I_2+I_5\]
This equation shows that
\[\sum I_{\text{in}} = \sum I_{\text{out}}\]
Summary
- Kirchhoff’s Current Law ensures that charge is conserved in a circuit.
- The sum of currents entering a node equals the sum of currents leaving it.
- It is widely used in both theoretical and practical electrical engineering.
Example of KCL with Solution
Example 1: In the given circuit find \(I_1\)
Solution
Apply KCL rule on node 1
\[\sum I_{\text{in}} = \sum I_{\text{out}}\]
at node 1
\[6 A+I_1=15 A\]
\[I_1=15 A - 6 A\]
Final answer
\[\boxed{I_1=9 A}\]
Example 2: In the given circuit find \(I_T\)
In given circuit we can select node 1 and node 2 where all currents are joining
Apply KCL rule on each node
\[\sum I_{\text{in}} = \sum I_{\text{out}}\]
at node 1
\[I_2=I_1+40 mA\]
here \(I_1=20 mA\)
\[I_2=20 mA+40 mA\]
at node 2
\[I_T=10 mA+I_2\]
\[I_T=10 mA+20 mA+40 mA\]
Final Answer
\[\boxed{I_T=70 mA}\]