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Kirchhoff's Voltage Law ○◂｜Definition｜1st｜20251119205401-00-⌔

Kirchhoff’s circuit laws - Wikipedia#Kirchhoff’s_voltage_law

Kirchhoff’s voltage law

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This law, also called Kirchhoff’s second law, or Kirchhoff’s loop rule, states the following:

The directed sum of the potential differences (voltages) around any closed loop is zero.

Similarly to Kirchhoff’s current law, the voltage law can be stated as:

$\sum _{i=1}^n{V}_i=0$

Here, n is the total number of voltages measured.

(A similar derivation can be found in Feynman’s lectures.)¹

Consider some arbitrary circuit. Approximate the circuit with lumped elements, so that time-varying magnetic fields are contained to each component and the field in the region exterior to the circuit is negligible. Based on this assumption, the Maxwell–Faraday equation reveals that

$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}=0$

in the exterior region. If each of the components has a finite volume, then the exterior region is simply connected, and thus the electric field is conservative in that region. Therefore, for any loop in the circuit, we find that

$\sum _i{V}_i=-\sum _i{\int }_{{\mathcal{P}}_i}\mathbf{E}\cdot \mathrm{d}\mathbf{l}=\oint \mathbf{E}\cdot \mathrm{d}\mathbf{l}=0$

where ${\mathcal{P}}_i$ are paths around the exterior of each of the components, from one terminal to another.

Note that this derivation uses the following definition for the voltage rise from $a$ to $b$:

$V_{a\to b}=-{\int }_{{\mathcal{P}}_{a\to b}}\mathbf{E}\cdot \mathrm{d}\mathbf{l}$

However, the electric potential (and thus voltage) can be defined in other ways, such as via the Helmholtz decomposition.

Printed 2026-06-28.

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Footnotes

1. Feynman, Richard; Leighton, Robert; Sands, Matthew (September 2013) [1964]. “Ch. 22: AC Circuits”. The Feynman Lectures on Physics. The Feynman Lectures on Physics. Vol. II (2013 online ed.). California Institute of Technology. 22–3: Networks of ideal elements; Kirchhoff’s rules. Retrieved 2018-12-06. ↩

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