Why does a voltmeter measure a voltage across inductor?

[Dale]

Well, then tell me which additional assumptions you need for the KVL

From Nilsson and Riedel "Electric Circuits" 5th ed. section 1.1 p. 5 under the heading "Circuit theory" the assumptions are:

"1. Electrical effects happen instantaneously throughout a system...

2. The net charge on every component in the system is always zero. Thus no component can collect a net excess of charge, although some components, as you will learn later, can hold equal but opposite separated charges.

3. There is no magnetic coupling between the components in a system. As we demonstrate later, magnetic coupling can occur within a component."

The same assumptions are worded slightly differently on Wikipedia at https://en.wikipedia.org/wiki/Lumped_element_model#Lumped_matter_discipline and I assume in all other good textbooks on circuit theory. Dr Lewin's example violates assumption 3, an inductor or a transformer does not.

E.g., of course in circuit theory you treat also transformers as "compact element", i.e., a "four pole"

Then why did you make the opposite claim and then falsely attribute such an obviously wrong claim to me when I said no such thing?

BTW Circuit theory is only a special case of the general "transfer approach" for the full Maxwell equations

And Maxwell's equations are only a special case of the general QED. And Newtonian gravity is only a special case of the EFE. Shall we claim to be using QED and GR to solve a "push a box up an inclined plane" problem? Shall we say that Newtonian mechanics is "for the birds" because there are some cases where it doesn't work?

 

[vanhees71]

From Nilsson and Riedel "Electric Circuits" 5th ed. section 1.1 p. 5 under the heading "Circuit theory" the assumptions are:

"1. Electrical effects happen instantaneously throughout a system...

2. The net charge on every component in the system is always zero. Thus no component can collect a net excess of charge, although some components, as you will learn later, can hold equal but opposite separated charges.

3. There is no magnetic coupling between the components in a system. As we demonstrate later, magnetic coupling can occur within a component."

The same assumptions are worded slightly differently on Wikipedia at https://en.wikipedia.org/wiki/Lumped_element_model#Lumped_matter_discipline and I assume in all other good textbooks on circuit theory. Dr Lewin's example violates assumption 3, an inductor or a transformer does not.
[/CITE]

Sure, that's what I said the whole time: (1) is the quasistationarity assumption; (2) is an immediate consequence of it ($\vec{\nabla} \cdot \vec{j}=0$); (3) transformers couple two subcircuits magnetically. This is what's behind the inductance matrix for the generator as a four-pole element.

[CITE]
And Maxwell's equations are only a special case of the general QED. And Newtonian gravity is only a special case of the EFE. Shall we claim to be using QED and GR to solve a "push a box up an inclined plane" problem? Shall we say that Newtonian mechanics is "for the birds" because there are some cases where it doesn't work?

That's nonsense. I've never said you should use QED to solve classical electrodynamics problems. I think we can end this discussion, which is just about semantics.

 

[Dale]

That's nonsense. I've never said you should use QED to solve classical electrodynamics problems. I think we can end this discussion, which is just about semantics.

Then why do you insist that circuit theory is the same as classical electrodynamics? Circuit theory bears the same relationship to classical EM as classical EM bears to QED. If you are justified is looking at any circuit theory problem and saying it is "just Maxwells" then why isn't someone else justified in doing the same thing when looking at any classical EM problem and saying it is "just QED"?

 

[Charles Link]

My input on this item if I may: Maxwell's equations is something most of the mainstream physics people along with the EE's should try and work through as undergraduate students. If it is explained well, including magnetostatics, most students should be able to understand it and use it quite routinely. QED (Quantum Electrodynamics), on the other hand, along with the Second Quantization formalism is rather abstract and extremely difficult to get any proficiency with. There are a handful of people, maybe a few more, who have mastered it, but I myself have tried to work through both textbooks by Bjorken and Drell, along with Schweber's book on relativistic quantum fields and they are next to impossible. Maybe if I dig them out and keep at it for about 5 more years, some of the things like the Feynman Diagrams will start to get a little easier. Another good text if a physics person really wants to challenge themselves is Fetter and Walecka, Quantum Theory of Many Particle Systems. In that textbook, (which also uses plenty of Feynman Diagrams), I think I have gotten farther than most people, but I am hardly proficient at it... Maxwell's equations, at least for physics people, should go hand-in-hand with electrical circuit theory. QED is something that the very ambitious can work at learning, but is hardly a prerequisite for doing good physics.

 

[vanhees71]

Then why do you insist that circuit theory is the same as classical electrodynamics? Circuit theory bears the same relationship to classical EM as classical EM bears to QED. If you are justified is looking at any circuit theory problem and saying it is "just Maxwells" then why isn't someone else justified in doing the same thing when looking at any classical EM problem and saying it is "just QED"?

Circuit theory is an approximate solution of the Maxwell equations. Basically it's just the quasistationary approximation + lumping the geometry and material constants of the linear response constitutive relations into the usual parameters of the "compact circuit elements". It's pretty simple. A nice elementary introduction can be found under

http://web.mit.edu/viz/EM/visualizations/notes/index.htm

chpt. 11. That's the standard way how this stuff is taught to physics students and pretty much what I'm doing in this thread the whole time. So I don't understand why we got into arguments in the first place ;-)).

 

[Dale]

So I don't understand why we got into arguments in the first place

We got into the argument because you (still?) misapplied the assumptions of circuit theory, particularly for Dr Lewin's example.

Basically it's just the quasistationary approximation + lumping the geometry and material constants of the linear response constitutive relations into the usual parameters of the "compact circuit elements". It's pretty simple

So do you now understand why Dr Lewin's example violates the assumptions of circuit theory? In particular do you see how the setup violates the "lumping" assumption you yourself have identified?

 

[vanhees71]

Ok, I don't see it. In Lewin's example you can clearly use the quasistationary approximation, which is basically the only approximation that gets into circuit theory. The coil provides the EMF, i.e., it's a magnetically coupled "voltage source". The circuit itself can be and in Lewin's writeup is completely treated with the correct KVL, including the EMF from the coil. That's all. I think, now one (you) really should close this thread, because everything is very clear!

 

[Dale]

Basically it's just the quasistationary approximation + lumping the geometry and material constants

the quasistationary approximation, which is basically the only approximation that gets into circuit theory.

You are contradicting yourself here. The first one describes circuit theory, the second one does not.

 

[vanhees71]

I've nothing to add. I don't worry, whether you call Lewin's analysis circuit theory or not. I do, because he's exactly doing what's written in the first quote, and also the 1st and 2nd quote don't contradict each other. For the derivation see the already above cited

http://web.mit.edu/viz/EM/visualizations/notes/index.htm

(Chpt. 11).

 

[Dale]

I don't worry, whether you call Lewin's analysis circuit theory or not. I do

You have been around this forum long enough that you know this is not how we work. You don't get to promote a private definition of "circuit theory" here. Circuit theory is what it is defined to be in standard professional circuit theory textbooks. It is more than just the quasistationary assumption, as I cited above.

 

[vanhees71]

Ok, I give up. Perhaps finally you can point me to a "standard textbook on circuit theory" that defines it the way you want me to use the term in future discussions.

 

[OnAHyperbola]

@Dale Ah, I understand the point you are making and it makes sense to me. It took me a little while to process this recent discussion (what with being in high school, and all :), but I have drawn a few conclusions:
--You are right that circuit theory has its own assumptions and domain of applicability.

the circuit shown by Dr Lewin, in contrast, shows magnetic induction outside a circuit element, which explicitly violates the assumptions of circuit theory.

Also in circuit theory we selectively (depending on occasion) neglect the time varying flux in Maxwell's-Faraday's Law. That's what makes KVL valid, and if we don't neglect the flux we make KVL valid by transferring the dΦdtdΦdt\frac{d\Phi}{dt} term in the left hand side of the equation and expressing it as LdI/dtLdI/dtLdI/dt and/or MdI/dtMdI/dtMdI/dt terms. That's what happens in the "internal" treatment of a transformer using circuit theory, or in the simpler case where an inductor is present in a circuit.

So,
a) since an assumption has been pre-made, as @Delta² says, we can't really say it has been violated
b) the confusion arose because there isn't supposed to be any potential drop between the ends of the inductor, yet there is. The understanding is that the coil created a closed electric field through the loop of the circuit, which would then account for the voltmeter's reading. However, as @nsaspook says:

It's implied that all fields are confined to the element so it's physical placement in relationship to other circuit elements in the circuit would be the same as moving a resistive element.

so within circuit theory, that explanation can't be used and we're back to what @Delta² said, which Dr. Lewin says "most physics books have right for the wrong reasons"
Elements of a circuit should keep their fields to themselves and not start radiating them to other places.
Dr. Lewin's video was an excellent introduction though, and I believed that KVL was for the birds (until I found all you enlightened folk)

I think one can go about pretending there really is a potential drop across the inductor as long as one knows what's really happening behind the scenes and the assumptions that one is going in with. Otherwise, as Charles Link says, you might not be able to bring your hot-air balloon back.

 

[Dale]

Ok, I give up. Perhaps finally you can point me to a "standard textbook on circuit theory" that defines it the way you want me to use the term in future discussions.

I did: Nilsson and Riedel "Electric Circuits" 5th ed. section 1.1 p. 5 under the heading "Circuit theory". Any good textbook on circuit theory will have these listed explicitly.

 

[vanhees71]

Ok, let's get this issue clear. I don't care, whether it's circuit theory or not. You only need Maxwell's equations in a very simple way. The setup is given by the following figure from Lewin's writeup:
http://th.physik.uni-frankfurt.de/~hees/tmp/lewin-setup.png
Now the only objection by Dale against calling the follosing circuit theory is, whether you call the "changing magnetic flux" (which can be realized by puting a long solenoidal coil with an alternating current there) a compact EMF source or not. As I said, it doesn't matter. The derivation is simply integrating Faraday's Law along the three loops in the circuit and to use the quasitationary approximation that the current is observed on every branching (this is what distinguishes for me circuit theory from the full Maxwell equations, which boils down to neglect any em-wave radiation effects).

Assuming the internal resistances of the volt meters to be $R_i \gg R_1,R_2$, you have (see also Lewin's writeup):
Left-most loop, neglecting the stray field of the coil:
$$R_i I_1+R_1(I_1-I)=0.$$
Middle loop
$$R_1(I-I_1)+R_2(I-I_2)=\mathcal{E}(=\pm \frac{1}{c} \dot{\Phi})$$
Right-most loop, neglecting the stray field of the coil:
$$R_i I_2 +R_2(I_2-I)=0.$$
Solving for the relevant currents you get
$$I_1=\frac{R_1(R_2+R_i)}{R_i(2R_1 R_2+(R_1+R_2) R_i)} \mathcal{E},$$
$$I_2=\frac{R_2(R_2+R_i)}{R_i(2R_1 R_2+(R_1+R_2) R_i)} \mathcal{E},$$
and for the readings of the volt meters, with their + label attached to the A side in the drawing above (following Lewin's writeup)
$$U_1=-R_i I_1 \simeq -\frac{R_1}{R_1+R_2}, \quad U_2=R_i I_2 \simeq \frac{R_2}{R_1+R_2},$$
where in the approximation are for a very large inner resistance of the voltmeters, i.e., making formally $R_i \rightarrow \infty$.

Now a funny thing is indeed problem 1. There the only difference to the above is that for the "left-most loop" we now have (with $N$ the numbers of winding the wire connecting volt meter 1 to the circuit around the coil)
$$R_i I_1+R_1(I_1-I)=N \mathcal{E}.$$
Then you get (again in the limit $R_i \rightarrow \infty$, which assumes that also $N R_1,N R_2 \ll R_i$)
$$U_1 \simeq-\frac{(N+1)R_1+N R_2}{R_1+R_2} \mathcal{E},$$
while $U_2$ doesn't change in this approximation.

Perhaps Dale could point to the place, where this analysis is beyond usual circuit theory. I still don't see it!

 

[vanhees71]

I did: Nilsson and Riedel "Electric Circuits" 5th ed. section 1.1 p. 5 under the heading "Circuit theory". Any good textbook on circuit theory will have these listed explicitly.

I see, so Lewin's example obviously only misses item 3, i.e., that magnetic couplings are only within a single element of the circuit. I.e., for calling the problem one of "circuit theory" it's not allowed to have a "magnetic flux" within a loop of the circuit as here. That makes sense as then such problems with the volt-meter readings in this example can't occur and they are independent of the way you connect them between any two points in the ciruit (modulo stray fields, which also must be small according to assumption 3).

Well, as I said, it's only semantics. Lewin's analysis, as expected, is of course correct although it goes beyond the strict definition of "circuit theory".

I hope, also my solution of the question 2 (for $N=1$ of course it also answers question 1) is correct :-).

 

[Charles Link]

A minor "typo" in post #103. In M.K.S. $\$ $\mathcal{E}=-d \Phi/dt$. The "c" in the denominator is only in c.g.s. units.

 

[Charles Link]

@OnAHyperbola Please read my post #68 if you haven't already. I think you might find it interesting. With so much other discussion going on, it is likely you missed it. Until just a moment ago I hadn't seen your post #101 amidst all of the other discussion. Note: The ac power lines use f=60 Hz in the U.S. $\$ In Great Britain and some other places they might use f=50 Hz, but hopefully you can follow the idea that Dr. Lewin's example shows precisely how a stray EMF can enter into your voltmeter and/or the circuit you are working with.

 

[Dale]

I don't care, whether it's circuit theory or not.

Then it is odd that you would repeatedly claim that it is circuit theory in the face of well reasoned objections. When I am in such a position (making a mistake on a topic that I don't care about) I just say "oops", learn, and move on. I have to say that this interaction with you has bothered me greatly.

You only need Maxwell's equations in a very simple way

Yes, I agree.

the quasitationary approximation that the current is observed on every branching (this is what distinguishes for me circuit theory from the full Maxwell equations

Again, you don't get to push personal definitions here. The fundamental assumptions of circuit theory are more than that and this has been explained and referenced in detail.

Perhaps Dale could point to the place, where this analysis is beyond usual circuit theory

The place pointed to by the line labeled "changing magnetic flux".

so Lewin's example obviously only misses item 3, i.e., that magnetic couplings are only within a single element of the circuit. I.e., for calling the problem one of "circuit theory" it's not allowed to have a "magnetic flux" within a loop of the circuit as here.

Yes, exactly!

Well, as I said, it's only semantics. Lewin's analysis, as expected, is of course correct although it goes beyond the strict definition of "circuit theory"

Yes, as I have agreed several times. His analysis is correct, the problem is exactly his semantics. The "for the birds" bit is just semantics, but it is such awful semantics that it needs to be objected to.

KVL is not for the birds, it is for circuit theory and it is valid whenever the assumptions of circuit theory are valid.

 

[Dale]

I think one can go about pretending there really is a potential drop across the inductor as long as one knows what's really happening behind the scenes and the assumptions that one is going in with.

Yes, I think that is the take home message. I will try to post a little about assumptions tomorrow.

 

[Charles Link]

One item that puzzled me here, [I think I have it figured out to my satisfaction, but it is somewhat complex and everyone (even the most astute ones) are likely to come up with interpretations that differ somewhat], is that the electric field E from the EMF points in the direction in the inductor (just like the EMF inside a battery) from the minus end to the plus end. For an electrostatic component, such as a capacitor, the electric field E points from plus to minus. A voltmeter (or oscilloscope) has no trouble reading the voltage of either of these components. For the inductor, $V=\int E \cdot dl$ and is very well defined, but the sign is actually opposite what you would get if the E as a function of position were an electrostatic field. Thereby the EMF E is a somewhat different entity than an electrostatic field. The voltage V is however very well defined from an EMF. The sign actually gets reversed when computing the voltage (please check this carefully=I do think I have it correct), but in any case it can be written as a potential (voltage) even if the curl E is non-zero. When sensed with a voltmeter or oscilloscope, these devices are unable to distinguish whether they are reading an EMF or reading an electrostatic voltage.

Repeating my post #25. Also @OnAHyperbola my post #68 I think you might find of interest. If you have figured out from all of this how changing magnetic fluxes can affect a circuit and can accurately predict what the voltmeter will read when there is a changing magnetic flux present, (and it will depend on how you string the wires from the voltmeter to the circuit), I think you have learned quite a lot.

 

[vanhees71]

A minor "typo" in post #103. In M.K.S. $\$ $\mathcal{E}=-d \Phi/dt$. The "c" in the denominator is only in c.g.s. units.

I never ever use SI units in electromagnetism, if I can avoid it. I use exclusively rationalized Gaussian units (aka Heaviside-Lorentz units) as in HEP and high-energy nuclear physics.

 

[vanhees71]

Then it is odd that you would repeatedly claim that it is circuit theory in the face of well reasoned objections. When I am in such a position (making a mistake on a topic that I don't care about) I just say "oops", learn, and move on. I have to say that this interaction with you has bothered me greatly.

Me too, because it's just a misunderstanding between a physics and engineering definition. I'll try to avoid the label "circuit theory" from now on, because it seems to have another meaning than what I learned in the E&M lecture. Sorry for any confusion this might have caused in this discussion.

Just a note concerning this book by Nielsson. From a physicist's point of view, it's pretty inaccurate and even dangerous in its language concerning percisely our debate. I don't know, what happens if you try to solve a problem with moving parts like in a generator or motor, but I guess that's also not "circuit theory" in this sense. As far as I can see, that's not even treated in the book...

My point of view is also found in at least one textbook for electrical engineers, which I like very much

K. Simonyi, Foundations of Electrical Engineering, Pergamon Press (1963)

although I know in better detail the German Edition ("Theoretische Elektrotechnik") of 1993. There's a large chapter on circuit analysis, and there everywhere the correct Kirchhoff rules, including the time derivative of the magnetic flux is used.

 

[OnAHyperbola]

@Charles Link Yes, I did read your post #68 and it is a very good example illustrating the effects of a changing magnetic flux on the voltmeter reading; one should be careful what the wires are going over/around and how it could leave you scratching your head if you haven't noticed :)

 

[Charles Link]

I never ever use SI units in electromagnetism, if I can avoid it. I use exclusively rationalized Gaussian units (aka Heaviside-Lorentz units) as in HEP and high-energy nuclear physics.

Then it isn't a typo. My mistake. 95% of the time or more, the voltage equations of the electrical engineer will be in SI units. The conversion from gauss to Webers/m^2 is an easy one, but the voltage, current/charge, and resistance can be rather cumbersome in converting from c.g.s. to SI. Maybe this might partly explain the different viewpoints between @Dale and @vanhees71 =a EE vs. someone doing more theoretical work. In E&M calculations, I also favor the c.g.s. units, since we used the book by J.D. Jackson Classical Electrodynamics in graduate school. However, it always takes me a few minutes, (basically by comparing the two Coulombs law formulas $v=q/r$ vs. $V=Q/(4 \pi \epsilon_o R)$ and using e=q= 4.801 E-10 e.s.u.'s and e=Q=1.602 E-19 Coulombs., and letting R=1 meter so that r=100 cm) , to write out the conversion factors when I am looking for the voltage that I'm going to measure with my SI voltmeter. The difference in voltages is a factor of 300, but I can never remember if you multiply or divide. :-) :-) :-)

 

[Dale]

Just a note concerning this book by Nielsson. From a physicist's point of view, it's pretty inaccurate and even dangerous in its language concerning percisely our debate.

But it isn't written for a physicist's point of view, it is written for electrical engineering students taking their first course on circuits with no background knowledge of Maxwell's equations. The assumptions are standard, although the wording does vary from text to text depending on the previous knowledge of the target audience.

 

[Dale]

as long as one knows what's really happening behind the scenes and the assumptions that one is going in with

I wanted to go back and highlight this comment of yours. I think it is fantastic that you have this level of scientific clarity in high school.

In any scientific endeavor you will use some sort of equation and some assumptions. For instance, in high school you may have been exposed to the "SUVAT" equations, such as $v=u+at$. This is a perfectly valid physics equation, but you need to understand its assumptions. There are some basic assumptions from Newtonian mechanics, such as that it is only good for velocities much slower than the speed of light, and (perhaps more importantly) there are some additional assumptions such as that the acceleration is constant. If you use the equation when the assumptions are valid, then you will get results that are physically meaningful. If you blindly "plug and chug" without consideration for the assumptions then you will get nonsense, not because the formula is wrong but rather because you applied it in the wrong situation.

Often, the assumptions will help you select the right formula to use, and the better assumptions you can make the simpler your formula can be. If you understand that, then you will have an academic and scientific advantage over your peers who focus exclusively on the variables.