Kirchhoff's Laws

Due to the 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}^{N}I_{n} = 0 \end{align}

where the total current \(\vec{I}\) can be decomposed into many currents of each branched path \(I_{n}\).

The Kirchhoff voltage law can be derived also from Maxwell's Equations, specifically the electrostatics equations that formulate the electric field as an electrostatic potential:

\begin{align} \label{} \vec{E} = -\vec{\nabla}V \end{align}

more specifically, the potential difference across a circuit element can be defined by \(\int \vec{E} \cdot d\vec{l} = V(b) - V(a)\), where \(a\) and \(b\) correspond to the positions before and after the circuit element. We know from electrostatics that:

\begin{align} \label{} \oint \vec{E} \cdot d\vec{l} = 0 \end{align}

and from the superposition principle we know:

\begin{align} \label{} V_{tot} = \sum V_{i} \end{align}

so the total voltage drop, or potential difference around the entire circuit must be zero:

\begin{align} \label{} \sum_{n=0}^{N}V_{n} = 0 \end{align}