Editing Voltage (section)

=== Circuit theory ===
In [[Network analysis (electrical circuits)|circuit analysis]] and [[electrical engineering]], [[lumped element model]]s are used to represent and analyze circuits. These elements are idealized and self-contained circuit elements used to model physical components.<ref name=":2">{{Cite web|last=A. Agarwal & J. Lang|date=2007|title=Course materials for 6.002 Circuits and Electronics|url=https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-002-circuits-and-electronics-spring-2007/video-lectures/6002_l1.pdf|access-date=4 December 2018|website=MIT OpenCourseWare|archive-date=9 April 2016|archive-url=https://web.archive.org/web/20160409071008/http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-002-circuits-and-electronics-spring-2007/video-lectures/6002_l1.pdf|url-status=live}}</ref>

When using a lumped element model, it is assumed that the effects of changing magnetic fields produced by the circuit are suitably contained to each element.<ref name=":2" /> Under these assumptions, the electric field in the region exterior to each component is conservative, and voltages between nodes in the circuit are well-defined, where<ref name=":2" />

:<math>\Delta V_{AB} = -\int_{\mathbf{r}_A}^{\mathbf{r}_B} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell} </math>

as long as the path of integration does not pass through the inside of any component. The above is the same formula used in electrostatics. This integral, with the path of integration being along the test leads, is what a voltmeter will actually measure.<ref>{{Cite journal|last=Bossavit|first=Alain|date=January 2008|title=What do voltmeters measure?|journal=COMPEL - the International Journal for Computation and Mathematics in Electrical and Electronic Engineering|volume=27|pages=9–16|doi=10.1108/03321640810836582|via=ResearchGate}}</ref><ref group="note">This statement makes a few assumptions about the nature of the voltmeter (these are discussed in the cited paper). One of these assumptions is that the current drawn by the voltmeter is negligible.</ref>

If uncontained magnetic fields throughout the circuit are not negligible, then their effects can be modelled by adding [[mutual inductance]] elements. In the case of a physical inductor though, the ideal lumped representation is often accurate. This is because the external fields of inductors are generally negligible, especially if the inductor has a closed [[Magnetic circuit|magnetic path]]. If external fields are negligible, we find that

:<math>\Delta V_{AB} = -\int_\mathrm{exterior}\mathbf{E}\cdot \mathrm{d}\boldsymbol{\ell}=L\frac{dI}{dt}
</math>

is path-independent, and there is a well-defined voltage across the inductor's terminals.<ref>{{Cite web|last1=Feynman|first1=Richard|last2=Leighton|first2=Robert B.|last3=Sands|first3=Matthew|title=The Feynman Lectures on Physics Vol. II Ch. 22: AC Circuits|url=https://feynmanlectures.caltech.edu/II_22.html|access-date=2021-10-09|website=Caltech}}</ref> This is the reason that measurements with a voltmeter across an inductor are often reasonably independent of the placement of the test leads.