Kirchhoff's second law and inductor's field

[DaTario]

Hi All,

The basis of Kirchhoff's second law is that an electric potential function is well defined, in such a way that, in a closed path, the potential difference between initial and final points (the same point) will be zero.

However, when we introduce in the circuit an inductor, we start producing electric fields which are not well described by potential function (as a closed line integral of E in general doesn't vanish).
Thus, why do we use Kirchhoff's second law in describing LC, RL and RLC circuits?
Is it a completely regular procedure in this theoretical context?

Best Regards,

DaTario

 

[stevenb]

Hi All,

The basis of Kirchhoff's second law is that an electric potential function is well defined, in such a way that, in a closed path, the potential difference between initial and final points (the same point) will be zero.

However, when we introduce in the circuit an inductor, we start producing electric fields which are not well described by potential function (as a closed line integral of E in general doesn't vanish).
Thus, why do we use Kirchhoff's second law in describing LC, RL and RLC circuits?
Is it a completely regular procedure in this theoretical context?

Best Regards,

DaTario

This question often comes up and is sometimes hotly debated. I hesitate to give you a direct answer, because I really don't want to get drawn into an ugly debate. However, what would become of science if we behave as cowards?

In my opinion, the best answer to this is to look up the definition of Kirchoff's Voltage Law in James Clerk Maxwell's Original "Treatise on Electricity an Magnetism".

In my opinion, his description is the most lucid statement with the least chance of misunderstanding and is historically closest to the work done by Kirchoff.

In my interpretation of his statement, Kirchoff's voltage law is just a less sophisticated (but no less general) version of Maxwell's version of Faraday's law. It does not require that "in a closed path, the potential difference between initial and final points (the same point) will be zero" at all. It works for nonconservative fields too, but the difference is that it does not distinguish the cause of an EMF, it simply states the following:

"In any complete circuit formed by the conductors, the sum of the electromotive forces taken around the circuit is equal to the sum of the products of the current in each conductor multiplied by the resistance of that conductor."

I've tried to track down the original paper by Kirchoff, but have been unable to do so. My guess is that Maxwell extended Kirchoff's earlier statement and made it more general based on his own later discoveries. In any event, I rely on Maxwell's definition. It clearly shows that EMF is summed and this can include EMF due to flux change or any other EMF (battery for example).

I've attached a copy of the relevant page to save you time.

Note that I'm fully aware that many intelligent and famous people don't always agree with what I've said here. In fact, I'm aware of this now because I lost a previous debate on this topic. If others disagree, feel free to post your opinions. I won't debate them. I do encourage anyone interested in the correct answer to read Maxwell's statement carefully, compare it to Faraday's law and then make their own informed opinion.

 

[DaTario]

Thank you for the reference. Is this debate open here? I mean, there were post on this subject and none of them presented a consistent conclusion?

Best wishes

DaTario

 

[stevenb]

Thank you for the reference. Is this debate open here? I mean, there were post on this subject and none of them presented a consistent conclusion?

Best wishes

DaTario

In my opinion the debate can be considered open. There are many books that express Kirchoff's law as you said. But, I find that older books state it in a way consistent with Faraday's law and you can see Maxwell's statement for yourself.


Also, your question highlights, very clearly, the basic problem with using a version that does not consider EMF due to flux change. It doesn't tell you anything about circuits with coils and transformers, or motors and generators.

It seems unwise (to me) to introduce concepts to beginning students of electricity with a so-called "law" that is inconsistant with real physics and doesn't describe the most basic circuits using coils.

 

[cortiver]

In my mind (and I'm sorry if I'm repeating something anyone's said on these forums before, I couldn't find any previous discussion), the solution to the problem is simply that Kirchoff's law refers to the conservative part of the electric field.

The electric field, like any vector field, can be written as the sum of a conservative part and a solenoidal part, $\mathbf{E} = \mathbf{E}_c + \mathbf{E}_s$. We can then define the electric potential V in terms of the conservative part so that
$$\int_a^b \mathbf{E}_c\cdot d\mathbf{l} = V_a - V_b$$
Thus Kirchoff's law applies perfectly well to the conservative part of the electric field. This definition of $V$ is in line with how we actually talk about circuits - for example, when we say that the voltage drop across an inductor is L di/dt we don't mean that
$$\int_a^b \mathbf{E} \cdot d\mathbf{l} = L \frac{di}{dt}$$
(actually if the inductor has no resistance then the electric field inside it must be zero!); we are actually referring only to the conservative part:
$$V_a - V_b = \int_a^b \mathbf{E}_c \cdot d\mathbf{l} = L \frac{di}{dt}$$.

 

[stevenb]

... the solution to the problem is simply that Kirchoff's law refers to the conservative part of the electric field...

I think it's important to identify what "problem" there is and make sure we are all talking about the same thing. I view the problem as, " how do you deal with inductors if flux change is ignored and only conservative fields are allowed".

With that problem identified, we can note that you are basically deciding on a definition of Kirchoff's Voltage law once you say it applies to conservative fields only. In principle there is nothing wrong with this as long as the definition is clear. This has been done by many people and many books do define KVL in the way you described. Once this is done, one is forced to do as you said and just use a voltage law equation for coils. The non-conservative nature of the EMF of the coil is then treated as internal to the coil and not important to the external circuit. In other words, it works even though the physics is not correct and even though this definition of Kirchoff's law is not at all consistent with Faraday's Law.

This view is exactly the opinion I held for a long time. Then, I got into a debate with someone and was forced to reconsider. It was clear that old books defined KVL in a manner consistent with Maxwell, while recent books were not consistent with the definition. Some new books used the old definition, some were worded in confusing ways and some were clearly the version that the line integral of electric field is zero. This left the issue open and really one of semantics. I then decided to try and track down the original paper by Kirchoff, but was unable to find it. Finally, I decided that Maxwell's definition should be used. He was writing in the same era of discovery as Kirchoff and he is the one who provided the basic final complete theory of electromagnetism. He was meticulous in presicely stating mathematical and verbal definitions that would stand the test of time. The benefit of Maxwell's definition is that it can be used by beginners in simple circuit analysis without getting into the various confusing ideas of non-conservative fields, but it does this in a way that is correct physically. In other words, it does not lie to the student.

I would pose this question on this subject as follows. If Maxwell clearly defined Kirchoff's Current Law to be consistent with charge conservation and clearly defined Kirchoff's Voltage Law to be consistent with Faraday's Law, why should we accept later definitions of either of these laws that dilute their validity and in effect bring them to a lower status than "Laws of Nature"? These two laws are very fundamental in physics. While charge conservation is simple to understand, Faraday's law is non-intuitive and not well-understood by most people, including an unacceptably high percentage of phycists and engineers. A simplified version of KVL does not help this problem, and indeed might even be a contributor to it.

 

[DaTario]

This definition of $V$ is in line with how we actually talk about circuits - for example, when we say that the voltage drop across an inductor is L di/dt we don't mean that
$$\int_a^b \mathbf{E} \cdot d\mathbf{l} = L \frac{di}{dt}$$
(actually if the inductor has no resistance then the electric field inside it must be zero!); we are actually referring only to the conservative part:
$$V_a - V_b = \int_a^b \mathbf{E}_c \cdot d\mathbf{l} = L \frac{di}{dt}$$.

If the charges in conductor are producing a time varying current, is it correct to say that the field inside is zero?

Best Regards

DaTario

 

[DaTario]

With that problem identified, we can note that you are basically deciding on a definition of Kirchoff's Voltage law once you say it applies to conservative fields only. In principle there is nothing wrong with this as long as the definition is clear. This has been done by many people and many books do define KVL in the way you described. Once this is done, one is forced to do as you said and just use a voltage law equation for coils. The non-conservative nature of the EMF of the coil is then treated as internal to the coil and not important to the external circuit. In other words, it works even though the physics is not correct and even though this definition of Kirchoff's law is not at all consistent with Faraday's Law.

But in this case the formalism we apply so often is an approximation, isn't it?

Best Regards

DaTario

 

[stevenb]

But in this case the formalism we apply so often is an approximation, isn't it?

Best Regards

DaTario

Yes, I think it is, but we should also keep in mind that circuit theory is inherently an approximation for any dynamic case. If I understand you correctly, I basically agree with your underlying point.

 

[cortiver]

If the charges in conductor are producing a time varying current, is it correct to say that the field inside is zero?

Best Regards

DaTario

Yes, it is. The motion of charges in a conductor is determined by the microscopic form of Ohm's Law:

$$\mathbf{E} = \mathbf{J}/\sigma$$

where E is the total electric field, J is the current density, and σ is the conductivity. In the approximation that the conductor has no resistance, σ goes to infinity and therefore E = 0.
(Of course, Ohm's Law breaks down at very high frequencies such that the charges in the conductor have no time to respond to changes in the electric field. This should not be important for frequencies normally encountered in circuits, and I don't think it was what you were referring to).

I think it's important to identify what "problem" there is and make sure we are all talking about the same thing. I view the problem as, " how do you deal with inductors if flux change is ignored and only conservative fields are allowed".

Well, this problem has no solution. If you introduce the unphysical assumption of conservative fields then of course you're not going to get a realistic result. But this is not what I'm assuming in the formulation I gave above. Once you stipulate that Kirchoff's law applies only to the conservative part of the electric field, there is no contradiction with Maxwell's equations at all. I don't understand why you say "the physics is not correct", "this definition of Kirchoff's law is not at all consistent with Faraday's Law", or that this formulation "lies to the student". If textbooks are stating Kirchoff's second law without explaining that it only applies to the conservative part of the field, then that would certainly be misleading.

If the question is simply which formulation is most useful pedagogically, then I think that, since the concept of electric potential is universally used in practice, a formulation which retains this concept is better than one which doesn't.

 

[stevenb]

If you introduce the unphysical assumption of conservative fields then of course you're not going to get a realistic result. But this is not what I'm assuming in the formulation I gave above. Once you stipulate that Kirchoff's law applies only to the conservative part of the electric field, there is no contradiction with Maxwell's equations at all. I don't understand why you say "the physics is not correct", "this definition of Kirchoff's law is not at all consistent with Faraday's Law", or that this formulation "lies to the student". If textbooks are stating Kirchoff's second law without explaining that it only applies to the conservative part of the field, then that would certainly be misleading.

I follow you here and agree. Yes, my problem is that I've found many textbooks are not clearly describing the situation as you just did, and I'm sorry I didn't pay better attention to your exact formulation. If one is careful in making definitions, then a theoretically and physically sound statement can be made.

I'm not really trying to debate here and change anyone's mind, but I would like to just make what I was saying clear, and it really isn't dependent on whether we can make sound definitions. I feel that Maxwell's statement of KVL is consistent with Faraday's law and is thus a fundamental physical law. I also feel (assuming I interpreted your statement correctly) that it differs from your formulation of KVL (or if not then it's the same and we agree). In principle this is OK if definitions are clear, but somehow I'm bothered that Maxwell's statement is there first and should have priority.

There is one flaw in this last feeling of mine. I still have not verified Kirchoff's original statement. My understanding is that his experiments were with batteries and resistors, but I don't have his paper, and even if I did, I don't understand German. The key question in my mind is whether he made the same statements as Maxwell published in his Treatise. If so, we have historical evidence that Kirchoff's two laws are circuit versions of fundamental laws: charge conservation and Faraday's law. These are both major physics laws and I'm not comfortable redefining them in a way that demotes this status.

If Kirchoff's original statement is not consistent with Maxwell's, then the question is much more wide open in my view. We then have a case of Maxwell trying to make a law or rule that is more general. I would still lean toward Maxwell, but I can't make a clear argument why others should necessarily follow my suggestion.

If the question is simply which formulation is most useful pedagogically, then I think that, since the concept of electric potential is universally used in practice, a formulation which retains this concept is better than one which doesn't.

OK, I won't try to change your mind on this. You are stating your opinion just as I stated mine above. I can't be sure which is less confusing and best to teach to students. I have no way to do an objective test.
EDIT: Actually, I'm wondering now if we are really saying the same thing. Can you state your version of KVL as an independent statement, so I can be sure I'm understanding you correctly? My main issue is with versions of KVL which are not consistent with Faraday's Law. One example is the statement that the integral of electric field around a closed path is zero. Another, is the sum of potential drops is zero. The stated assumption in the first post was "in a closed path, the potential difference between initial and final points (the same point) will be zero."

 

[Antiphon]

I agree with stevenb. I haven't seen the debate hot or otherwise but will add only this: never perform field operations on circuits. Although it can be intuitive it is not valid. For example E.dl along the wire in a schematic is always zero, not only because it has no resistance but because it is a topological abstraction with zero electrical length. The same is true of r,l,c. They have zero size and therefore are abstractions rather than idealizations. Maxwells equations do not permit inductors to exist without interwinding capacitance for example, and no circuit with a finite number of elements can represent a simple loop of wire as another example. Circuit theory is in truth a standalone discipline wherein it is imposed as a condition that a schematic loop's voltage totals to zero.

 

[cortiver]

I agree with stevenb. I haven't seen the debate hot or otherwise but will add only this: never perform field operations on circuits. Although it can be intuitive it is not valid. For example E.dl along the wire in a schematic is always zero, not only because it has no resistance but because it is a topological abstraction with zero electrical length. The same is true of r,l,c. They have zero size and therefore are abstractions rather than idealizations. Maxwells equations do not permit inductors to exist without interwinding capacitance for example, and no circuit with a finite number of elements can represent a simple loop of wire as another example. Circuit theory is in truth a standalone discipline wherein it is imposed as a condition that a schematic loop's voltage totals to zero.

Sure, you can treat circuit theory as an abstract mathematical discipline, but are you saying it's not important to understand how those abstractions relate to the real world? After all, the original post was asking a question about how to reconcile Kirchoff's loop rule (basically an axiom of circuit theory) with Maxwell's equations, which describe the actual electromagnetic fields in space.

 

[Phrak]

However, when we introduce in the circuit an inductor, we start producing electric fields which are not well described by potential function (as a closed line integral of E in general doesn't vanish).

You don't get to use Kichoff's voltage law inside an inductor. It is not, in general true, in the presence of magnetic fields. Kichoff's current law isn't true in half of a capacitor either. The current law isn't generally true in electric fields. These laws don't work well outside their domains of applicability.

 

[cortiver]

EDIT: Actually, I'm wondering now if we are really saying the same thing. Can you state your version of KVL as an independent statement, so I can be sure I'm understanding you correctly? My main issue is with versions of KVL which are not consistent with Faraday's Law. One example is the statement that the integral of electric field around a closed path is zero. Another, is the sum of potential drops is zero. The stated assumption in the first post was "in a closed path, the potential difference between initial and final points (the same point) will be zero."

Basically, what I am saying is that the statement of KVL as often given in textbooks, "the sum of potential differences around a closed loop is zero", is correct, and not inconsistent with Faraday's law, as long as you define the potential in terms of the conservative part of the electric field (but actually this is the only way you can define it! You can't define a scalar potential for a nonconservative field). It's this last part which tends not to be well explained. You could also state it as "the line integral of the conservative part of the electric field around a closed loop is zero", though this is basically a tautology since that's precisely the definition of conservative.

I'll also point out that the conservative part of the electric field obeys
$$\begin{align*} \nabla \cdot \mathbf{E}_c &= \rho/\epsilon_0 \\ \nabla \times \mathbf{E}_c &= 0 \end{align*}$$
Thus it is basically the electrostatic field arising from charge separation. The solenoidal part obeys
$$\begin{align*} \nabla \cdot \mathbf{E}_s &= 0 \\ \nabla \times \mathbf{E}_s &= -\frac{\partial\mathbf{B}}{\partial t} \end{align*}$$
Thus the solenoidal part arises entirely from changing magnetic fields. Hence the two components have completely different origins and it does make a lot of sense to treat them separately.

 

[Antiphon]

Sure, you can treat circuit theory as an abstract mathematical discipline, but are you saying it's not important to understand how those abstractions relate to the real world? After all, the original post was asking a question about how to reconcile Kirchoff's loop rule (basically an axiom of circuit theory) with Maxwell's equations, which describe the actual electromagnetic fields in space.

No it's very important. I'm saying if you look even at the well-written post preceeding this one, there is a seamless discussion of going around a circuit loop and going around a closed line integral in space. I'm saying that there is a conceptual blurring in that is the root of the confusion. Circuits do not contain electric fields or magnetic fluxes so it doesn't make sense to talk about conservative fields and totaling potentials around a circuit as concepts that should correspond. It's interesting but circuit theory is mathematically as distinct from Maxwell's equations as integers are from the real numbers. Sure there's a resemblence but they are not the same animals.

 

[cortiver]

No it's very important. I'm saying if you look even at the well-written post preceeding this one, there is a seamless discussion of going around a circuit loop and going around a closed line integral in space. I'm saying that there is a conceptual blurring in that is the root of the confusion. Circuits do not contain electric fields or magnetic fluxes so it doesn't make sense to talk about conservative fields and totaling potentials around a circuit as concepts that should correspond. It's interesting but circuit theory is mathematically as distinct from Maxwell's equations as integers are from the real numbers. Sure there's a resemblence but they are not the same animals.

I don't see why you draw such a firm distinction. The laws of circuit theory are interesting only in as much as they approximate the behaviour of real-world devices. The laws of circuit theory are perfectly self-consistent, but the question of whether they apply to the real world requires consideration of the actual behaviour of the electromagnetic fields. We apply Kirchoff's loop rule to circuits because the properties of the electromagnetic fields ensure that it correctly describes realizations of those circuits.

EDIT: I'm already starting to get a headache trying to separate my conceptions of a circuit in abstraction, as represented in a circuit diagram, and its realization. In general there's nothing wrong with slightly sloppy thinking if it doesn't lead you to incorrect conclusions, so could you explain how exactly you think the "conceptual blurring" is leading to confusion?

 

[Antiphon]

We're in full agreement. Circling back to the OP, they should not be puzzled by the "mismatches" between field theory and circuit theory. We shouldn't be surprised when anomolies show up in trying to take the analogies over into analogs.

I'll give an example of how circuit theory is distinct from field theory.

Consider the "missing fundamental element" of circuit theory, the Memristor. This device completes a symmetry in circuit theory that had been missing. It's a variable resistance with an integral history of the current that's passed through it.

Not only is this device not necessary to complete a symmetry in Maxwell's equations but it's not even possible to realize the element using Maxwell's equations and linear materials. This inability to track back to field equations hasn't stopped circuit theorists from proclaiming this one of the four fundamental circuit elements.

My bias, like the physicists posting here is toward field theory not circuit theory. My idea of completing a missing fundamental symmetry is to build three new circuit elements based on the magetic monopoles. There would be magnetic capacitors, inductors and resistors that where the moving charges were all magnetic.

 

[cortiver]

So are you trying to suggest that there is a discrepancy between Kirchoff's loop rule and Faraday's law, but it doesn't matter because the analogies between circuit theory and Maxwell's equations are not perfect? Because the explanation I gave seems a lot more satisfactory.

 

[DaTario]

Circuits do not contain electric fields or magnetic fluxes so it doesn't make sense to talk about conservative fields and totaling potentials around a circuit as concepts that should correspond. It's interesting but circuit theory is mathematically as distinct from Maxwell's equations as integers are from the real numbers. Sure there's a resemblence but they are not the same animals.

Could you please elaborate a little further on this part?

Best Wishes

DaTario