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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. Or, put another way, is is sometimes claimed that KVL is equivalent to the equation:

$$\oint_{\Gamma}\vec{E}\cdot d\vec{\ell} = 0$$

According to this claim, if a given electric field is not conservative, then KVL is said not to 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 '21 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:

Edit:

The original question states that

It is sometimes claimed that Kirchhoff's Voltage Law (KVL) is equivalent to the statement that a given electric field is conservative.

Sredni Vashtar, in an lengthy answer, defends exactly that claim of equivalence. S.V. asserts that.

in modern notation the original form of KVL can also be written as:

$$\oint_{\Gamma}\vec{E}\cdot d\ell = 0$$

S.V. supports his case by citing texts such as the second volume of Berkeley Physics: Electricity and Magnetism 3rd edition by Purcell and Morin, and Ramo Whinney VanDuzer, "Fields and Waves for Communication Electronics" 3rd edition, p. 179. The are numerous other texts that S.V. could have cited as well.

It is agreed upon by all sides, that this re-cast version of Kirchhoff's voltage law is physically inaccurate in the presence of a time varying magnetic field. But I believe that the text of Kirchhoff's formulation makes it clear that the modern version is a re-casting, and not simply the original formulation of KVL by Kirchhoff.

It thus behooves us to examine whether or not this recasting is faithful to the original, or whether it introduces some innovation, that cannot be properly ascribed to the original. In particular, we need to examine in what ways we may re-cast the terms $I_CR_C$ and $\mathscr{E}_C$, where the latter term is electromotive force along the path $C$

S.V. re-arranges Kirchhoff's equation into the form:

(R1 I1 - emf1) + (R2 I2 - emf2) + ... + (Rn In - emfn) = 0

which is completely justifiable. However, he then proceeds to claim that

In the twentyfirst century we recognize that each term in parenthesis represents the path integral of the total electric field along each of the segments 1, 2,... n, in which the closed path (the loop) has been partitioned. Therefore, in modern terms, the original formulation of KVL says

Sum of path integral of Etot.dl along the branches of our closed figure = 0

Although this is in agreement with modern texts, one must ask if there is any justification for re-casting in this way.

I do not take issue with recasting $I_CR_C$ in the form of an $\int_C\vec{E}\cdot d\ell$ term. Indeed, I think such a recasting is an improvement over Kirchhoff. It removes the dependence of KVL in it's original formulation on Ohm's law.

However, the recasting of $\mathscr{E}$ in the form of an $\int_C\vec{E}\cdot d\ell$ term is objectionable.

Gauss's law tells us that an electric field has a non-zero divergence only in the presence of a net non-zero charge density. It is thus reasonable to take the EMF produced by a time varying magnetic field along some curve, not as the total electric field, but as the rotational or solenoidal component of the electric field. [See Helmholz decomposition for an understanding that a 3D vector field can be decomposed into a rotational/solenoidal/divergence-free component and an irrotational/conservative/curl-free component.] That is,

$$\mathscr{E}_{induced} = \int_C\vec{E}_{rotational} \cdot d\ell$$

Note that for closed loops

$$\mathscr{E}_{induced} = \oint_C\vec{E}_{rotational} \cdot d\ell = \oint_C\vec{E}_{total} \cdot d\ell$$

but the above does not apply for open paths.

Although the change in subscript appears minor, it means the difference between a law which is correct in general, and a law which is correct only in the absence of time varying magnetic fields.

[It is easy to see that if $\mathscr{E}_{induced}$ is recast as $\mathscr{E}_{induced} = \int_C\vec{E}_{rotational} \cdot d\ell$ rather than $\mathscr{E}_{induced} = \int_C\vec{E}_{total} \cdot d\ell$ then the resulting KVL is correct, even in the presence of time-varying magnetic fields. $\vec{E}_{total} = \vec{E}_{rotational} + \vec{E}_{irrotational}$. So, if one subtracts $\vec{E}_{total}$ from $\vec{E}_{rotational}$ one is left with an irrotational, i.e. conservative vector field, and therefore the sum of any closed loop contour integral of this latter field will always be 0.]

Is there any justification for re-casting KVL in a form that is broken, when it could have been re-cast in a form that is not broken? Is there any evidence that the broken form is more faithful to Kirchhoff than the unbroken form?

S.V. makes the case that Kirchhoff did not base his formulation of KVL on experiments that employed induction of EMF via a time-varying magnetic field. The unstated implication seems to be that Kirchhoff was perhaps ignorant of this source of EMF. Perhaps a more egregious unstated implication is that, being unaware of this source of EMF, the more faithful re-casting of KVL is the one which is broken.

I do not have at hand evidence that Kirchhoff was aware of induction of EMF via a time-varying magnetic field in 1845. However, Maxwell wrote in "A Treatise on Electricity and Magnetism" Volume 2, 1873, chapter XVIII p358, section 759, that

The first determination of a resistance of a wire in electromagnetic measure was made by Kirchhoff*. He employed two coils...

  • 'Bestimmung der Constanten von Welchar die Intensitat inducirter elektrischer Stromme abhungt' Pogg. Ann. lxxvi (April 1849)

including an illustration of Kirchhoff's apparatus:

enter image description here

So, we know that no later than 4 years after publishing KVL, Kirchhoff was not only aware of Faraday induction, but actively employed it in his experiments.

If, we suppose, as S.V. seems to, that Kirchhoff did not learn of Faraday induction at the University of Konigsberg, then it surely would have come as a major surprise to him, no more than 4 years later. If he didn't consider induced EMF in formulating KVL, would it not be surprising that he didn't reconsider or amend KVL upon learning of it's existence?

One final comment regarding S.V.'s reply.

I agree with Sredni Vashtar that Kirchhoff's voltage law and Faraday's law of induction are different. KVL says nothing about the origin of EMFs, Faraday's law of induction does. KVL expresses the relationship between EMFs and resistive voltage drops in any arbitrary loop in any arbitrary circuit, but Faraday's does not. So, we can all agree that Kirchhoff's voltage law, and Faraday's law of induction are different. Where I disagree is with S.V.'s claim that Faraday's law (in it's original form) is an extension of KVL (in it's original form).

Edit:

In another stack exchange answer S.V. calls the electric field induced by a time varying magnetic field $E_{ind}$ and uses the following formula and caption:

enter image description here

voltage = induced voltage + scalar potential difference

From which, one can safely infer that

$$\mathscr{E}_{ind} = \int_{\gamma_{A\rightarrow B}} \vec{E}_{ind} \cdot d\vec{\ell}$$

(or possibly the negative of that).

S.V.'s $\vec{E}_{ind}$ is the same as my $\vec{E}_{rotational}$, that is, the rotational or solenoidal component of the total electric field.

So, by rights, S.V. ought to use

$$\mathscr{E}_{ind} = \int_{\gamma_{A\rightarrow B}} \vec{E}_{ind} \cdot d\vec{\ell}$$

for the magnetically induced EMF when recasting KVL. That is, he should acknowledge that his claim that

In the twentyfirst century we recognize that each term in parenthesis represents the path integral of the total electric field along each of the segments 1, 2,... n, in which the closed path (the loop) has been partitioned. Therefore, in modern terms, the original formulation of KVL says [emphasis added]

Sum of path integral of Etot.dl along the branches of our closed figure = 0

is wrong. The original formulation uses the specific term "emf", rather than "voltage", and by S.V.'s own account, the "emf" induced along a branch of a circuit by a time-varying magnetic field is not the path integral of Etot.dl, but the path integral of Eind.dl

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  • $\begingroup$ I based my answer on what Kirchhoff had written at the time, based on what he had used at the time. I believe that from the point of view of an historian it is better to avoid speculations about what Kirchhoff could have thought had he known what we know today, or even four or eigth years later. See the comment in your original question. $\endgroup$ Dec 19 '21 at 14:53
  • $\begingroup$ What he wrote at the time was that the sum of the products of current times resistance is the sum of the emfs. You constantly conflate/confuse emf with E field, which was unknown at the time. $\endgroup$ Dec 19 '21 at 15:10
  • $\begingroup$ "sum of the emfs located on the path". In order to know that the induction emf is 'distributed' along the path and in what manner you need fields. And that was not common knowledge at the time Kirchhoff wrote this paper. I analyze what Kirchhoff wrote, not what he could have meant knowing the corpus of knowledge developed after that time. $\endgroup$ Dec 19 '21 at 22:37
  • $\begingroup$ @SredniVashtar Faraday's 1831 result that the motion of a magnet relative to a conductor induced an emf in that conductor was well known. K did not know F's law either as emf being proportional to rate of change of flux, nor the curl of E being proportional to the rate of change of B. But that is analogous to him perhaps not knowing how to predict the emf produced by an novel galvanic/voltaic cell. Knowledge that allows one to predict the magnitude of an emf is not necessary to usefully use the magnitude of an emf to determine further results. (cont.) $\endgroup$ Dec 19 '21 at 22:57
  • $\begingroup$ It seems your position is that the statement "the total of all IR drops in a loop equals the the total emfs in that loop" is in fact correct, but because K. allegedly did not use induced emfs in his experiments from which he derived KVL, therefore he could not have meant what he actually wrote, even though what he actually wrote is fairly accurate. But rather than give him credit for postulating a law that was not only validated by his own experiments, but also by experiments that came after him, you argue that his law says something quite different than what the words actually say. $\endgroup$ Dec 19 '21 at 23:03
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We can read the original article that Kirchhoff published on Annalen der Physik und Chemie 1845, Band LXIV with the title "Ueber den Durchgang eines elektrischen Stromes durch eine Ebene ins besondere durch eine kreisformige" ("On the passage of an electric current through a plane, through a circular shape in particular")

Kirchhoff original paper 1845
source: Annalen der Physik und Chemie LXIV 1945 - p. 497

Unfortunately I could not find the pictures referenced in the paper, but I believe the information in the text alone can still be used to infer the following two points:

  1. Kirchhoff formulated his laws on the base of experiments that used localised and non-inductive forms of electromotive force.
  2. Kirchhoff Voltage Law, as orginally formulated, is fundamentally different from Faraday's law. In particular, the inductive EMF due to a changing magnetic flux linked by the loop could not appear among the 'electromotorischen Krafte' along the path ('auf dem Wege') as considered by Kirchhoff.

On the sources of electromotive force considered by Kirchhoff

Right from the start, Kirchhoff mentions a constant galvanic current that flows through a metallic disc, entering and exiting it by means of wires:

"Leitet man einen constanten galvanischen Strom durch eine Metallscheibe , so wird sich die Elektricität in die ser auf eine bestimmte Weise vertheilen . Die Art der Vertheilung kann man nach den von Ohm aufgestellten Principien theoretisch ermitteln . Ich habe die dazu no thige Rechnung unter der Voraussetzung , daſs der Zu stand der Scheibe ein stationärer geworden sey , in dem Falle durchgeführt , daſs die Scheibe eine kreisförmige ist , und daſs die Elektricität durch einen Draht in sie hinein , durch einen zweiten aus ihr heraustrete."

The automatic translation from a well-known service provider was a bit approximative so I asked a German native speaker with a scientific formation to fix it. In the following the bold is mine:

"If a constant galvanic current is passed through a metal disc, the electricity into said disc will distribute in a certain way. The type of distribution can theoretically be determined according to the principia established by Ohm. I have carried out the necessary calculation on the condition that the state of the disk has become a stationary one, for the case that the disk has circular shape, and that the electricity enters it through one wire, and exits through a second one."

The sources of electricity used by Kirchhoff are mentioned in several points in the paper, for example on page 509:

kirchhoff 1845 on sources

"Ich leitete durch die Scheibe der Strom einer constanter Hydrokette, und beruhrte sie an zwei Punkte mit den Enden zweier Drahte, in deren Schliefsung aufser dem Multiplicator eine schwache, auf Kupfer und Zink gebildete, Thermokette eingeschaltet war."

Again, the automatic translation was too uncertain, but with the help of the same native German speaker and after changing 'multiplier' into 'galvanometer' (because that is what the Schweigger and Poggendorff type of galvanometer was called at the time) we get:

"I passed through the disc the current of a constant Hydrokette [battery], and contacted it at two points with the ends of two wires, in the loop of which a weak Thermokette [thermoelectric element] formed on copper and zinc was switched in, onto a [galvanometer]."

A sentence that helps in casting some light on the exact nature of the electromotive forces used in the experiments. 'Hydrokette' and 'thermokette' were also the sources of electricity used by Georg Ohm, who is mentioned at the beginning of Kirchhoff's paper, in his experiments. Ohm experimented with galvanic cells and thermopiles in the early twenties of the nineteenth century; his theory of galvanic circuits, published in 1827 as "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"), was harshly criticized by Georg Pohl and ultimately led to Ohm's resignation from his academic position in 1828. (Ohm was later 'rediscovered' ex-patria in 1841 when the Royal Society in London awarded him the Copley medal as a recognition for his past accomplishments). At the time, the sources of electricity were galvanic cells, chemical batteries and thermopiles, like the stable 'state of the art' electricity source used by Ohm to estabilish his law: a thermocouple based on the Seebeck effect.

Seebeck thermocouple
Seebeck thermocouple with inbuilt primitive galvanometer in the form of a magnetized needle (source: http://dingler.culture.hu-berlin.de/article/pj315/ar315128)

As a matter of fact, thermokette translates into thermopile (German wikipedia - English Wikipedia). For hydrokette, which some dictionaries translate as 'hydrocell' or 'hydro element', I have not been able to find a satisfying English definition, but I am told that "Hydrokette" or "hydroelektrische Kette" is a cell made of two metal plates of different "order" immersed into a conductive fluid. A "Constante Hydrokette" has each metal plate immersed in a different conductive fluid, the two fluids being separated by a diaphragm that allows charge carriers migration but doesn't allow the fluids to mix. It basically is an electrochemical cell. Here is a picture from a Czech presentation showing Ohm's galvanometer and comparing Thermokette with Hydrokette in regard to Ohm's experiment (the drawing on the right might not exactly illuminating, I know, but is compatible with the assumption made)

Thermokette vs Hydrokette
Thermokette vs. Hydrokette in Ohm's experiment (source: https://image2.slideserve.com/4698073/hled-n-kvantitativn-ch-vztah-l.jpg)

The following shows a photograph of the galvanometer (back then called 'a multiplier') used by Ohm with a "thermokette" in place, along with a sketch of the device in use. The voltage was 'fetched' with wires connected to the two copper 'cups' at the base of the the instrument:

Ohm's multiplier with thermokette
Actual apparatus used by Ohm and pictorial representation during its use (sources: left https://www.oocities.org/bioelectrochemistry/ohm.htm, right: https://www.youtube.com/watch?v=fk_BpXlfZ8U )

From the same source of the picture on the left:

"This apparatus was used by Ohm. Current flowing through the metal bar in the center cylinder deflects a magnetized needle suspended above it. The deflection angle is proportional to the current. The source of electric potential is a thermocouple (discovered by Seebeck in 1821). The ends of the thermocouple are heated by steam and cooled by ice-water in the small containers on the tripods."

Not only the EMFs sources mentioned by Kirchhoff in his paper are localised, but he also made sure that the supplied electricity was constant and stable as explained from the very start of the 1845 paper (bold mine):

"I have carried out the necessary calculation on the condition that the state of the disk has become a stationary one..."
source: Annalen der Physik und Chemie LXIV 1845 - p. 497

Use of a constant source of galvanic current in steady state conditions strongly deposes against the possibility that said paper was intended to include time-varying sources of EMF, or even the side effects of self-inductance. Moreover, there are no references to inductive electromotive forces or inductive elements anywhere in the text.

In contrast to that, when in a later paper ("Ueber die Bewegung der Elektricität in Leitern", 1857) Kirchhoff analized a circuit with self-inductance, he was very clear in mentioning "Weber's law of induction". In this 1857 paper, translated in English by P. Graneau and A. K. T. Assis for APEIRON Nr. 19 June 1994 (pp. 19-25) , Kirchhoff finds the local form of Ohm's law as (in modern notation)

formula for Ohm's law

where he clearly considers the total electric field Etot = - grad phi - dA/dt, as in j = sigma Etot and not the conservative component alone Ecoul = -grad phi. In the foreword to the translation the translators comment that "This is essentially Ohm’s law generalized by Kirchhoff to three dimensional conductors and to take into account the effects of self-induction." It is also worth noting that the concept of magnetic vector potential A was introduced by Neumann in a paper published in January 1846 on volume I of the Annalen Der Physik 143 with the title “Allgemeine Gesetze Der Inducirten Elektrischen Ströme” (from this answer).

All thing considered it seems plausible that, had Kirchhoff employed inductive sources of EMF in his experiments leading to his 1845 paper, such a novel form of (time-varying) electromotive force would at least be mentioned explicitly, if not discussed at length. But this is not the case. On the other hand, on p. 510 of the 1845 paper Kirchhoff mentions again "the electromotive force of a thermopile", and on p. 513, right before stating his two celebrated laws, he writes:

"Der Strom einer starken Hydrokette theilte sich in die beiden Arme ACB und ADB..."

That can be translated as

"The current of a strong [battery] parted its way into both arms ACB and ADB..."

I believe I have made my point that in his original 1845 paper Kirchhoff only mentioned and used localized, stationary, and non-inductive sources of electromotive force. Why this is relevant to the question posed here? Because localized EMF sources such as galvanic cells, voltaic batteries, thermocouples, and thermopiles, develop a voltage that can be precisely located at the device terminals (two wires) inside the branch of the circuit they are part of. This is not the case for the delocalized EMF originated by the changing magnetic flux linked by the loop in its entirety.

On what the original formulation of KVL by Kirchhoff says about the location of the sources of EMFs

This is the the original formulation of what today is known as KVL (Kirchhoff Voltage Law) or Kirchhoff Loop Rule, as written by Kirchhoff in his 1845 paper (page 513 of the linked Annalen):

KVL in original 1845 paper

Modernizing the notation (by using R instead of omega to denote the resistance, and n instead of nu for the indexes), the text reads:

2) wenn die Drahte 1, 2, n, eine geschlossene Figur bilden R1 I1 + R2 I2 + ... + Rn In = der Summe aller elektromotorischen krafte, die sich auf dem Wege: 1, 2, n befinden; wo R1, R2, Rn die Widerstande der Drahte, I1 I2,... die Intensitaten der Strome bezeichnen, von denen diese durchflossen werden, alle nach einer Richtung als positiv gerechnet.

Which translates into (bold mine)

2) if the wires 1, 2, n, form a closed figure R1 I1 + R2 I2 + ... + Rn In = = the sum of all electromotive forces located on the path: 1, 2, ...n; where R1, R2, ... Rn are the resistances of the wires, and I1 I2, ...In denote the intensities of the currents that flow through them, all calculated as positive in one direction.

Automatic translation is getting better and better every day, thanks to AI progress. Google Translate's output for the original German text did not need any tweaking:

this translation did not need any tweaking

The locution "located on the path: 1, 2, n" points to sources of EMF located on the path itself, as if Kirchhoff was expressely treating localized forms of EMF. Further confirmation of this comes from a subsequent article, published in the December 1847 volume 72 of the Annalen der Physik und Chemie. The paper, published with the title "Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuching der linearen Vertheilung Galvanisher Ströme geführt wird" has been translated by J. B . O'Toole and republished in IRE Transaction on Circuit Theory (IEEE Trans. CT-5, 4–7, 1958). Here Kirchhoff writes (bold and italics mine):

"Let there be given a system of n wires: 1, 2, . . . , n joined to one another in an arbitrary fashion. If an electromotive force is in series with each of them, the necessary number of linear equations for the determination of the currents I1, I2, . . . , In, flowing through the wires is found by using..."

"It follows from this, among other things, that if we choose two wires from an arbitrary system, the current caused in one by an electromotive force in the second is exactly equal to the current produced in the second by an equal electromotive force in the first."

Thereby confirming he is talking about EMFs localised as series elements inserted in each branch and not about an EMF delocalized along the whole loop. Decomposing the whole loop EMF in the partial contributions along various parts of the loop would require the knowledge of the configuration of the induced electric field in the space occupied by the conductors (knowledge that was not available at the time) or the line integration of the time derivative of the vector magnetic potential A along the various curvilinear segments. Again, while it might be argued that knowledge of a mathematical entity corresponding to the vector potential A transpires from Kirchhoff's 1857 paper, such a feat would have at least required a mention, if not a full explanation.

Finally, I have also found at least one additional source confirming that Kirchhoff formulated his rules addressing localized, lumped, forms of EMF. In his Physics Education paper "Explaining electromagnetic induction: a critical re-examination. The clinical value of history in physics" (J Roche 1987 Phys. Educ. 22 91), science historian J. Roche writes:

"It was shown by G Kirchhoff (1824-87) in 1849 that localised EMFs generally set up auxiliary electrostatic forces, by means of surface charges, in order to establish a uniform current around the circuit (Kirchhoff 1879 pp49-55, 1514). Since these additional fields are conservative, the sum of the net potential differences around the whole circuit will be exactly equal to the sum of the PDs across the localised underlying EMFs only. This is the substance of Kirchhoff’s second network law (Kirchhoff 1879 pp 15-16)"

In summary, not only the experimental setup employed by Kirchhoff, but also the way he expressed his laws constitute evidence that the right hand side of Kirchhoff original KVL equation collects (with due sign) the contribution of all the lumped EMF sources that are present along the path ("auf dem Wege").


Appendix: On what the localized nature of the EMF sources means for the original formulation of KVL by Kirchhoff

A common objection raised by people who believe KVL always applies, even when the electric field is not conservative, is that Kirchhoff Voltage Law was intended from its first inception to encompass all forms of electromotive forces, including the nonlocalized inductive EMF linked by the loop itself. The reason behind this belief is that in most high school and introductory university textbooks KVL is expressed in a form applicable to lumped circuits where the sum of all kind of localized EMFs (including those from lumped transformers and induction generators) is equated to the sum of all voltage drops (including those attributed to lumped inductors).

The problem is that the inductive EMF is a very special kind of EMF that is not like other forms of EMF. For example, the voltage that is measured across the terminals of an inductor (such as a multi-turn coil) is different from the voltage computed along the conductor the coil is made of (something that by itself violates KVL in the loop formed by the coil filament and the jump at its terminals). For further information on this see Ramo Whinnery VanDuzer referenced below.

Original KVL with modern terminology
Fast forward to the twentyfirst century, where we tend to express the laws of electrodynamics in terms of relations between fields, we can see how Kirchhoff formulation of localised EMFs would be expressed today.

First, let's rewrite Kirchhoff's equation by making whatever EMFs that are on the paths 1, 2, ... n explicit by calling them emf1, emf2, ... emfn. If one branch has no localized emf in it, then the corresponding emf will be zero; the same can be said for the resistance. In its expanded form, Kirchhoff's original KVL becomes:

R1 I1 + R2 I2 + ... + Rn In = emf1 + emf2 + ... + emfn

where the emf1, emf2,... emfn are the localized electromotive forces located on the segments 1, 2, ..., n of the closed path.

Now we take the localized EMFs of each branch that are on the right hand side and we bring them on the left hand side. The change in sign reflects that different sign convention we use for generators and for passive elements.

(R1 I1 - emf1) + (R2 I2 - emf2) + ... + (Rn In - emfn) = 0

This is the modern equivalent form of KVL as Kirchhoff formulated it.

In the twentyfirst century we recognize that each term in parenthesis represents the path integral of the total electric field along each of the segments 1, 2,... n, in which the closed path (the loop) has been partitioned. Therefore, in modern terms, the original formulation of KVL says

Sum of path integral of Etot.dl along the branches of our closed figure = 0

The right hand side is zero because we have already accounted for all "der electromotorischen Krafte, die sich auf dem Wege" - all the electromotive forces on the branches 1, 2,... n - and because Kirchhoff did not consider any delocalised source of inductive EMF. Since the path integral along a closed path is the circulation, in modern notation the original form of KVL can also be written as:

circulation of Etot is zero
circulation of Etot.dl = 0
Modern notation for KVL as formulated by Kirchhoff (i.e. using only localized EMF sources)
Note: all localized EMFs appear in the path integral on the left hand side.

Faraday's law introduces an EMF that is not on the path
Faraday discovered, in modern terms, that the circulation of Etot.dl can be nonzero, if there is a variable magnetic flux cut by the surface formed from the closed figure 1, 2, ... n Kirchhoff was arguing about. According to Faraday's law (not Kirchhoff's law), the circulation of the total electric field along the circuit path is equal to minus the flux of the magnetic field B cut by the surface delimited by said closed path:

faraday's law in integral form
circulation of Etot.dl = -d/dt flux of B
Modern notation for Faraday's law in integral form showing the addition of the flux on the righ hand side. Faraday's law introduces an EMF source that does not fit into the path integral of the (total) electric field on the left.

Note that the integral on the left accounts for all the localized electromotive forces (batteries, solar cells, peltier cells, thermocouples...) present in the branches of the loop, but not for the nonlocalized inductive EMF due to the flux cut by your closed circuit, which is the novelty introduced by Faraday. We don't need a new law for every kind of non-inductive EMF out there: they are all accountable for in the circulation integral on the left-hand side. The term on the right, on the other hand (pun intended), is a new addition that breaks the original Kirchhoff's law and considerably extends our knowledge of the electromagnetic field. In its local, differential form

Faraday's law differential form
curl E = - dB/dt

Faraday's law states that the electric field curls (i.e. ceases to be irrotational) in the presence of a time-changing magnetic field. This is not 'just another kind of EMF': it is a fundamental property of the electromagnetic field that has changed our understanding of electromagnetism. It is not a coincidence that Faraday's law is one of the four celebrated Maxwell equations.

The breaking of Kirchhoff's loop rule is confirmed, for example, in the second volume of Berkeley Physics: Electricity and Magnetism 3rd edition by Purcell and Morin:

Purcell and Morin on Faraday vs Kirchhoff
Source: Purcell, Morin third edition, section 7.5

"...Kirchhoff Loop rule (which states that path integral of E.ds = 0 around a closed path) is no longer applicable when there is a changing magnetic field. Faraday has taken us beyond the comfortable realm of conservative electric fields. The voltage difference between two points now depends on the path between them."

Extended KVL and its limits:
KVL is such a useful tool that it has been extended to work with AC and general dynamic circuits. Still, even in the 'modified', or 'generalized', or 'extended' version, in order to hold (and not to break other circuital laws, such as Ohm's law in its own generalized form) KVL requires that the sources of inductive EMF be lumped into components that will be located along the circuital path. In its extended or generalized form, KVL can accept both the contributes of lumped inductive EMFs and 'voltage drops' of lumped self-inductors in the path integral on the left of the equation.

In order for extended KVL to work in a loop, it is necessary that voltages for any two points in the loop be uniquely defined. This demands for no changing flux linked by the loop itself, and in turn requires that all changing magnetic flux be confined inside the magnetic components with no appreciable leakage in the loop itself. For further information, the reader can refer to Ramo, Whinnery, VanDuzer, "Fields and Waves in Communication Electronics", third edition (chapter 4: the electromagnetics of circuits, in particular pp. 174-179). Here is a most relevant excerpt:

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Source: Ramo Whinney VanDuzer, "Fields and Waves for Communication Electronics" 3rd edition, p. 179

Extended KVL works when the sources of EMF can be lumped (and therefore localised on a particular branch) and the changing magnetic flux can be hidden inside the component, but we cannot make it work when the EMF is due to the changing flux linked by the circuit path itself. That will break KVL for good, and we are forced to consider the more general law: Faraday's Law.

Note that trying to bring the delocalized inductive EMF contribution represented by the surface integral of B on the right, incorporating them inside the path integral of E on the left side in order to distribute it along the branches will alter the integrand field changing it from the total electric field Etot to the conservative part alone Ecoul of such field. This partial constituent of the (total) electric field will admit a potential function (the scalar electric potential phi) that will obey KVL but will break Ohm's law in its local form.

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  • $\begingroup$ Your reply is too lengthy to critique in the comments section. Please read the edit within my own reply. $\endgroup$ Dec 7 '21 at 19:29

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