Insights A New Interpretation of Dr. Walter Lewin's Paradox - Comments

[arydberg]

I did this experiment and it shows what Dr. Lewin stated. I took a 500 foot roll of wire used for dog fencing. ( it measured 5 ohms). Unspooled it, Cut it in half and rewound 250 feet back on the spool. I brought out a pigtail and connected it to a 1meg resistor. The other end of the resistor went to the 2nd 250 foot wire. I then wound the 2nd piece on the spool in the same direction. and connected a 100 K resistor to the beginning and end of the coil.

I put 2 strong magnets in the center of the coil and snatched them away and got a pulse on my Oscilloscope. The pulse on the 1 meg resistor was 10 times that on the 100 K resistor. I have 2 pictures and hopefully included them.

 

[rude man]

I did this experiment and it shows what Dr. Lewin stated. I took a 500 foot roll of wire used for dog fencing. ( it measured 5 ohms). Unspooled it, Cut it in half and rewound 250 feet back on the spool. I brought out a pigtail and connected it to a 1meg resistor. The other end of the resistor went to the 2nd 250 foot wire. I then wound the 2nd piece on the spool in the same direction. and connected a 100 K resistor to the beginning and end of the coil.

I put 2 strong magnets in the center of the coil and snatched them away and got a pulse on my Oscilloscope. The pulse on the 1 meg resistor was 10 times that on the 100 K resistor. I have 2 pictures and hopefully included them.

Dr. Lewin's data was never in contention. That was not the issue. It's his explanations that were wrong, in particular the statement that "Kirchhoff was wrong". Kirchhoff's laws hold in all cases. They refer to voltage drops, which is not necessarily the same as measurements using voltmeters.

A voltmeters always correctly measures the voltages it "sees" but this voltage can be artificially induced by the voltmeter and its leads and is thus not the voltage in the absence of the voltmeter and its leads. For example, in Lewin's setup there is a voltage between any two points along a wire not including a resistor, yet the voltmeter reads zero.

The sum of voltages along any closed path is always zero irrespective of the nature of the emf generating them.

 

[anorlunda]

Kirchhoff's laws hold in all cases.

That's an overly broad statement. There are important assumptions. If those are violated, then Kirchoff's Laws can not be used. See the Insights Article.

https://www.physicsforums.com/insights/circuit-analysis-assumptions/

 

[rude man]

I think I disagree with your #1. The Lewin setup includes time rate of change of magnetic flux outside a conductor being non-zero if I'm interpreting your statement per your intention, yet there Kirchhoff's laws certainly hold.

I agree with the rest. Quasi-stationariness must be assumed, ortherwise lumped-circuit anaylis laws have to be superseded by Maxwell's equations. A radiating circuit is one example, as is a distributed circuit.

But that is not the discussion here.

Reference https://www.physicsforums.com/insights/circuit-analysis-assumptions/

 

[anorlunda]

The Lewin setup includes time rate of change of magnetic flux outside a conductor being non-zero if I'm interpreting your statement per your intention, yet there Kirchhoff's laws certainly hold.

At minutes 30-35 in the video, he says that Kirchoff's Laws only apply when the external magnetix flux is zero. And since the flux is nonzero in his experiment (assumption #1) you can't use Kirchoff's Laws or circuit analysis to describe that experiment. Well duh.

Bottom line, you can't say that KVL and KCL apply always.

By the way, be careful when you say you don't agree with those assumptions. They are repeated in many standard textbooks. Peer reviewed journals and standard textbooks are the bible here on PF.

 

[rude man]

At minutes 30-35 in the video, he says that Kirchoff's Laws only apply when the external magnetix flux is zero. And since the flux is nonzero in his experiment (assumption #1) you can't use Kirchoff's Laws or circuit analysis to describe that experiment. Well duh.

Bottom line, you can't say that KVL and KCL apply always.

By the way, be careful when you say you don't agree with those assumptions. They are repeated in many standard textbooks. Peer reviewed journals and standard textbooks are the bible here on PF.

Voltage is the line integral of the electrostatic field. And the circulation of that field is zero.
I suggest perusal of the two papers by Princeton's K. McDonald I cited in my Insight article on this subject.
And if I may counter with my own "standard textbook": Fundamentals of Electric Waves by Stanford's H H Skilling. Let me know if you need chapter & pages.

 

[Charles Link]

@anorlunda and @rude man I think you both may be arguing the very same thing, and it is open to debate whether the EMF generated in a loop by a changing magnetic field is part of Kirchhoff's voltage laws (KVL), or if it happens to be an exception that Professor Walter Lewin has highlighted. Others have previously argued this fine detail: See https://www.physicsforums.com/threa...ge-across-inductor.880100/page-5#post-5533643 . $\\$ Right around post 83 @Dale and @vanhees71 went back and forth on this a couple of times, but I think everyone is in agreement on how this problem gets solved, and it is very useful that Professor Walter Lewin has pointed out this special case, even if he says a couple of things that perhaps also aren't 100% accurate.

 

[anorlunda]

I remember that thread. It got really heated among several people who really know their stuff. You may be right that it's semantics.

I really don't care enough about Professor Lewin to go down that rabbit hole, so I'm going to exit this conversation.

 

[rude man]

Pace nobiscum.

 

[vanhees71]

Well, there's nothing to fight about. I think there's no paradox at all (I don't like the word "paradox"; it just indicates a lack of careful analysis based on "common knowledge", which is contrary to the very basic principles of basic science). Just use Maxwell's equations, and everything is fine. Also avoid to talk about "voltage" as soon as emf's from time-varying magnetic fields are involved. BTW Contrary to the statement in the above cited Insight article, of course there are both electric and magnetic fields in stationary circuits (in fact there's only one electromagnetic field in nature, but that's another story). The only thing is that one can eliminate them from the considerations using the stated assumptions and lump everything in currents, voltages and emf's. That's because Kirchhoff's laws are nothing else than the integrated version of Maxwell's equations under the simplifying assumptions made.

 

[Charles Link]

@vanhees71 makes a very good point, you can no longer use the term "voltage" when you include a circuit loop that has a changing magnetic field inside of it. $\\$ The "voltmeter" does not measure a "voltage" difference in this case, between the two points on the circuit that it probes.$\\$ Instead, the voltmeter needs to be considered for what it actually is=a couple of wires with a large resistor through which a small current flows. In this case, the voltmeter really doesn't "measure". Instead, it gives a reading which is the (multiplicative) product of the small current times the large resistor. The placement of the wires that form the leads of the voltmeter can yield different results depending on whether the circuit loop that they form encloses the changing magnetic field, in which case there is an EMF around that circuit loop.

 

[vanhees71]

Another equivalent view is that the EMF drives the electrons in the wires making up the volt meter (think of an old-fashioned galvanometer for simplicity), leading to the current @Charles Link mentioned in the previous posting.

This becomes clear if one uses the complete (!) integral form of Faraday's Law of induction. Its fundamental form is, as anything in electromagnetism, the local form in terms of derivatives (SI units):
$$-\partial_t \vec{B}=\mathrm{\nabla} \times \vec{E}.$$
Now if you integrate this over an arbitrarily moving area $A$ with boundary $\partial A$ you can first use Stokes's theorem. The only correct version of this simple treatment is
$$-\int_{A} \mathrm{d}^2 \vec{f} \cdot \partial_t \vec{B} = \int_{\partial A} \mathrm{d} \vec{r} \cdot \vec{E}.$$
Now one likes to express in terms of the magnetic flux through the area
$$\Phi_{\vec{B}}=\int_A \mathrm{d}^2 \vec{f} \cdot \vec{B}.$$
Now, if the area and its boundary are moving, you cannot take the partial time derivative out of the integral in the previous equation but you get an additional line integral along the boundary curve of the surface, which you can lump to the integral on the right-hand side. Taking Gauß's Law for the magnetic field, $\vec{\nabla} \cdot \vec{B}=0$ into account the resulting equation gets
$$-\dot \Phi_{\vec{B}} = \int_{\partial A} \mathrm{d} \vec{r} \cdot (\vec{E}+\vec{v} \times \vec{B})=:\text{EMF},$$
where $\vec{v}$ is the velocity field along the moving boundary of the area we've integrated over.

Now if you choose the area such that its boundary is along the wires connecting the volt meter with the rest of the circuit, what it measures is in fact the electromotive force, i.e., the line integral along the closed (!) boundary. It's obviously the line integral over the force on a unit charge $\vec{E}+\vec{v} \times \vec{B}$, and this shows that indeed that's the physical picture on what's measured given above: The force on the charges inside the wires connecting the volt meter with the rest of the circuit (including the wires making up the coil in the volt meter, if you take the model of a old-fashioned galvanometer setup).

This considerations also explain why the reading of the volt meter is beyond the simple Kirchhoff circuit theory: It's reading cannot be understood without taking into account the correct geometry of the connection of the volt meter with the rest of the circuit since this you need to calculate the line integral defining the EMF, which is what the volt meter measures. The Kirchhoff theory becomes applicable only if you make the wires connecting the volt meter very short, so that the magnetic flux through the corresponding current loop becomes negligibly small. Then the reading is what you expect according to the Kirchhoff circuit theory, i.e., the EMF through the element of the circuit you want to measure (which may be an Ohmic resistor, a capacitor, or coil).

Note: Another source of confusion is the very name "EMF" for the line integral: Here force is obviously not the modern notion of "force" (which is represented by the Lorentz force per unit charge, $\vec{E}+\vec{v} \times \vec{B}$) but its meaning is more in the sense of "energy". Indeed the EMF is a line integral of the force along a closed loop. The very fact that the quantity is a line integral along a closed loop shows that it is NOT a "voltage". If there'd be a potential for the force integrated over, the integral over any closed loop is 0 (modulo the caveat that the region under consideration is simply connected!).

 

[Henryk]

There is no paradox whatsoever!. I admit, it took me a while to understand what is going on.
First thing first, the loop with two resistor is a red hearing. So, let's remove it and we get a circuit like that:

Now, we have a loop containing two voltmeters encircling flux change of 1 Wb/s. Obviously, the induced EMF is 1 Volt and the direction is indicated by the circular arrow. With the way the voltmeters are connected, the one on the right would show a positive voltage, the other negative voltage, just like in the video.
How much will each of them show?. That would depend on the internal resistance of the voltmeters. Portable meters have resistance of 10 Mohm, if both have this value, one will show 0.5 V, the other -0.5 V. Change the internal resistance of the left voltmeter to 1 Mohm and the other one to 9 Mohm and you will get -0.1 v and 0.9 V. No paradox, just a red herring.
However, Dr Lewin makes a statement in his video that I would disagree. He says that the Kirchhoff (second) law is not valid. The way I was thought physics, it is still valid. I understand that the Kirchhoff law says that for a loop $\sum I_k R_k = \sum EMF$ and that actually agrees with the Faraday law.
Now, I would also like to point out that the supposed tutorial contains some false statements. One of the false statement is

"which is non-conservative in the sense that its circulation is non-zero. E_m can be created by a chemical battery, magnetic induction, the Seebeck effect, and others."

This statement is not correct. The non-conservative electric field can only be created by changing magnetic flux. The field inside a battery is conservative. How is it created.

The key to understand operation of a battery is thermodynamics and equilibrium condition for particle exchange. Thermodynamics tells us that a system is at equilibrium with a reservoir with respect to particle exchange if the chemical potentials are equal. Let's take, for example, an alkaline battery. It consist of a zinc cathode, MnO anode and KOH electrolyte. KOH in solution dissociates into K⁺ and OH^- ions. At the cathode, the following reaction takes place ( see https://en.wikipedia.org/wiki/Alkaline_battery )

Zn(s) + 2OH⁻_(aq) → ZnO(s) + H₂O + 2e⁻

The reaction of solid Zn with OH^- ions produces ZnO, water and free electrons. Where do the free electrons go? they go to the Zn metal charging it up negatively, i.e. increasing the chemical potential of electrons in the metal. The reaction stops when the chemical potential of electrons in the Zn metal become equal to the chemical potential of the electrons attached to OH^- ions. The net result is formation of a potential difference at the electrode/electrolyte interface. This is not unlike creation of the depletion layer in a p-n junction of a semiconductor.
Similarly, there is a potential step created at the anode. The total voltage of an (open circuit) battery is algebraic sum of the two voltage steps.
Seebeck effect, photovoltaic cell EMF can also be understood considering the thermodynamics, that is, EMF is created by a gradient of chemical potential of electrons and the field is conservative.

 

[rude man]

@vanhees71 makes a very good point, you can no longer use the term "voltage" when you include a circuit loop that has a changing magnetic field inside of it. $\\$ The "voltmeter" does not measure a "voltage" difference in this case, between the two points on the circuit that it probes.$\\$ Instead, the voltmeter needs to be considered for what it actually is=a couple of wires with a large resistor through which a small current flows. In this case, the voltmeter really doesn't "measure". Instead, it gives a reading which is the (multiplicative) product of the small current times the large resistor. The placement of the wires that form the leads of the voltmeter can yield different results depending on whether the circuit loop that they form encloses the changing magnetic field, in which case there is an EMF around that circuit loop.

Yes. Your resistive voltmeter shows the field equivalent of Ohm's law which is ir = d(E_s + E_m) with d the length of r. This reduces to ir = dE_s if d << voltmeter wire lead lengths. In my various posts I had made this assumption.

 

[rude man]

BTW Contrary to the statement in the above cited Insight article, of course there are both electric and magnetic fields in stationary circuits (in fact there's only one electromagnetic field in nature, but that's another story)..

E_m is also an electric, not a magnetic field. Two E fields: E_s and E_m. One begins and ends on charges; the other does not.