Why Voltage is NOT just Potential Difference 

Feline minds think alike. 🐈😀

Thank you. I remember reading your excellent comments on Electroboom's channel. I believe we shared some fight together, but at the time I used a different handle.

Thanks for the feedback. Yes, all of this is pretty basic physics, and that is why I was surprised to witness such a fierce opposition to the idea that voltage (a path integral!) could be a path dependent quantity. I had a look at the English, French, Spanish, German, Italian pages of Wikipedia where voltage is defined, and so far only the German page has the correct, complete definition that agrees with the IEC definition of voltage (next, upcoming video).

Well, the last part was a bit too condensed since the full explanation of what is meant by quasi-static - and in particular electro-quasistatic and magneto-quasistatics would have lengthen the video to a total duration of about 45 minutes. And that would have been too much. I plan to make a dedicated video on quasistatics and the orders of approximation in the solution of Maxwell's equations, but that won't be anytime soon. I have three videos almost ready (IEC notation, what a voltmeter measures, how a voltmeter measures what it measures) before even considering that. So, don't be worried by the fact that it wasn't clear, because due to excessive summarization, it ended up not being clear. Maybe if I added Maxwell's equations for each case it would have been better - but I figured "these are just concluding observations..."

@@copernicofelinis Mate, you have done wonders for me. I was wondering about "EMF around a circle", and I fully got that now. I have seen a professor talk nonsense, and you made perfect sense. I really do appreciate your efforts. I trust you. Subscribed. I learn so much. Thank you!

Actually, AC circuits are covered by the quasi-static approximation of Maxwell's equations (Look at the figure in the third note at the end: they are firmly under the quasi-static column). Translation: this video is about AC circuits.

Glad it was helpful!

I thought I had already added them in the video description, but I didn't. I did it a few minutes ago. Here's the list: Edward M. Purcell, David J. Morin Electricity and Magnetism 3e 2013, Cambridge University Press Branko D. Popovic Introductory Engineering Electromagnetics 1971, Addison Wesley Zoya Popovic, Branko D. Popovic Introductory Electromagnetics 1999, Prentice Hall Kenneth R. Demarest Engineering Electromagnetics 1998, Prentice Hall Markus Zahn Electromagnetic Field Theory: A Problem Solving Approach 1979 Wiley, 2003 Krieger Publishing Company Herman A. Haus, James R. Melcher Electromagnetic Fields and Energy 1989, Prentice Hall J. A. Brandão Faria Electromagnetic Foundations of Electrical Engineering 2008, Wiley David J. Griffiths, Introduction to Electrodynamics 3e 1999, Prentice Hall Wolfgang K. H. Panofsky, Melba Phillips Classical Electricity and Magnetism 2e 1962, Addison Wesley (reprinted 1983, Dover) Hans C. Ohanian Classical Electrodynamics 2e 2007, Infinity Science Press John D. Jackson Classical Electrodynamics 3e 1999, Wiley Guest Stars: John D. Kraus Electromagnetics 2e 1973, McGraw Hill William H. Hayt, John A. Buck Engineering Electromagnetics 4e 1981, McGraw Hill

@@copernicofelinis great thanks!

The next video ("The IEC definition of voltage") will still be on the boring side, but the one after that ("What a voltmeter measures") will be really interesting with actual practical implications and little to no math.

The problem with the analogies required to reduce the concept to laymen's terms is that they always leave out some key aspect, or worse incur into contradiction. In my opinion the surefire way to understand voltage is from the definition I gave in my previous video (Voltage for grown-ups): it is work (energy, if you will) per unit charge, and this work is in general path dependent. Here is an analogy: the "gaseolage" is the amount of gasoline per unit passenger that a car requires to go from A to B. As you well know, this amount depends on the road you choose. And even if all roads from A to B had the same length, in a world where there is friction and air resistance, "gaseolage" like voltage will still depend on the particular path. Now, this analogy can be made more stringent by considering spring powered cars, running on smooth rails so that in the absence of air resistance "gaseolage" would reduce to height difference (or rather a difference in gravitational potential energy per unit passenger) - much like voltage would reduce to a difference in electric potential energy per unit charge. The path dependency comes into play when we introduce air vortexes (analogous to the circulating lines of the solenoidal electric field associated with changing magnetic fields) but... It's getting a bit complicated and I am not sure this explanation is simpler than the original one...

The potential difference is defined between any two points, it is simply the difference in the electric field strength at the two points. The voltage between two points, however, is non-sensical. You must define voltage along a path, the journey matters if you will. This is why there can be a voltage across a loop, even though it ends and starts at the same point. This all only comes into effect in the dynamic case (ie. changing electric fields), and in the static case this "correction" vanishes and potential difference equals voltage.

Yes, understood. Unless I continue on from Pre-calculus I may have to use Mathematica or some other math simulator to follow along. Your measurement at two different points on the same conductor was an interesting complement to your articulated points. I also found it helpful to dive into the transcript to tackle the vocabulary I lack practice with.

@@gary.richardson The electric field is a vector field, with an arrow at each point. This field (when no charge is moving) has the remarkable property that it is conservative. This means there is no "curl" in the field (a term perhaps to google as an image speaks a thousand words: vector field curl). This allows us (although not always, we get lucky with the e field) to create a potential function for the e field. We have to do integration and all sorts to find this function but simply put, there exists a function of (x,y,z) position such that it spits out a single number for each position. The gradient of this surface at a point gives us back the vector e field at that point. I'm sure there are very good videos on this, I think 3b1b has done great videos on vector calculus. The short of it is that as we can have this function that is uniquely defined at each point, it doesn't matter how we travel between points, the change in this function will just depend on the start and end point. This difference in potential is named as such: potential difference. Analogously we can think of a mountain or other landscape. When we climb we move to a spot with higher gravitational potential, and when we descend we move to a position with lower gravitation potential. As this potential is a function of the spot only, the change in potential does not matter on the journey, but just the start and end points. Unfortunately, in the case with moving change (hence magnetic fields) these potential analogies break down. This is as we can now get curl in the field (it is this curl that induces current in coils in changing magnetic fields and upon which all AC electronics depends) and it is no longer conservative. This means we cannot make a potential from it, and the whole landscape goes out the window. Instead we must define a voltage, which is path dependent. It is defined with the complex integral in the video, and equates to adding up the small bits of the field along the path. When you take this formula and set the change in magnetic field to zero, you get out a function that is identical in value to the values from the potential function; the voltage with no moving charge is the potential difference. I hope that helps give a broader picture of the maths, if you have any questions do ask. It helps me to better understand it when I explain it!

Thanks!

In quasi-static there is no propagation of potentials (or fields). So, there are no waves to which associate a wave vector. Can you be more specific about what you mean?

Fourier transform in space, the wavenumber is just the transform variable of position. I didn't mean to imply a Fourier transform in time.

Did you mean at 9:45? Yeah, I agree they are concomitant in that they come out of the particular decomposition of the electromagnetic field in a given frame of reference. Sometimes concise expressions can be imprecise, as in this case. I also consider voltage and current to be two sides of the same coin but from time to time I find it more concise to say "the current flowing in the resistor R will produce a voltage..." or vice versa.

@@copernicofelinisYes I would say that the change of a magnetic field correlates to an induced voltage in a conductor perpendicular to the field, but I’m not sure if it’s [always] the cause. Say for example we have an ideal toroid transformer of very high permeability. We could expect some leakage flux at the primary winding with little to no leakage flux at the secondary. Yet we still have an induced voltage in the secondary proportional to the rate of change of B, even though no line of magnetic flux touches the secondary coil. Is this action at a distance? Or does it involve the magnetic vector potential, and if so does that correlate to it being a physical reality?

@@cosmicyoke I tend not to read cause and effect in any physical equation unless there is an explicit reference to time in it. I consider them to be just relationships between physical quantities. So I do not consider changing electric fields to cause magnetic fields or changing magnetic fields to cause electric fields (even if sometimes - actually often - I use colloquial imprecise language for brevity). For example, in the case of the toroid: there is essentially zero magnetic field outside if it, but there is an associated curling electric field all around it and it's that induced electric field that is responsible for the current flowing in a loop of copper around the core. We tend to see this as a chain of cause +effects ("the changing flux produces the induced field and this in turn produces a current") when in reality these are concomitant effects and not really one the cause of the other. Regarding the last part of your comment, in classical physics there are three ways to describe EM phenomena: via fields, via potentials and ultimately via the position, velocity and acceleration of charges. I like to see this last one to be more physical; fields are an expression of how charges are disposed and move, while potentials are a mathematical aid that lead to simplified equations. But all three ways are essentially equivalent. In quantum physics it seems that the potentials are the most relevant descriptor of reality, but I'm not getting into that black magic here :-)