Why does a voltmeter measure a voltage across inductor?

[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.