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Apocalympics 2016 at Rio

*Formerly "Apocalympics2016".* A subreddit to document happenings surrounding the 2016 Olympics in Rio de Janeiro, Brazil - and to take note of any disasters, foreseen or unforeseen - surrounding the event.
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Apocalympics2016

A subreddit to document happenings surrounding the 2016 Olympics in Rio de Janeiro, Brazil - and to take note of any disasters, foreseen or unforeseen - surrounding the event.
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Kirchhoff's voltage law, Faraday's law, and electromotive forces

This post is about ElectroBOOM's disagreement with Walter Lewin. Spoiler alert: both are right under their respective premises, both sides misrepresent the other side. The misrepresentation is implicit in ElectroBOOM's video, which ignores the objective of Lewin's experiment. ElectroBOOM raises a valid point, but is dismissed by Walter Lewin as incompetent. This is unfortunate since a response to ElectroBOOM's video complements Walter Lewin's demonstration, provides further insight into the underlying physics, and is an opportunity to discuss some misconceptions about Faraday's law.

Lewin's experiment

The debate is about Lewin's demonstration in his MIT lecture 8.02–16, which is based on an article by Robert H. Romer. Voltmeter measurements on a simple electric circuit surrounding the changing magnetic field of a solenoid are predicted and experimentally verified. It is concluded that the electric potential between two points in the circuit is undefined, and that the loop integral over the electric field along the circuit does not vanish. For the history behind the experiment see the extensive references given in Kirk McDonald's notes on Lewin's experiment.

ElectroBOOM's objection

Measured voltages between two points in the circuit are well defined when we take the effect of the induced electric field on the probe wires into account. Kirchhoff's voltage law still holds for these voltages.

Kirchhoff's law

Lewin and ElectroBOOM use different versions of Kirchhoff's law. Lewin's version is the condition for conservative fields that the loop integral over the electric field vanishes. It is Faraday's law for the special case of a constant magnetic flux through the loop. In accordance with Lewin, I will call this version Kirchhoff's loop rule (KLR).
ElectroBOOM's version is the one given in Feynman's lecture notes for AC circuits (including the special case of steady currents), which sums over voltages (including electromotive forces) that are measurable with a voltmeter over the circuit elements. I will call this version Kirchhoff's voltage law (KVL).
Lewin calls KVL a reformulation of Faraday's law that is done to make it look like KLR. Lewin seems to have a strong opinion about his claim that KLR is the only real Kirchhoff's law. I think it is safe to say that only KVL is relevant for applied circuit analysis. I would like to see a justification for the idea that KLR is used anywhere outside some elementary textbooks. Kirchhoff's original publication mentions only voltages over batteries and resistors, and is therefore a special case of both KVL and KLR, although his formulation involves no line integrals and is closer to KVL. Remarkably, Kirchhoff (correctly) calls the voltage over a battery an "electromotive force" (emf). This is consistent with KVL, but inconsistent with Lewin's (correct) use of the term "emf" for the induced emf in his circuit. This emf does not enter his KLR.

The lumped element model

KVL is formulated for the lumped element model, which assumes that changing magnetic fields are well contained inside the circuit elements. This assumption is obviously not fulfilled for Lewin's circuit, but it should also be obvious that the effect of the external magnetic field on the circuit in this case is equivalent to that of a series of secondary coils of a transformer, distributed over the circuit. KVL still applies and the only problem is the measurement of the voltages with probe wires influenced by the uncontained induced electric field.

Lewin's experiment explained with Faraday's law

In Lewin's experiment, the voltage over a resistor is measured with a measurement loop that has a vanishing magnetic flux. Faraday's law implies that the measured voltage must coincide with the voltage drop over the resistor, because it is the only element in the measurement loop and the resistance of the wires can be neglected (the total electric field inside an ideal conductor is zero). Note that the induced electric field surrounding the solenoid is irrotational in a simply connected region containing the measurement loop (for example everything left to the solenoid), so we don't worry about undefined voltages.
The voltages for both resistors in this circuit is measured this way and the results are −0.1V and −0.9V for a chosen orientation of the circuit loop. These results coincides with the prediction from Faraday's law for the circuit loop. It predicts that there is a total induced emf of 1V and with Ohm's law the current and the individual voltage drops over the resistors can be calculated. We conclude that the loop integral over the electric field along the circuit is −0.1V+(−0.9V)=−1.0V and KLR does not hold.

KVL still holds for Lewin's experiment

KVL takes all voltage drops and electromotive forces in the circuit into account, so we have −0.1V+(−0.9V)+1.0V=0V. Up to here all possible disagreement should be only due to a difference in terminology.

Measuring electromotive forces

For the lumped element model we have no problem measuring the voltage drop over an inductor. The voltmeter reading gives us the emf in the inductor. In Lewin's experiment, the emfs is distributed over the circuit and when we try to measure the emf in a segment of the circuit, we have the problem that there is also an induced emf in the probe wires that contributes to the measurement. In fact, Faraday's law tells us that in Lewin's measurement the emf in the probe wires must cancel the emf in the probed circuit segment, so the emf in the circuit segment remains invisible.
In order to measure the emf (together with the other voltage drops) in the probed circuit segment, we have to arrange the probe wires so that the total emf in both wires is zero. If we can achieve this, then the voltmeter reading gives us the voltage over the circuit segment that enters KVL. We can not do that without additional information about the induced electric field. This is a crucial difference from Lewin's measurement. Lewin only needs one of Maxwell's equations (in its integral form) to predict and understand his measurements. ElectroBOOM needs some details about the solution of Maxwell's equations for the induced electric field. Lewin only needs loop integrals over the electric field, ElectroBOOM needs the induced field vector for every point along the probe wires in order to calculate the emf as the line integral along the wires.
Fortunately, the induced electric field of a solenoid has simple geometric properties that can be exploited to find a proper arrangement of probe wires. Since we know that the field lines are concentric circles around the axis of the solenoid [citation needed], we also know that the field is perpendicular to every plane that contains the axis. Let's call these planes "zero flux planes" (they happen to have zero magnetic flux everywhere). A probed point on the circuit is contained in exactly one zero flux plane. If we run the probe wires from the probed points inside their respective zero flux plane until they meet on the axis, we know that the emf in both wires is zero everywhere up to the meeting point (the wires are perpendicular to the induced electric field). From the point on the axis where they meet onward, the two probe wires shall follow the same path, so that both contributions to the total emf cancel (alternatively choose a meeting point far away where the field can be neglected). Examples for such measurements have been conducted by Cyriel Mabilde:
(See also the measurements by Dirk Van Meirvenne.)

Can the electromotive force be localized and what does a voltmeter measure?

[This section is somewhat speculative. Comments and criticism are welcome.]
The title of Romer's article poses the question 'What do "voltmeter" measure?', and he gives the answer that a voltmeter measures the line integral over the electric field along the probe wires through the voltmeter itself. For ideal probe wires this amounts to "a voltmeter measures the voltage inside of it". I will make the case that this unsatisfactory answer is related to Romer's unexplained claim that the question "Where is the emf located?" is poorly defined. As we can clearly see, Romer's approach is at odds with our measurements of emfs over wire segments, which gives us information over the local distribution of the emfs in the circuit.
We have defined the emf in a wire segment as the line integral over the induced electric field along the segment. This is in accordance with the definition of the emf as it is used in Faraday's law, but note that the integral in Faraday's law is usually over the total electric field, which includes the conservative field coming from the charge distribution in the conductors. The contributions from this field do not cancel if we don't integrate over a loop. This field is not the source of an emf, so it must not be taken into account when calculating the emf in a wire segment.
This definition is consistent with the results of voltage measurements over inductors in the lumped element model, which localizes the emf in a lumped element. We have also seen that local emfs can be consistently applied to Lewin's circuit to predict measured voltage drops over wire segments to check KVL. Together with the fact that the emf is caused by the effect of the induced electric field on the charges inside the conductor, which is defined in local terms, this gives us reason to believe that we are dealing with a well defined physical quantity.
There is an objection to our method of observing local emfs that will make us take a closer look at how to define and observe them. If one is used to rely only on Faraday's law to predict and explain measurements, one might be under the impression that the emf is only defined as a property of a loop and cannot be localized. In fact, every measurement of local emfs we have described above can be explained in terms of magnetic flux changes and loop emfs. Apparently, voltmeter measurements alone cannot help us to convince a strict follower of Faraday's law that local emfs represent an underlying reality.
The electric field on the inside of an ideal wire is assumed to be zero, any deviation from zero is compensated (almost) instantly by a rearrangement of charges inside the wire. In the presence of an induced electric field we should see the corresponding emf represented in the charge distribution. On the outside of the wire, the charges create a measurable electric field that can be probed with a fieldmeter. In the case of Lewin's circuit, where ideal wires connect resistors, we expect to observe a change of field strength along the outside of the wires. The size of the local emf in a short piece of wire is determined by the rate at which the field changes as we go along this piece. This can be considered a continuous version of more familiar emf sources like batteries, where we observe a jump in charge density over the electrodes. With this insight, we can understand the definition of the electromotive forces in an ideal wire segment as the line integral over the negative of the electrostatic field of the charges, as it is given, for instance, by Reitz, Milford, Christy, Foundations of Electromagnetic Theory, 1960, p. 135, see also the Wikipedia definition (which doesn't give a valid source).
Considering the field of the charge distribution also gives us a answer to the question what a voltmeters measures. An answer should describe a physical quantity that is a property of the circuit alone and exists independently from the measurement device. It must be measurable by a voltmeter without disturbing the probed system. Feynman's derivation of KVL gives us the hint that the voltmeter reading is given by the line integral over the electric field of the displaced charges near the surface of the wire where the probes are connected. This field is the only contribution to Feynman's loop integral over the electric field along the circuit. When we connect the probe of a voltmeter to the circuit, it becomes an extension that transfers the field to the inside of the voltmeter where it is compared to the field coming form the other probe. (Note that Feynman's loop integral applied to Lewin's circuit has a contribution from the induced electric field surrounding the circuit. This contribution accounts for the induced emf in the circuit, which is not represented by a lumped element.)
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Ingresso com média de 153, (provas de ingresso) acabadinho de sair de um curso de Ciências e Tecnologias, pior decisão que já tomei até agora. I survived tho!
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