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It is sometimes claimed that Kirchhoff's Voltage Law (KVL) is equivalent to the statement that a given electric field is conservative. When a given electric field is not conservative, it is claimed that KVL does not apply and/or it is violated.

How did Kirchhoff actually explain KVL?

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    $\begingroup$ Kirchhoff stated his laws for wire circuits, electric fields did not even enter the picture. So-called "Kirchhoff's voltage law for electrostatics", i.e. for loops in a field, is a later development after Maxwell, see integral form of the Maxwell–Faraday equation. $\endgroup$
    – Conifold
    Aug 30 at 0:46
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Kirchhoff's Voltage Law appears in his 1845 paper "Ueber den Durchgang eines electrischen stromes etc."

  1. wenn die Draehte $1,2,...\nu$ eine geschlossene Figur bilden, $$I_1\omega_1 + I_2\omega_2+ ... +I_{\nu}\omega_{\nu}$$ = der Summe alle elektromotorischen Kraefte die sich auf Wege:

Which I translate as:

  1. when the wires $1,2,...n$ form a closed figure, $$I_1R_1 + I_2R_2 + ... I_nR_n$$ = the sum of all electromotive forces that are on the way:
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