Check the attached pdf from Electromagnetics by Notaros, pages 279-280, example 6.6.
It is a curious coincidence that Notaros seems to have been mentored by the late Branko Popovic (he is the first one he thanks in the preface, and the age difference means he must have learned from him), the author I point to for the best definition of voltage from a didactic point of view. In fact, Popovic does a better job of defining voltage in a way that does not lend itself to confusion. Let me describe you the "
Popovic manouver" using my convention for symbols, and by calling the electric scalar potential phi to make it crystal clear which is which.
On voltage and scalar potential differenceThis is the definition of
voltage as work done per unit charge to move from point A to point B along a given path \gamma. It is also the convention adopted by the IEC and it requires one to compute the path integral of the TOTAL electric field along the path \gamma from start point A to endpoint B (fun fact: the reason for the inversion of B and A in V
A->B = V
BAgamma is that when the electric fields admits a potential function we have
E = -
grad phi and the two minus signs cancel themselves, the rest is basic integral calculus)
Since the TOTAL electric field is in general non conservative, the path integral will in general depend on the particular path joining A and B. Therefore there is not a potential function V(P) such that the path integral can be expressed as the difference between the values assumed by V in the endpoint and startpoint (which, considering the minus sign
would have given an expression of the form V(B) - V(A) = V
BA ) .
In this context I am using the notation "V
ABgamma" enclosed in quotes with the addition of a reference to the path \gamma to remind us what that voltage would be called
IF we could treat it as a potential difference.
By the sheer power of superposition, we can mathematically decompose the total electric field
Etot into its two components
Ecoul and
Eind.
Etot =
Ecoul +
EindThe first is the conservative (or irrotational, or laminar) part that is due to the coloumbic force, while the latter is the nonconservative (or rotational, or solenoidal) induced electric field associated with the changing magnetic flux. Since integrals are linear operators, we can split the path integral that expresses voltage in two contributes
Now, the conservative part of the electric field
Ecoul admits a potential function that we call the
electric scalar potential phi (I am using this notation on purpose to make it clear it is not the non-existent potential function for the total electric field Etot), so we can express that component of voltage as a scalar potential difference phi(B) - phi](A) = phi
B - phi
A = phi
BAHere I made the signs explicit, so that it is clear that the actual path-dependent voltage V - the one Ohm's law works with - is the sum of two components: the path-independent difference in electric scalar potential phi
BA and the contribute of the inducted electric field (that can be expressed as a function of the magnetic vector potential
A).
The partial component phi
BA is the voltage that KVLers believe is the real and true voltage, and if we express it in this way
it is clear that we are subtracting a partial component from the actual 'full' voltage. V
A->B = V
BAgamma is the voltage that gives the full picture as it is expression of the one and only total electric field
Etot that electrons and other charges can sense. While phi
BA = phi(B) - phi](A) is just a partial component that gives you only half of the story.
As a side note, if we wish to make use of the magnetic vector potential
A, the formulae become:
and (spoiler alert!) this is the key to understand where that 'tiny batteries' model comes from.
Yeah, yeah, it's voltage sources since they are generally time-varying, but if we fix a particular instant in time we can represent them with battery symbols, they make it easier to see the polarities without cluttering the drawings.
The source(s) of confusionSome textbook authors define voltage in the chapters for electrostatics as the difference in electric potential (because when there are no changing magnetic fields, using the conventions above we have V
BAgamma = V
BA = phi
BA ). The problem with this approach is that instead of using a dedicated symbol such as phi for the electrostatic potential, they use V from the start - thereby allocating the symbol V for the path-independent voltage of electrostatics. Then, when they reach the quasi-static electrodynamics chapters they can't use V for the more general path-dependent voltage, and they are therefore forced to give two physical quantities to fully describe their systems (either V and
A, or
Ecoul and
Eind). This is not a big deal, as long as one understands that in this context the path-independent component of voltage only gives a partial description of the physical system. The Helmoltz decomposition is well known and used in electrodynamics, but to completely describe your system you need to specify both the electric scalar potential \phi (that some authors call V) and the magnetic vector potential
A.
- Notaros says that explicitly on p. 277 of his 2010 "Electromagnetics" textbook, formula 6.43: E(t) = - dA/dt - grad V. He writes: "We see that both potentials are needed for E..." and what V is we can see from eq. 6.18 on page 269: Eq = - grad V, where Eq is the conservative part of the electric field (Ecoul in my notation). In Notaros' notation V is the scalar electric potential , i.e. what I called phi above.
- McDonald does so in his note on page 6: "In time-dependent situations, such as the present example, the electric field E is related to both the scalar potential[] V as well as to the vector potential A according to E = -grad V -dA/dt", same formula for decomposing the total electric field into the conservative and solenoidal components E = EV + EA. McDonald also goes the extra mile in acknowledging that he suggest to call 'voltage' the scalar potential difference (but he gets a little muddy with the terminology):
"While some people designate the integral of E · dl from a to b as a 'voltage drop', we advocate calling this the EMF between points a and b, and that the “voltage drop” between points a and b be reserved to mean simply Va − Vb, the difference in the scalar potential between the two points". And one day, when Arrow, Farnell, Mouser, and the likes will sell voltmeters capable of reading the scalar potential difference, instead of the actual voltage, I am sure everybody will be happy to follow his advice.
- Belcher does not even get into defining potentials, but he makes use of the Helmoltz decomposition because he gives both fields on the ring: Ecoulomb and Einduced. One field component alone is not enough to describe the system. Just look at the pictures. He just does not compute the voltage, possibly to keep a neutral position and not expose himself to any of the backlash that had hit Lewin by that time. Personally, I think he should have shown more courage.
Source: Belcher's note available on Electroboom's channel
Also note that Belcher, like Lewin, agrees that the resultant electric field in the wires is zero (for perfectly conducting wires): "This will leave us with no electric field in the wire, but electric fields in both resistors, and these are the electric fields we found above in (4), where we begin with the assumption that the current is the same everywhere in the circuit."
- Feynman uses the scalar and vector potentials in a more general setting on p. 15-16 he writes:
"once we have A and phi , we get B from curl A, as before, and E from −grad phi − dA/dt."
Like all the others he knows you need both potentials to determine the electric field E. The one and only electric field that a charge can experience, and that obeys the local form of Ohm's law.
KVLers, on the other hand, believe that the scalar potential difference phi
BA is the only one and true voltage between points A and B, and that if you obtain a different value then it must mean you are committing a probing error. No, probing has nothing to do with the fact that
the scalar potential difference alone only gives a partial description of your physical system when variable magnetic fields are present. This partial description is also what is represented by the 'tiny batteries' model of inductive components. This model and why it represents only half of the story, is the subject of another post, but a sneak preview is contained in this picture: