Why does a voltmeter measure a voltage across inductor?

[Charles Link]

In my opinion the EMF that originates from any sort of structure with $-d\Phi/dt$ drives the buildup of charge densities at the end points , so it creates a scalar potential and then this scalar potential is responsible for what is happening in the rest of the circuit.

I think this could be a possibility. Although I've had plenty of E&M courses and also been through the circuit theory in college, (1974-1981), this is the first time I've looked this closely at the inductor and observed the $V=\int E \cdot dl$ with a plus sign, i.e. that the Faraday E behaved differently from the electrostatic in computing a voltage. It has proved for interesting discussion in any case. The other item that surfaced was back around post #25, where @vanhees71 was able to find a "link" to the item I mentioned a day or two ago about a changing magnetic field inside a continuous resistor ring and connecting a voltmeter. The way the wires are wrapped from the voltmeter to the ring can put the voltmeter circuit in a loop that has an EMF from the changing magnetic field and thereby it gets a different reading, for the same two points, etc.

 

[OnAHyperbola]

@Charles Link You are absolutely right in that the voltmeter won't distinguish between an EMF type driving voltage and an electrostatic one. I think I may have found an answer to our conundrum. And, yes, it is incorrect to call this EMF a potential difference; it is an electromotive force, but much, much more importantly, it is a change in magnetic flux THROUGH A SPECIFIC LOOP wrt time. I think this conversation is very enlightening:

(look at the newest comment on the YouTube page)My question was, how is there a drop in potential if there is no field inside the inductor? There isn't. The voltmeter doesn't care that there isn't. It can't think. The wires of the voltmeter and inductor make an Amperian loop, through which the magnetic flux changes in time. The changing flux through this loop creates an emf around the loop and a feeble current in it. EMF/R=I , which the voltmeter interprets as a p.d.
That seeming drop in potential/voltage/EMF/whatever didn't even exist until the voltmeter came along to measure it and make its own loop. It is our measuring it that forced it to exist in a measurable way. (This is just my take)
Also, I was wrong when I said the voltmeter should read a p.d when attached to any wire in the circuit. No changing flux there==> no emf around the loop==> 0 reading

 

[vanhees71]

It describes one of the conditions under which circuit theory can be applied. The induced E field does not need to be negligible everywhere, just on some surface surrounding the inductor.

I'm really puzzled about what you mean. Of course, the induced E field is not neglible. What's neglected in quasistationary circuit theory (i.e., for the case that the typical wavelength of the fields involved is large compared to the geometrical extensions of your circuit) is the displacement current. Faraday's Law is fully implemented, i.e.,
$$\vec{\nabla} \times \vec{E}=-\frac{1}{c} \partial_t \vec{B}$$
is integrated along an arbitrary surface. For circuit theory the best choice is to use a surface with the wires, resistances, coils, capacitors, and sources as its boundary. Then you use Stokes's Law. If the circuit is at rest this results in Kirchhoff's Law including $-\dot{\Phi}/c$, where $\Phi$ is the magnetic flux along the surface.

Take the above example of a coil $L$ and a resistor $R$ in series connected to some voltage source/battery $U$. Integrating in direction of the current density, you get
$$\int_{\partial A} \mathrm{d} \vec{r} \cdot \vec{E}=R i-U=-L \dot{i}.$$
As you very clearly see the closed-path integral of $\vec{E}$ is not 0 if the current is not stationary, and thus the electric field is not conservative in this case. See also Levin's example where the source is given by an induced EMF and the practical importance for the path dependence when measuring "voltages" in such cases.

For Kirchhoff's other law you use the Ampere-Maxwell law, reduced to the Ampere Law by neglecting the "displacement current", leading to the quasistationary condition
$$\vec{\nabla} \cdot \vec{j}=0,$$
leading to "current conservation" at branchings within the circuit. It's nothing else than the conservation of electric charge.

 

[Delta2]

Take the above example of a coil $L$ and a resistor $R$ in series connected to some voltage source/battery $U$. Integrating in direction of the current density, you get
$$\int_{\partial A} \mathrm{d} \vec{r} \cdot \vec{E}=R i-U=-L \dot{i}.$$

Essentially you using faraday's law in integral form here right? If so why do you evaluate the integral of left hand side as only $Ri-U$?? Shouldn't you include the term $\int E\cdot dl$ for the path inside the coil wire?Do you let the E-field to be zero inside the wire of the coil? Or the path of integration does not include the spirals of the coil and it goes as straight line through them?

 

[vanhees71]

Essentially you using faraday's law in integral form here right? If so why do you evaluate the integral of left hand side as only $Ri-U$?? Shouldn't you include the term $\int E\cdot dl$ for the path inside the coil wire?Do you let the E-field to be zero inside the wire of the coil? Or the path of integration does not include the spirals of the coil and it goes as straight line through them?

That's the very point! Of course, my path on the left-hand side goes through the wire of the coil, but (assuming $R_{\text{coil}}=0$, i.e., lumping its resistance into the overall resistance $R$) there's no "voltage drop" across it. The inductance enters the equation from the right-hand side, i.e., the flux of the magnetic field inside the coil (I neglected the inductance of the rest of the circuit as much smaller than that of the coil).

I insist that here you DO NOT measure a potential difference. To the contrary one has a closed loop here! It's an EMF, which is more general than electrostatic potentials.

Also it's true that there is of course the four-vector potential, but in my experience for physics discussions it is wise not to refer to it. In classical electrodynamics you can very often avoid it and discuss everything in terms of the observable (and thus physical) fields $\vec{E}$ and $\vec{B}$ (and in macroscopic electrodynamics also $\vec{D}$ and $\vec{H}$). Introducing the four-potential usually involves debates about their physical interpretation, which is pretty tricky, because they are gauge-dependent quantities.

 

[Dale]

I'm really puzzled about what you mean. Of course, the induced E field is not neglible.

It doesn't matter, it is not relevant to the thread.

The surface described is not a surface enclosing the whole circuit, it is a surface enclosing a single component, such as an inductor. The induced E field on such a surface certainly can be negligible, even for a strong inductor.

 

[vanhees71]

Stokes's Law is also very clear about the surface and the boundary involved, including the mutual orientation of both. In my example the path was along the circuit. Now you can choose any surface, for which this path is the boundary.

 

[Charles Link]

Essentially you using faraday's law in integral form here right? If so why do you evaluate the integral of left hand side as only $Ri-U$?? Shouldn't you include the term $\int E\cdot dl$ for the path inside the coil wire?Do you let the E-field to be zero inside the wire of the coil? Or the path of integration does not include the spirals of the coil and it goes as straight line through them?

Perhaps @vanhees71 needs to answer this, but I believe the integral that he is doing is the Faraday $E$ over the path of spiral path of the inductor. This is the EMF of the inductor and it acts as a voltage in the circuit. Also notice the sign on his integral: it gives a $V=\int E \cdot dl$ just as my claim in post #43. Editing... And in fact, I think he answered your question in post #57.

 

[vanhees71]

I do the integral along the entire "wire", including the spiral path of the inductor. As I said, I used the usual approximations or circuit theory (ideal coil, negligible self-inductance of all parts of the circuit except the coil as well as the magnetic field outside the coil), and again THERE IS NO $V$! It's $-\dot{\Phi}/c=-L \dot{i}$!

For more details, see p. 102 ff of

http://th.physik.uni-frankfurt.de/~hees/physics208/phys208-notes-III.pdf

For more basic E&M (including worked-out examples of elementary circuit theory)

http://th.physik.uni-frankfurt.de/~hees/physics208.html

 

[Charles Link]

I do the integral along the entire "wire", including the spiral path of the inductor. As I said, I used the usual approximations or circuit theory (ideal coil, negligible self-inductance of all parts of the circuit except the coil as well as the magnetic field outside the coil), and again THERE IS NO $V$! It's $-\dot{\Phi}/c=-L \dot{i}$!

I'm using the letter V to indicate the EMF. Perhaps I should use $\mathcal{E}$.

 

[Charles Link]

@OnAHyperbola Thank you for the video of post #52. I watched the first 20 minutes of it and it is a very good one. I looked for your comment, (on Youtube), but I think it might need to get approved. (I will check for it again later). editing... Question: Are you "marcandrin"? If so, yes, I see the comments.

 

[OnAHyperbola]

@Charles Link Yes it's the conversation between Walter Lewin and marcandrin that I was referencing. It is very illuminating. It was when Lewin said that pulling either voltmeter over to the other side (by 180 degrees) would make it show an entirely different reading (the readings will switch with each other, in fact) that blew my mind and really set it in that geometry matters. I am not marcandrin, by the way :) although I have their curiosity to be grateful for.

 

[Charles Link]

@Charles Link Yes it's the conversation between Walter Lewin and marcandrin that I was referencing. It is very illuminating. It was when Lewin said that pulling either voltmeter over to the other side (by 180 degrees) would make it show an entirely different reading (the readings will switch with each other, in fact) that blew my mind and really set it in that geometry matters. I am not marcandrin, by the way :) although I have their curiosity to be grateful for.

The video is very much the problem that was discussed earlier in the thread: (See posts #8,10, 23, 27, 28, etc.). I first saw this puzzle in approximately 1979. I don't know if Professor Lewin is the one who originated the problem back then, but he may have been. And thank you so much for posting the video!

 

[Dale]

The problem that I have with Dr Lewin's presentation is that it is a scenario which explicitly violates the assumptions of circuit theory. Of course the equations of circuit theory don't work when you violate the assumptions of circuit theory!

It makes no sense to claim that there is anything wrong with KVL in this case. It is like saying that there is something wrong with the conservation of momentum by showing that it doesn't work for a system acted on by an external force.

The problem isn't KVL, the problem is using circuit theory where it doesn't apply.

 

[vanhees71]

Why does this example violate the assumptions of circuit theory? Lewin's writeup correctly uses circuit theory to explain the measured result!

 

[Charles Link]

Professor Lewin's problem is very much a part of the circuit theory as I know it. I first saw this problem in 1979 when a Mr. Jerry Davis, M.S.E.E. showed it to me when I was a graduate physics student. He didn't know what was causing the dilemma, but I figured it out within a week or so. I later saw it in a college E&M textbook approximately 2002 where a co-worker at my workplace, a B.S. in physics who was getting his M.S. asked me for assistance with his homework problem because he was stuck on it. It seems to have become part of the curriculum. Professor Lewin has evidently helped to make it somewhat well-known. I didn't know he had videos and printed notes of the problem and its solution until this current thread.

 

[OnAHyperbola]

Well, I think this thread has been one of the finer learning experiences I have had. Precisely because the solution was so simple, right there in Faraday's law and I kept getting all muddled in complicated reasons to account for that deceptive little voltage drop. (Note to self: Find the loop, find the flux through it, get the induced emf)

Many thanks to all of you! I hope you had as great a time as I did.

 

[Charles Link]

One additional comment in regards to Professor Lewin's instruction. Besides being a good lesson in EMF's, it is also a good lesson in stray EMF's. e.g. When measuring an electronic circuit with a voltmeter or oscilloscope, it is always good to run your two wires as closely connected as possible to each other in traveling from the voltmeter to the circuit being measured. Often a coaxial cable is used for these two wires, but a twisted pair can also work quite well. You certainly don't want to run your two wires in such a way that they happen to encircle an active power transformer that is part of your laboratory equipment (such as the ac to dc converter inside a piece of laboratory electronics, etc.). You would wonder where all the stray 60 Hz signal is coming from... :-) :-) Editing... Two things could actually happen here: 1)You observe a false 60 Hz signal on your oscilloscope that isn't in the circuit. 2) You disrupt your circuit by sending in a 60 Hz signal from your measurement probes! :-) :-)

 

[Dale]

Why does this example violate the assumptions of circuit theory? Lewin's writeup correctly uses circuit theory to explain the measured result!

No, it doesn't.

First, one of the three basic explicit assumptions of circuit theory is that there is no magnetic coupling between circuit elements, only within a element. So as soon as he starts talking about an induced EMF he has violated that explicit assumption. The other violations are implicit

Second, there are no fields in circuit theory, as soon as he starts drawing fields he is using Maxwell's equations. Circuit theory uses voltages, currents, and circuit elements, not fields. Think about it, there is no standard circuit element for representing a B field, because it is not part of the theory.

Third, a circuit diagram does not have any sense of spatial position. The loop drawn on the diagram may be a coaxial cable, a twisted pair, or some other configuration with minimal area for flux. Or it could have multiple large loops. There is no way to perform an integral over space within circuit theory because position isn't part of the theory.

Dr Lewin is using classical EM, not circuit theory. This is fine, he is a physics professor, not a EE professor. But I think it is poor form to give students the false impression that there is something wrong with KVL instead of giving them the correct understanding of the assumptions and limitations when circuit theory can and cannot be used.

 

[Charles Link]

I think Professor Lewin does a very good job in presenting both the concept of EMF and the solution to this puzzle. The puzzle I had actually seen and solved in 1979. I don't know whether it originated with Professor Lewin, but he does illustrate a couple of concepts extremely well. Professor Lewin's example also helps to illustrate the physics behind how an inductor works. I do think it is a very useful thing to be able to tie circuit theory with classical E&M. In this case, the circuits used are simple ones. For more complex circuits, you generally want to ensure that you don't have stray EMF's interfering with the circuits. And one other item that came out of a careful study of the physics involved here is the concept that the Faraday EMF is virtually indistinguishable from an electrostatic potential by common laboratory apparatus such as an oscilloscope or a voltmeter. I think most of the OP's questions were answered through the course of he discussion, and I found the discussion very worthwhile as well.

 

[Dale]

he does illustrate a couple of concepts extremely well.

I agree.

I just think that he inappropriately maligns KVL. Whenever the assumptions of circuit theory are met KVL is valid.

 

[anorlunda]

I had trouble following much of this thread, too much verbiage and too few equations and diagrams for me. But the thigs @Dale said about the assumptions of circuit theory were very interesting. I, along with many others on both the physics and EE sides, have never stopped to think of which explicit assumptions those are.

I looked them up, and found the slides of a lecture on that subject. Starting at slide 18, it becomes very interesting, touching on just the things that @Dale urged us to consider.
http://web.ewu.edu/groups/technology/Claudio/ee209/f09/Lectures/physics.pdf

One can study electricity at the QED level, or Maxwell's equations, or Circuit Theory, or RF propagation levels, and each makes sense as if they were isolated domains. However, spanning the boundaries between the levels I always thought exceedingly difficult. Thank you @Dale for showing me that at least one of those transitions is orderly and approachable.

 

[vanhees71]

No, it doesn't.

First, one of the three basic explicit assumptions of circuit theory is that there is no magnetic coupling between circuit elements, only within a element. So as soon as he starts talking about an induced EMF he has violated that explicit assumption. The other violations are implicit

(1) So you are saying, that it's not allowed to have transformers and even simple coils in your circuits? I'm sure electrical engineers wouldn't buy that.

(2) All you use in ciruit theory are the Maxwell equations in integral form, lumping the geometry (i.e., boundary conditions) of the setup into various constants (resistivities, capacities, self and mutual inductance).

(3) To analyze a circuit you need to specify the sense of the underlying line integrals. The orientation of the corresponding surface integrals is then determined by the right-hand rule, which is used to define the quantities of the theory to begin with. Only then you get the various signs in a problem right, also in cases where this is not so obvious from intuition.

 

[vanhees71]

I looked them up, and found the slides of a lecture on that subject. Starting at slide 18, it becomes very interesting, touching on just the things that @Dale urged us to consider.
http://web.ewu.edu/groups/technology/Claudio/ee209/f09/Lectures/physics.pdf

One can study electricity at the QED level, or Maxwell's equations, or Circuit Theory, or RF propagation levels, and each makes sense as if they were isolated domains. However, spanning the boundaries between the levels I always thought exceedingly difficult. Thank you @Dale for showing me that at least one of those transitions is orderly and approachable.

For ciruit theory you need classical electromagnetism, i.e., Maxwell's equations, and the above link is to be taken with great care since the concepts are not always accurate but commits several didactical sins you often find even in textbooks. They are not plain wrong but forget to explain the context and meaning when they are used. Just some examples

On slide 7: They claim there's a "Gauss's Law for magnetic fields". The only Gauss's Law of the magnetic field (there's only one magnetic field in nature!) is explained correctly on the slide before. On slide 7 they discuss Ampere's circuital law, which is valid only for static fields and stationary currents. In SI units it reads in differential form (which always the save starting point for all electromagnetic theory) it reads
$$\vec{\nabla} \times \vec{H}=\vec{j}.$$
Together with the constitutive equation $\vec{B}=\mu \vec{H}$ and in the vacuum (where $\mu=\mu_0$ becomes
$$\vec{\nabla} \times \vec{B}=\mu_0 \vec{j}.$$
First of all this tells you that you must have
$$\vec{\nabla} \cdot \vec{j}=0$$
everywhere. This is nothing else than the equation for charge conservation for stationary currents. This is very important to keep in mind. Thus to get corre t results from the Biot-Savart Law, which is derived by taking the curl of the Ampere law and using the true Gauss's Law for magnetic fields, $\vec{\nabla} \cdot \vec{B}=0$, you have to integrate along closed circuits. The explanation with "current elements" has thus to be taken with a grain of salt.

On slide 8 they are very sloppy concerning the order of integration and time derivative on the right-hand side. The basic law is again the differential form, which is Faraday's Law. In SI units it reads
$$\vec{\nabla} \times \vec{E}=-\partial_t \vec{B}.$$
Integrating over a surface $S$ with boundary $\partial S$ and using Stokes's Law you get
$$\int_{\partial S} \mathrm{d} \vec{r} \cdot \vec{E}=-\int_S \mathrm{d}^2 \vec{S} \cdot \vec{B}.$$
Note that the time derivative is taken BEFORE the integral here. To get it out for the general case is not so easy. Of course, if the surface including its boundary is at rest in the calculational frame, the equation on slide 8 is correct. Otherwise you get the correct law in integral form as
$$\mathcal{E}=\int_{\partial S} \mathrm{d} \vec{r} \cdot (\vec{E}+\vec{v} \times \vec{B})=-\mathrm{d}{\mathrm{d} t} \int_S \mathrm{d}^2 \vec{S} \cdot \vec{B}=\dot{\Phi}_B.$$
So on the left-hand side you get the FULL EMF, including the magnetic force along the boundary ($\vec{v}=\vec{v}(t,\vec{r})$ is the velocity of the boundary of the surface). In any case if the right-hand side (i.e., the time derivative of the magnetic flux through the surface) is not vanishing there's no potential for $\vec{E}$ for the static case, and this is important, also in circuit theory which of course also includes magnetic effects like self inductances and mutual inductances (as well as generators which produce the AC current in our households in the first place) as needed to analyze transformers. Lewin's example is nothing else of an inductively coupled source, i.e., a transformer of an AC current. On page 9 it must read that $\vec{E}$ is the total electric field along the path of the surface boundary (and again that it's valid only for the case of surface and its boundary at rest).

On slide 13 they precisely give handwaving (note that there are no equations!) explanation for the very case that they do not treat correctly in the slides before, namely the case of moving surfaces/boundaries.

One must say that the discussion from slide 19 on is pretty accurate, particularly the discussion on p. 21. Also on page 25 it is correctly stated that there's "a voltage", BECAUSE the magnetic flux is vanishing (negligible) in the case discussed.

 

[Charles Link]

@vanhees71 Very good reading=your posts #73 and #74. I agree it is important to include the physics of Maxwell's equations with the circuit theory or some of the finer details can be very ambiguous. One correction on post #74. (Hopefully you concur.) Ampere's law (in complete , non-steady-state) is $\$ $\nabla \times B=\mu_o \, J_{total}+\mu_o \epsilon_o \, dE/dt$ $\$ where $\$ $J_{total}=J_{free} +J_m+J_p$. $\$ Now $\nabla \times M=\mu_o \, J_m$ and $J_p=dP/dt$ .$\$ This gives $\$ $\nabla \times H=J_{free}+d D/dt$ $\$ where $\$ $B=\mu_o \, H+M$ $\$ and $\$ $D=\epsilon_o \, E+P$. $\$ The reason for the extra detail is it is important to include the non-steady state $dE/dt$ term in analyzing a capacitor in the case where the electric field in the capacitor is non steady state. (Note: One vector identity that is used in analyzing the above is $\nabla \cdot \nabla \times A =0$ for any vector $A$. Also $\$ $- \nabla \cdot P=\rho_p$ where $\rho_p$ is polarization charge density. The continuity equation applies $\nabla \cdot J_p+ \frac{\partial {\rho_p}}{\partial t}=0$ ) . In order to have consistency, the complete form of Ampere's law (rather than the steady state form) is necessary. (In the case that a capacitor is being charged, $\nabla \cdot J_{free}$ is not equal to zero) (Besides the inductor, the capacitor is also an interesting circuit element to analyze using Maxwell's equations). ... I'm going to need to study much more carefully your write-up for the Maxwell's equations with the moving surface, but I'm very glad you presented it. I always like seeing the complete details. It is important to know the complete details to be able to assess whether simplified equations that are used in steady-state and/or cases without any moving boundaries will apply in more complex cases.

 

[nsaspook]

(1) So you are saying, that it's not allowed to have transformers and even simple coils in your circuits? I'm sure electrical engineers wouldn't buy that.

The transformers and coils are circuit elements in circuit theory as per Dale. 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.

If we needed to examine a transformer as a circuit element then a process similar to this would be used.
http://www.electrical4u.com/equivalent-circuit-of-transformer-referred-to-primary-and-secondary/

 

[Dale]

(1) So you are saying, that it's not allowed to have transformers and even simple coils in your circuits? I'm sure electrical engineers wouldn't buy that.

A transformer is a single circuit element with four terminals. There is no magnetic coupling between circuit elements, only within an element.

(2) All you use in ciruit theory are the Maxwell equations in integral form, lumping the geometry

No, what you use in circuit theory are KVL and KCL, which deviate from Maxwell's equations outside of circuit theory's domain of applicability. Similarly, Maxwell's equations deviate from QED, and Newtonian gravity deviates from the EFE outside their respective domains of applicability. Every simplified theory is derived from the more complicated theory, but the simplified equations are different and make different predictions outside of their domain of applicability.

 

[Charles Link]

The Maxwell equation approach is also very important in things like transformer design and trying to determine if a magnetic material with a given $\mu$ and other magnetic properties is suitable for a transformer. It is important for a complete physicist to know how these things work, instead of just using a couple of simple formulas that provide an answer for the voltage and current. (Otherwise, you may very well wind up like the wizard in the movie the Wizard of Oz who accidentally takes off in his hot-air balloon, and Dorothy hollers to him "Come back, come back", and he responds "I can't come back-I don't know how it works" !). :-) :-) :-)

 

[Dale]

The Maxwell equation approach is also very important in things like transformer design and trying to determine if a magnetic material with a given μ and other magnetic properties is suitable for a transformer.

Agreed. Circuit theory has a more limited domain of applicability than Maxwell's equations. Similarly, Maxwell's equations have a more limited domain than QED, which is important for lasers and semiconductors and so forth.

 

[Charles Link]

And for @OnAHyperbola I think you might also find it of interest how Ampere's law comes into play (besides Coulomb's law), in the charge that builds up in a capacitor. Please see also post #75. The magnetic material terms $M$ and $J_m$ are zero for a capacitor. In addition, if the capacitor doesn't contain a dielectric, (for the simplest case), then $P=0$ $\$ and $J_p=0$. You can take the divergence of both sides of Ampere's equation, and (using $\nabla \cdot \nabla \times A=0$ for any and all $A$), you get $0=\nabla \cdot (\mu_o \, J_{total})+\mu_o \epsilon_o \, \nabla \cdot (dE/dt)$. Using $\nabla \cdot E=\rho_{total}/\epsilon_o$, you get the continuity equation: $$ \nabla \cdot J_{total} +\frac{\partial{\rho_{total}}}{\partial t}=0. $$ This shows how the extra non-steady state ($\mu_o \epsilon_o \, dE/dt$) term in Ampere's law is necessary in order to have the electrical charge obey the continuity equation. (Note: $\int \frac{\partial{ \rho_{total}}}{\partial t} dv=dQ/dt$ where $Q$ is the charge on one face of the capacitor and the volume integral is over one plate). Maxwell's Ampere's law is not needed to solve for the capacitor voltage (that can be done simply with Coulomb's law/Gauss's law), but the capacitor example actually gives the correct non steady state correction to Ampere's law, (the $\mu_o \epsilon_o \,dE/dt$ term), that is used along with Faraday's law to derive the electromagnetic wave equation.

 

[Delta2]

1) In circuit theory we neglect the displacement current term in Maxwell's-Ampere's law, and this is what makes KCL valid and also makes it easy to express the time varying magnetix flux as a $LdI/dt$ or $MdI/dt$ terms.

2) 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 $\frac{d\Phi}{dt}$ term in the left hand side of the equation and expressing it as $LdI/dt$ and/or $MdI/dt$ 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.

Whether we have the right to do 1) and 2) depends on the problem. Usually 1) is valid if the wavelength of the currents is much larger (10x) the dimensions of the circuit, and 2) is valid if the magnetic fields are weak or the circuit loops are too small or a combination of both.

 

[vanhees71]

@vanhees71 Very good reading=your posts #73 and #74. I agree it is important to include the physics of Maxwell's equations with the circuit theory or some of the finer details can be very ambiguous. One correction on post #74. (Hopefully you concur.) Ampere's law (in complete , non-steady-state) is $\$ $\nabla \times B=\mu_o \, J_{total}+\mu_o \epsilon_o \, dE/dt$ $\$ where $\$ $J_{total}=J_{free} +J_m+J_p$. $\$ Now $\nabla \times M=\mu_o \, J_m$ and $J_p=dP/dt$ .$\$ This gives $\$ $\nabla \times H=J_{free}+d D/dt$ $\$ where $\$ $B=\mu_o \, H+M$ $\$ and $\$ $D=\epsilon_o \, E+P$. $\$ The reason for the extra detail is it is important to include the non-steady state $dE/dt$ term in analyzing a capacitor in the case where the electric field in the capacitor is non steady state. (Note: One vector identity that is used in analyzing the above is $\nabla \cdot \nabla \times A =0$ for any vector $A$. Also $\$ $- \nabla \cdot P=\rho_p$ where $\rho_p$ is polarization charge density. The continuity equation applies $\nabla \cdot J_p+ \frac{\partial {\rho_p}}{\partial t}=0$ ) . In order to have consistency, the complete form of Ampere's law (rather than the steady state form) is necessary. (In the case that a capacitor is being charged, $\nabla \cdot J_{free}$ is not equal to zero) (Besides the inductor, the capacitor is also an interesting circuit element to analyze using Maxwell's equations). ... I'm going to need to study much more carefully your write-up for the Maxwell's equations with the moving surface, but I'm very glad you presented it. I always like seeing the complete details. It is important to know the complete details to be able to assess whether simplified equations that are used in steady-state and/or cases without any moving boundaries will apply in more complex cases.

Of course, you are right. The complete Maxwell equations include the "displacement current" in the Ampere-Maxwell Law. Otherwise the equations are inconsistent, and that's how Maxwell introduced the displacement current into Ampere's Law. I also agree to the treatment of "macroscopic electrodynamics" as you describe it.

In the usual circuit theory, however, one assumes that the quasistationary approximation is valid, i.e., the displacement current can be neglected. Roughly speaking that's the case, if the relevant geometrical extent of the entire circuit is small compared to the typical wave length of the involved electromagnetic field. Then you can lump the entire geometry and the constitutive parameters $\epsilon$, $\mu$, and $\sigma$ into some constants like $C$, $L$, and $R$ of the "compact elements" built into the circuit. Of course, this also includes magnetically coupled sources like transformers or the very illuminating example by Lewin discussed above.

The greatest mess is done with Faraday's Law, if it comes to moving parts as in generators. In the case of homopolar generators you need even relativity although nothing is moving at speeds close to the velocity of light. For a derivation of the complete integral form of Faraday's Law, see the appendix of

http://th.physik.uni-frankfurt.de/~hees/pf-faq/homopolar.pdf

 

[vanhees71]

@vanhees71 Very good reading=your posts #73 and #74. I agree it is important to include the physics of Maxwell's equations with the circuit theory or some of the finer details can be very ambiguous. One correction on post #74. (Hopefully you concur.) Ampere's law (in complete , non-steady-state) is $\$ $\nabla \times B=\mu_o \, J_{total}+\mu_o \epsilon_o \, dE/dt$ $\$ where $\$ $J_{total}=J_{free} +J_m+J_p$. $\$ Now $\nabla \times M=\mu_o \, J_m$ and $J_p=dP/dt$ .$\$ This gives $\$ $\nabla \times H=J_{free}+d D/dt$ $\$ where $\$ $B=\mu_o \, H+M$ $\$ and $\$ $D=\epsilon_o \, E+P$. $\$ The reason for the extra detail is it is important to include the non-steady state $dE/dt$ term in analyzing a capacitor in the case where the electric field in the capacitor is non steady state. (Note: One vector identity that is used in analyzing the above is $\nabla \cdot \nabla \times A =0$ for any vector $A$. Also $\$ $- \nabla \cdot P=\rho_p$ where $\rho_p$ is polarization charge density. The continuity equation applies $\nabla \cdot J_p+ \frac{\partial {\rho_p}}{\partial t}=0$ ) . In order to have consistency, the complete form of Ampere's law (rather than the steady state form) is necessary. (In the case that a capacitor is being charged, $\nabla \cdot J_{free}$ is not equal to zero) (Besides the inductor, the capacitor is also an interesting circuit element to analyze using Maxwell's equations). ... I'm going to need to study much more carefully your write-up for the Maxwell's equations with the moving surface, but I'm very glad you presented it. I always like seeing the complete details. It is important to know the complete details to be able to assess whether simplified equations that are used in steady-state and/or cases without any moving boundaries will apply in more complex cases.

Of course, you are right. The complete Maxwell equations include the "displacement current" in the Ampere-Maxwell Law. Otherwise the equations are inconsistent, and that's how Maxwell introduced the displacement current into Ampere's Law. I also agree to the treatment of "macroscopic electrodynamics" as you describe it.

In the usual circuit theory, however, one assumes that the quasistationary approximation is valid, i.e., the displacement current can be neglected. Roughly speaking that's the case, if the relevant geometrical extent of the entire circuit is small compared to the typical wave length of the involved electromagnetic field. Then you can lump the entire geometry and the constitutive parameters $\epsilon$, $\mu$, and $\sigma$ into some constants like $C$, $L$, and $R$ of the "compact elements" built into the circuit. Of course, this also includes magnetically coupled sources like transformers or the very illuminating example by Lewin discussed above.

The greatest mess is done with Faraday's Law, if it comes to moving parts as in generators. In the case of homopolar generators you need even relativity although nothing is moving at speeds close to the velocity of light. For a derivation of the complete integral form of Faraday's Law, see the appendix of

http://th.physik.uni-frankfurt.de/~hees/pf-faq/homopolar.pdf

 

[vanhees71]

A transformer is a single circuit element with four terminals. There is no magnetic coupling between circuit elements, only within an element.

No, what you use in circuit theory are KVL and KCL, which deviate from Maxwell's equations outside of circuit theory's domain of applicability. Similarly, Maxwell's equations deviate from QED, and Newtonian gravity deviates from the EFE outside their respective domains of applicability. Every simplified theory is derived from the more complicated theory, but the simplified equations are different and make different predictions outside of their domain of applicability.

In a transformer, of course you couple two circuits magnetically. In the spirit of circuit theory everything is lumped into a inductance matrix, describing this magnetic coupling. It's derived from Faraday's Law, which is one of Maxwell's equations.

What is KVL and KCL? I guess it's Kirchhoff's Rules, but of course these are nothing than forms of Maxwell's equations applied to a special case (and the assumption that the quasistationary approximation, i.e., the negligence of the displacement current is valid). I guess "KVL" means "Kirchhoff's voltage law", which is nothing else than the integral form of Faraday's Law with the integrals lumped into the constants of the elements in the circuit like $R$, $C$, $L$. Then "KCL" may stand for "Kirchhoff's current law", which is nothing else than the integrated continuity equation for electric charge in the quasistationary approximation in accordance with the use of Ampere's instead of the full Ampere-Maxwell Law.

I think our debate is merely about semantics rather than physics, but in my experience to have a good intuitive understanding of the Maxwell equations and the physics behind them is also very helpful for circuit theory.

 

[vanhees71]

The transformers and coils are circuit elements in circuit theory as per Dale. 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.

If we needed to examine a transformer as a circuit element then a process similar to this would be used.
http://www.electrical4u.com/equivalent-circuit-of-transformer-referred-to-primary-and-secondary/

Sure! That's why I was surprised by the claim that transformers, i.e., magnetically coupled circuits, are not included in circuit theory. That's for sure wrong. When I was a physics student many of my school mates studied electrical engineering and they liked to ask me about their problem sets. So I know, what electrical engineering students have to struggle with in the first semesters, concerning circuit theory ;-).

 

[Dale]

It's derived from Faraday's Law, which is one of Maxwell's equations.

More precisely, it can be derived from Faraday's law plus some simplifying assumptions. The simplifying assumptions are every bit as important as the underlying equation because they define the domain of applicability.

The situation is exactly analogous to Newtonian gravity and GR. Newtonian gravity can be derived from GR plus some simplifying assumptions. However, I think that you would be hard pressed to find anyone who would say that a first semester physics student doing their first gravity problem is doing GR.

Also, Maxwell's equations can be derived from QED plus some simplifying assumptions. So if we took your approach and applied it consistently then we would never say that we are using Maxwell's equations either, we would just say that we are doing QED.

I think our debate is merely about semantics rather than physics

This is probably true, but in my opinion Dr Lewin's "KVL is for the birds" semantics is so bad that it deserves to be argued against.

That's why I was surprised by the claim that transformers, i.e., magnetically coupled circuits, are not included in circuit theory. That's for sure wrong

I don't know why your own claim surprised you. It is for sure wrong, but it is your claim and not mine.

 

[vanhees71]

My claim is that transformers ARE in circuit theory. Yours was it isn't, but maybe I misunderstood you statement. My original claim was that it is important to stress that some "voltages" are not the same as "potential differences" but EMFs, and there Lewin's example is a good one to demonstrate that, and also Lewin's example is of course fully treatable by circuit theory, which he uses himself to analyze it in his writeup, which I linked in some previous posting in this thread. I'm still not sure what "KVL" means in the usual slang. For me it's one of Kirchhoff's rule, and it's just the integral form of Faraday's Law, and then it includes non-potential EMFs as in Lewin's example. If understood in this (imho usual) sense then KVL is NOT for the birds but can be successfully used to analyse Lewin's demonstration (which is nothing else than a cricuit with a magnetically coupled source EMF).

 

[Charles Link]

One of the more interesting parts of the physics to come out of this (at least for me) is the observation that the electric field from the Faraday term, whose voltage is the term $LdI/dt$ of the KVL theory actually points from minus to plus, unlike the electrostatic E of a capacitor. I had probably observed it previously, but never paid much attention to it... Professor Lewin's claim in the video that KVL is not 100% accurate I think is something he used to get the viewer's attention as well as to show his own interest at a rather fascinating problem. Even the physics student who asked me to help him with his homework of this same problem that appeared in an E&M textbook was very puzzled by it as I was myself when I first saw in in 1979. How could that be that your voltmeter theoretically reads 1/4 volt in one case and then you hook it up the other way (encircling the changing magnetic field) and it theoretically reads 3/4 volt? Perhaps it reads zero all the time was one possibility until the puzzle was finally resolved. It's a very good illustration of EMF concepts. I do think Professor Lewin does a reasonably good job of illustrating the concept. I had mentioned the problem in post #8 (not knowing Professor Lewin to have his famous video solution.) Whether it's R.P. Feynman or Professor Lewin, most of these fellows are a little eccentric. Vanhees71 presented a pdf form of Professor Lewin's solution for which I didn't observe any inaccurate claims. The thing I was focused on was if he have a good solution to the puzzle, which he did. The study of circuit theory is much more complete by having seen this puzzle and having a sound explanation for it. The video made for a much easier way to discuss the solution. Most viewers probably didn't even see Vanhees71 pdf link from Professor Lewin. I do think the video has its merits and I don't think anyone else has made a well-known video of the same solution.

 

[Dale]

My claim is that transformers ARE in circuit theory. Yours was it isn't, but maybe I misunderstood you statement

That is not my claim. You misunderstood my statement despite my clarifying it. A transformer is in circuit theory. I never claimed otherwise.

In circuit theory an ideal transformer is modeled as a single circuit element with four terminals. All of the magnetic coupling is inside the transformer, which is within the assumptions of circuit theory.

and also Lewin's example is of course fully treatable by circuit theory

No, the circuit shown by Dr Lewin, in contrast, shows magnetic induction outside a circuit element, which explicitly violates the assumptions of circuit theory.

I'm still not sure what "KVL" means in the usual slang. For me it's one of Kirchhoff's rule, and it's just the integral form of Faraday's Law,

KVL is Kirchoff's voltage law, and as discussed repeatedly it is not just the integral form of Faraday's law, it is that plus some simplifying assumptions which you seem to consistently neglect.

Neglecting the assumptions is a huge conceptual error, IMO far worse of a "didactic sin" than anything else that I have seen you complain about from others. You really need to reexamine your approach here.

 

[vanhees71]

Well, then tell me which additional assumptions you need for the KVL, and I still don't see, where Lewin's setup needs more than the usual quasistationary approximation to be explained. I guess you only need the full Maxwell theory, if the entire setup becomes larger or of the order of magnitude of the typical wavelength of the applied source, and that's not the case in Lewin's experiment, but as I said, I think this whole discussion is pretty empty since it's pure semantics. E.g., of course in circuit theory you treat also transformers as "compact element", i.e., a "four pole" (I don't know if this is a correct English translation of the German "Vierpol"). In the formalism it's given by the inductance matrix.

BTW Circuit theory is only a special case of the general "transfer approach" for the full Maxwell equations, which has been developed by J. Schwinger to simplify the calculation concerning wave guides.

 

[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.