"What about the other forms of EMF, uh, what about 'em(f)?"A common objection raised by KVLers is that Kirchhoff Voltage Laws was originally intended to include all forms of electromotive forces, including the nonlocalized inductive EMF. They mention the high school version of KVL that puts all EMFs on one side of the equation and all voltage drops on the other. From their point of view, considering the inductive EMF as a special case, would imply having a new law for every new kind of localized EMF in the loop. The problem is that the inductive EMF is a very special kind of EMF that is not like other forms of EMF.
If we go back to the original formulation by Kirchhoff in the 1845 Annalen der Physik und Chemie,
link
https://i.postimg.cc/5tzVMxGx/screenshot-2.pnghttps://books.google.de/books?id=Ig8t8yIz20UC&pg=PA494&hl=de&source=gbs_toc_r&cad=3#v=onepage&q&f=false:
he indeed wrote (I am using R instead of omega to denote the resistance, and n instead of nu):
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. Here is Google Translate's output for the original German text:
link
https://i.postimg.cc/CLJMvxyX/screenshot-3.pngThe locution "located on the path: 1, 2, n" points to sources of EMF located on the path itself and I have found at least one source confirming that Kirchhoff formulated his rules by using 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)"
At the time the sources of EMF were galvanic cells, chemical batteries and thermopiles, like the state of the art electricity source used by Ohm to estabilish his law: a thermocouple based on the Seebeck effect (Ohm experimented in 1827, was basically discredited by Georg Pohl, and later 'rediscovered' in 1841, when the Royal Society (in London) awarded him the Copley medal in recognition of his accomplishments).
According to the above text, the right hand side of Kirchhoff's original KVL equation collects (with due sign) the contribution of all the lumped EMF sources that are present along the path (on the path, on the way, "
auf dem Wege"). He even specifies "
all electromotive forces located on the path 1, 2, ... n", the segments into which the path (the CLOSED FIGURE) has been partitioned.
Original KVL with modern terminologyFast forward to the twentyfirst century, where we know something more about fields and we can even extend Kirchhoff's law to work in the presence of variable magnetic fields (but only when we can lump the inductive EMF along the path).
First, let's rewrite Kirchhoff's equation by making all EMFs located 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 resistances. 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 has been partitioned. Therefore, in modern terms, the original formulation of KVL says
Sum of path integrals 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 (basically, he was experimenting with batteries and thermopiles). 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.dl = 0 (this is KVL as formulated by Kirchhoff)
(all localized EMF appear in the path integral)
Faraday's law introduces an EMF that is not on the pathFaraday 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 time derivative of the flux of the magnetic field B cut by the surface delimited by said closed path:
circulation of Etot.dl = -d/dt flux of B ( this is Faraday slaying Kirchhoff)
(introduces an EMF that does not fit into
the path integral)
Note that the integral on the left still accounts for all
der electromotorischen krafte --- I mean,
all the localized electromotive forces you can imagine (batteries, solar cells, peltier cells, thermocouples...), but it does not include the nonlocalized inductive EMF due to the flux cut by your closed circuit. Therefore, we don't need a new law for every kind of non-inductive EMF out there: they already are all accounted for in the circulation integral on the left.
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 states that the electric field curls (i.e. ceases to be irrotational) in the presence of a time-changing magnetic field. This is a general property of the EM field, not just 'another kind of EMF'!
The breaking of Kirchhoff's loop rule is not something Lewin came up with on his own. It's part of standard classical electrodynamics and is commonly acknowledged in EM books for physicists and engineers, for example in Haus and Melcher, Ramo Whinney vanDuzer, Brandao Faria, Rosser...
In particular, a respected introductory textbook that makes this explicit is the second volume of Berkeley Physics: Electricity and Magnetism 3rd edition by Purcell and Morin:
link:
https://i.postimg.cc/kGXzD9jR/purcell-faraday-2.jpgSource: Purcell, Morin third edition, section 7.5
...
Kirchhoff Loop rule is no longer applicable when there are variable magnetic fields inside the circuit's premises.
Extended KVL and its limits: We can even push it a little further and accept in the path integral on the left even lumped inductive EMFs: it's what most textbooks (like Hayt, for example) call "extended KVL". It works when the source of EMF can be lumped and the changing magnetic flux can be hidden inside the component: the key is to alter the circuit path so as to exclude the variable magnetic region from it. Still, this trick cannot possibly work when the EMF is due to the changing flux linked by the circuit path itself (like in the example proposed by Romer and popularized by Lewin). That will kill KVL for good, and we are forced to consider the more general law: Faraday's Law.
(And no, trying to bring the delocalized inductive EMF contribution on the left side and distributing it along the branches won't work and I'll explain why in the post about the 'tiny batteries' model)
I already posted a solved example: Lewin's ring with a battery.
Next stop: the tiny battery model and how it related to the McDonald Manouver.