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:PROPERTIES:
:ID: 1d586d6b-bd97-4c59-ad57-8894ae4ac8ba
:END:
#+title: Kirchhoff's Laws
#+author: Preston Pan
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* Introduction
Kirchhoff's Laws, along with Ohm's law, create the axioms of circuit analysis. The two laws are the Kirchhoff Voltage Law
(KVL) and Kirchhoff's Current Law (KCL). They can be derived from an approximation of [[id:fde2f257-fa2e-469a-bc20-4d11714a515e][Maxwell's Equations]].
** KCL
:PROPERTIES:
:ID: 9f7e61fa-a6ed-4d9b-8cdf-7f4ffdd80f06
:END:
Due to the [[id:a871e62c-b4a0-4674-9dea-d377de2f780b][continuity equation]] for electrodynamics, current is always conserved locally. In an ideal one-dimensional
wire, the surface integral can be reduced to a simple line integral, given the current only moves in one direction
(which we will assume for circuits).
\begin{align}
\int I \cdot d\vec{l} = -\frac{\partial Q_{enc}}{\partial t}
\end{align}
If the total amount of charge in the wires are conserved:
\begin{align}
\label{}
\int \vec{I} \cdot d\vec{l} = 0
\end{align}
Therefore:
\begin{align}
\label{}
\sum_{n}I_{n} = 0
\end{align}
where the total current $\vec{I}$ can be decomposed into many currents of each branched path $I_{n}$.
** KVL
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