# Loop Rule

Why does the Loop Rule arise as a consequence of conservation of energy?

## Answers and Replies

Related Other Physics Topics News on Phys.org
Tide
Homework Helper
Kirchoff's loop rule simply states that if you traverse a loop and return to given point then the potential at that point remains the same, i.e. the electrical potential is single-valued!

rbj
Tide said:
Kirchoff's loop rule simply states that if you traverse a loop and return to given point then the potential at that point remains the same, i.e. the electrical potential is single-valued!
what's the deal with the terminology here?? are we talking about Kirchoff's Voltage Law (KVL)?

if there is a net changing magnetic field inside the loop (of any reasonable quantity), it won't be a single electrical potential. this is why 60 Hz AC hum gets induced into audio circuits. but it should be small.

if there is no net changing magnetic field, then taking a small test charge from point "A" around the loop and back to point "A", then the electrostatic field is "conservative" and the integral or sum of all of the work done to that test charge will be zero and that is why, assigning the polarities consistently going around the loop clockwise, the sum of all of the voltages is zero.

Kirchoff's Current Law (KCL) for every node (less the "ground" node), Kirchoff's Voltage Law (KVL) for every loop (there are also redundant loops that need no separate equation), plus the volt-amp characteristics of every device connect between the nodes (that are also in the loops) are exactly the information one needs to analyze an electrical or electronic circuit.

Last edited:
Tide
Homework Helper
rbj,

I presume that's what hidayah is asking about.

I'm not sure what you mean when you say there's not a single electrical potential when an oscillating magnetic field is present. If the electrical potential is multivalued at any given point then it is unphysical. Perhaps you meant there are different frequency components?

rbj said:
what's the deal with the terminology here?? are we talking about Kirchoff's Voltage Law (KVL)?

if there is a net changing magnetic field inside the loop (of any reasonable quantity), it won't be a single electrical potential. this is why 60 Hz AC hum gets induced into audio circuits. but it should be small.

if there is no net changing magnetic field, then taking a small test charge from point "A" around the loop and back to point "A", then the electrostatic field is "conservative" and the integral or sum of all of the work done to that test charge will be zero and that is why, assigning the polarities consistently going around the loop clockwise, the sum of all of the voltages is zero.

Kirchoff's Current Law (KCL) for every node (less the "ground" node), Kirchoff's Voltage Law (KVL) for every loop (there are also redundant loops that need no separate equation), plus the volt-amp characteristics of every device connect between the nodes (that are also in the loops) are exactly the information one needs to analyze and electrical or electronic circuit.
Are you implying that 'loop' or 'mesh' analysis does not work for AC circuits? If you are implying this, you might want to rethink your statement.

rbj
Are you implying that 'loop' or 'mesh' analysis does not work for AC circuits?
no, i am not. (and i am not sure what i said to be construed to mean that.)

If you are implying this, you might want to rethink your statement.
i am not sure if this is the offending concept, but one of Maxwell's Equations, expressed in integral form (i think it's Faraday's Law) says that when there is a changing magnetic field, going around that changing magnetic field in a closed loop gives you an induced potential (voltage) that is proportional to the rate of change of the magnetic field. do you agree with that? if so, then there is a deviation from Kirchoff's Voltage Law, the voltages don't all add up to zero.

like the deviation of Kirchoff's Current Law (where the currents going into a node don't sum to exactly zero resulting in a charge buildup), this KVL deviation is not normal and is not what we do when we analyze circuits (using the so-called "node-voltage" or "loop-current" methods) because we model that non-zero current or voltage as an additive source (and it's some kind of noise or hum or error source).

are we on the same page? i know what i'm typing about here.

Last edited:
I still don't see how this is not saying that KVL, KCL, Loop analysis and Nodal Analysis does not work for AC circuits.

"Hum and buzz (50Hz/60Hz and it's harmonics) occur in unbalanced systems when currents flow in the cable shield connections between different pieces of equipment. Hum and buzz can also occur balanced systems even though they are generally much more"

http://www.epanorama.net/documents/groundloop/

I don't think that what you're trying to tell me is coming across correctly.

Regards,

Tide
Homework Helper
rbj,

You're thinking about induction and, since it's caused by varying magnetic flux, it's not quite correct to talk about electrical potential (vector potential will do, however!). In any case, the potential is single valued so traversing a loop will get you back to the same potential when you return to your starting point. Kirchoff works!

Induced fields are taken into account as the inductance (mutual and self) of the circuit or its components.

rbj
i dunno what to say to you guys. i dunno if it's a communication problem or whatever. maybe i'm just trying too hard to dot the i's and cross the t's, but this is a real thing in real circuit design on real printed circuit boards. if you have a bunch of components laid out in a loop that is big enough in area, and in an environment where there is some 50 Hz or 60 Hz electromagnetic radiation about, this causes this hum to get into the voltage signal of those components laid out in a loop. this is precisely because of Faraday's Law and that, in fact traversing this loop will get you back to a different potential when you return to your starting point. but, what we do so we can use our normal node-voltage or loop-current analysis methods, is we model that difference in potential (that should be zero if we were more fortunate) as a "virtual" component: an additive independent voltage source. it might be a couple of microvolts.

this is not controversial among practicing electrical engineers. but it is something we don't put into the first sophmore physics and EE texts, because we don't want to confuse the hell out of students and we are able to model this added voltage as a lumped voltage source in the loop and then say all of the voltages add up to zero.

are we still in disagreement?

Tide
Homework Helper
rbj,

The hum is real but your explanation is not.

traversing this loop will get you back to a different potential when you return to your starting point
Please explain how the electrical potential at a point can have two different values?

rbj
one more thing:

forget about circuits for the time being...

given just Coulomb's Law and a static electric field, then you have a conservative potential field and doing a line integral of work moving a test charge around any closed loop will get you zero. always.

this is not the same thing as a having a static electric field along with a changing magnetic field.

rbj
I still don't see how this is not saying that KVL, KCL, Loop analysis and Nodal Analysis does not work for AC circuits.
this is not directly related to the issue at hand, but, in fact, the KVL, KCL, Loop and Node analyses we do for circuits at low frequencies does not work for extremely high frequency (like microwaves and above) AC circuits. you get to go to grad school in electrical engineering and take a sh1tload of really hard classes to learn how to do that stuff.

think of an antenna. there is current at one part of the radiating element where it is driven, but no current at the ends. that sure as hell does not satisfy KCL.

edit: just to be clear KVL, KCL, Loop and Node analyses work well when the wavelength of the signal in the circuit is much longer than the dimensions of the circuit and components. when that is the case, you gotta do physics rather than just circuit analysis.

"Hum and buzz (50Hz/60Hz and it's harmonics) occur in unbalanced systems when currents flow in the cable shield connections between different pieces of equipment. Hum and buzz can also occur balanced systems even though
hum and buzz can occur inside of a single box where the power supply components are not adequately shielded from the very low voltage analog signal processing components. you get some kind of nasty large varying E and M fields and the changing M fields induce spurious voltages in loops of components strung together. when the signal going into the op-amp is only a few microvolts (say it's coming out of an un-preamped microphone), that induced 60 Hz voltage can add up to something that competes well with the desired signal. and it is precisely because the voltages are not adding to exactly zero.

what happens when you have a single circular closed loop of wire in the presence of a non-zero and changing magnetic field? what happens when you apply

$$\oint_{s} \mathbf{E} \cdot d\mathbf{s} = - \frac {d\Phi_{\mathbf{B}}} {dt}$$ ?

that first integral, on the left hand side, is a voltage. do you get zero?

Last edited:
rbj
Tide said:
The hum is real but your explanation is not.
i'm sorry you don't like it.

Please explain how the electrical potential at a point can have two different values?
the purely electrical potential at a point does not have two different values at the same instance of time.

but KVL isn't just that. KVL says that, assigning a consistent sense of polarity going around a loop of electrical components, that the voltages of all of the components of the loop add to zero. but if there is a changing magnetic flux that this loop of components is strung around, those voltages will not add to zero.

KVL works. but sometimes to make it work (when analyzing these noisy or hummy situations) we lump that non-zero induced voltage into a single virtual voltage source in that loop. sometimes we model it as several little voltage sources in series with each component in that loop (that might be more accurate, but is often not necessary).

Tide
Homework Helper
The loop rules apply to an instant of time.

rbj
Tide said:
The loop rules apply to an instant of time.

i never implied anything different. the left hand side of

$$\oint_{s} \mathbf{E} \cdot d\mathbf{s} = - \frac {d\Phi_{\mathbf{B}}} {dt}$$

is a line integral in space applied at a instant of time and the right hand side is an instantaneous rate of change at the same instant of time.

yet, if the right hand side is non-zero (that is my premise for the case of induced hum), then the left side is non-zero.

also, back to terminology: by "loop rules" i presume you mean Kirchoff's Voltage Law. if this is not the case, then all bets are off.

Last edited:
Rbj is right again. I have taken all those hard classes and then some.

When you analyze a circuit, it's an idealization where the
inductance, capacitance and resistance are concentrated in the components.

In reality they are distributed everywhere. So KVL works for an idealized
circuit because the idealization takes into account all the inductance. You can't treat
a real wire as if it were an ideal wire. Then KVL can't be used and you
must use the full induction relation for a physical wire.

The confusion with AC vs DC is because in a DC circuit the time dependent terms are
gone so you are only left with resistances. These are distributed too in
a real circuit but you can still apply KVL by integrating the differential VI
drop around the actual circuit- because different parts of the circuit are
decoupled at DC.

But you cannot do this in an AC circuit because the equivalent schematic will change.
Parts of the circuit will couple to other parts and in different ways depending
on the frequency.

A simple example is the power line in your house. At 500 MHz, the schematics
won't look anything like they do at 60 Hz if they are to accurately model the
circuit. And at 500 MHz, the component values of the equivalent circuit
will change depending on where I sit in my house. This is not the case at 60 Hz.

In sum,
-KVL, KCL work only for schematics, not physical circuits.
-KVL, KCL can still work for a DC but not AC physical circuit by integrating the differential VI drops around the circuit

Edit: Oh yes, and as for the OP, it has nothing to do with conservation of energy.

Last edited:
Tide
Homework Helper
rbj,

Regarding my comment: "The loop rules apply to an instant of time." you said

i never implied anything different
Well, yes, you did. You said

traversing this loop will get you back to a different potential when you return to your starting point
followed by

the purely electrical potential at a point does not have two different values at the same instance of time
If the electrical potential at a point has two different values and if those two different values cannot be at the same instant then you must have returned to the starting point at a different time than when you started.

In any case, I think we should put the burden back onto the original poster whose question is ambiguous and unclear.

rbj
Tide said:
rbj,

Regarding my comment: "The loop rules apply to an instant of time." you said

"i never implied anything different."

Well, yes, you did. You said

"traversing this loop will get you back to a different potential when you return to your starting point"
here i made the mistake of using the word "when". i do not offhand know how else to concisely word it, but "when" did not mean at a later time than starting. you can still, conceptually traverse the loop with a test charge in an instant of time (getting back to your starting point at the same instant of time) and you will have a line integral of the total amount of work done on that test charge. if there is no time-varying magnetic field, then the line integral done at that instant of time is zero because an electrostatic field is conservative. but if there is a non-zero

$$\frac {d\Phi_{\mathbf{B}}} {dt}$$

at that instant of time, then

$$\oint_{s} \mathbf{E} \cdot d\mathbf{s}$$

is also non-zero and the sum of all of the work done to the test charge is not zero meaning that the instantaneous sum of all of the voltages of the components in the loop do not add to zero and that this is a practical deviation from KVL.

i think you're stretching it a little to imply that by using "when" i meant that it had to be two different instances of time for starting and ending the closed loop. in fact, i think i proved to you that i was correct all the time and you're trying to save face. that's okay.

but i never once said that KVL was not meant to be applied at an instant of time. what i always said is, that under some kind of adverse circumstances, KVL is not always precisely valid. sometimes the voltages (at an instant of time) do not always add to zero, going around the closed loop and that circumstance can only be during an instant of time that there is a net changing magnetic field flowing through the hole of the loop. i also said that the practice often is to say KVL is exactly true anyway, but then introduce a phantom voltage source in the loop that would be equal to the line integral above.

followed by

"the purely electrical potential at a point does not have two different values at the same instance of time."

If the electrical potential at a point has two different values and if those two different values cannot be at the same instant then you must have returned to the starting point at a different time than when you started.
i said "purely electrical" when i could have been more clear by saying "purely electrostatic". the point i was making was that at an instant of time there is one unambiguous electrostatic potential for that point because classical electrostatic fields are conservative fields (i think the OP was sorta referring to that when he says "Conservation of energy". in that case KVL is completely accurate. but a more general electromagnetic field is not necessarily conservative and in that case KVL is not perfectly valid.

In any case, I think we should put the burden back onto the original poster whose question is ambiguous and unclear.
if the OP meant "KVL" when he said "Loop rules", i think it was reasonably clear. it was the other thread (Convervation of charge) that i didn't get right away.

anyway, perhaps i shouldn't have offered these arcane caveats to KCL and KVL (we basically always take them to be exactly true when we do circuit analysis), but i just didn't want some other EE taking issue with what i said if i didn't bring up these caveats. i was trying to cover my butt and got it spanked (unjustly, i think) anyway.

Tide
Homework Helper
rbj,

i think you're stretching it a little to imply that by using "when" i meant that it had to be two different instances of time for starting and ending the closed loop.
I am stretching nothing and I understood exactly what you meant by the word "when." Reread what I wrote in my previous post.

Your first statement says (a) the potential at a given point has two different values - at the same time - while your second says (b) the different values at a point can only occur at different times. Which is it?

If (a) then the potential is multivalued and nonphysical which makes it moot.

If (b) then you were in fact implying some temporal element in the loop theorem.

We've already spent more time on the semantics than is warranted and, again, I suggest we return the burden to the original poster who posed an ill-stated and ambiguous question.

rbj
Tide said:
I am stretching nothing and I understood exactly what you meant by the word "when."
i'm glad to read that. then there is no misunderstanding. nowhere did i mention anything about KVL at different times. (by "changing magnetic field" i mean the instantaneous rate of change at some instant of time, what the calculus prof. calls "the derivative".)

Reread what I wrote in my previous post.
i've been very careful (except for saying "when" in a context of an instant of time and "purely electrical", meaning no magnetic component, when i should have said "purely electrostatic"). you said that i implied that this deviation from KVL had something to do with different times or that my explanation had something to do with different times and i never said that. it is you, Tide, that needs to carefully read what the other is saying and not inject something unsaid into it.

Your first statement says (a) the potential at a given point has two different values - at the same time - ...
only if there is a changing magnetic field and then comparing the potential if there was no movement of the test charge to the resulting potential if the test charge was moved around a closed curve that encircled some of that changing magnetic field. same point in space, same instance of time, but different potentials if you include the magnetic effects. (of course they can't be different if you are only considering electrostatic effects.)

to be more clear, scaler "potential" doesn't have a well-defined meaning in cases other than purely electrostatic. with pure electrostatic situations, there is this unambiguous property of each point in space called its electrical potential and then we can call that electrostatic field a "conservative" field. and if you move a test charge around in that field and move it back to the starting point, no net work is done to it.

but it is not well-defined as a scaler field for situations with changing magnetic fields. you can take a test charge, move it around, return it to the original point (all in an instant of time) and been forced to do a net non-zero amount of work doing it. however if you just left that charge alone at that point (fixed it so it can't move), you would have done no work on it. same situation, identical starting and ending point, but different paths and different amounts of work done. that is not a "conservative field" and KVL will not be valid in that situation.

... - while your second says
(b) the different values at a point can only occur at different times.
i never said anything about "different times" (until you brought it up and only to say that i ain't saying anything about "different times"). you are the first (and only) person to bring this up. i am saying something about different circumstances (for comparison purposes). like "compare scenario A at some point in space and some instance of time to scenario B at the same point in space and the same instance of time." that is what is different. i never said anything about different times.

Which is it?

If (a) then the potential is multivalued and nonphysical which makes it moot.
ah! now you said something meaningful that we can talk about. i respectfully disagree with what you say here (it's not "nonphysical" at all, perhaps a nonconservative field, but certainly not nonphysical). and i have stated my alternative above. this might be something tangible to argue about. maybe not.

If (b) then you were in fact implying some temporal element in the loop theorem.
no, only you are.

We've already spent more time on the semantics than is warranted and, again, I suggest we return the burden to the original poster who posed an ill-stated and ambiguous question.
fine by me. (it wasn't just semantics. you made a factual statement, i said i never implied otherwise, then you made another factual statement saying that i did imply otherwise which was not true.)

Last edited:
Tide
Homework Helper
rbj,

My original, central and persistent statement:
the electrical potential is single-valued!
it won't be a single electrical potential
and
the purely electrical potential at a point does not have two different values at the same instance of time.
and now you add:
it's not "nonphysical" at all
referring to a multivalued potential. Then, of course, you add
now you said something meaningful
in reply to my
then the potential is multivalued and nonphysical which makes it moot.
I'm sure glad you recognize that I am making progress! Thanks!

So, while we're at it, please address my original question to you: How is a (pointwise) multivalued potential physical?

rbj
Tide said:
My original, central and persistent statement:

"the electrical potential is single-valued!"

"it won't be a single electrical potential"
wow! i said that? well, i guess i did, but you didn't quote the entire context. there's a qualification:

"if there is a net changing magnetic field inside the loop (of any reasonable quantity), it won't be a single electrical potential."

but i should have left out the word "electrical" (since i have since used it to mean "electrostatic") or replaced it with "electromagnetic" (but then, as i have said before, and i'm sure you know, "potential" is not a well-defined concept in the context of changing magnetic fields or anything outside of pure electrostatics, for the very same reason that it is multi-valued. but, hey, let's run with it.)

so we're both bad. you left out some salient context and i shouldn't have used the word "electrical" especially since i later used the same word for "electrostatic" and the above statement of mine would be both untrue and self-contradictory if "electrostatic" was substituted for "electrical". my apologies.

and

"the purely electrical potential at a point does not have two different values at the same instance of time."
and, if you had quoted the clarification later, that we all have read, i had already said that "electrostatic" would be a better word, but "purely electrical" has the same connotation. it means to exclude magnetic effects. so i think i've been immunized against any liability there.

and now you add:

"it's not 'nonphysical' at all"
referring to a multivalued potential.

...

So, while we're at it, please address my original question to you: How is a (pointwise) multivalued potential physical?
it's not non-physical at all. we can certainly conceive of a situation where there are both electrostatic fields and non-zero changing magnetic fields.

i'm pretty sure that the term "potential" at some point in space is defined as the amount of work (energy) per unit charge that it takes to bring a test charge (a "test charge" is a analagous to "test mass" in gravitational analyses, it has enough charge to be affected by the ambient fields but not enough to affect the fields measureably) from a point at infinity to that point in space.

for purely electrostatic situations, it doesn't matter what path you take to bring that test charge from infinity to "point A". the work is the same and independent of the route you took, including a route where you bring the test charge to point A and continue to move the charge away from A and then back again to point A. the simple route that brought the charge to point A took V(A) joules/coulomb of work and the more circuitous route that brings it to point A (the same way) and then around some closed loop and then back to point A takes the same amount of work (because V(A) is a function only of the coordinates of point A not of how we got the charge to point A). so then the additional work of taking the damn charge around that extra loop must be zero.

in the case where there is a non-zero and changing magnetic field, that is not the case as we know from Faraday's Law. bringing the charge from the same point at infinity to point A via those two different routes takes a different amount of energy because Faraday's Law says that there is a net amount of work required to move the charge around the loop. so then the "potential", the amount of work per unit charge needed to bring the charge from infinity to point A depends on what route is taken to get it to point A. it is not simply a function of the cooridinates of point A. the value of that potential is not single-valued since there is not (in general) a single route from the point at infinity to point A. it is multivalued because there are multiple routes and the work per unit charge (the "potential") depends on which route. it's a physical situation, but the concept of a scaler potential is not well-defined.

nonetheless, the well-defined concept of adding up the voltages of each component going around the loop is still well defined. it's just that, in the latter case (changing magnetic field in the loop) the voltages don't add to zero.

Tide
Homework Helper
Specifically, you're interested in the potential difference from one point to the next and, more to the point, how can the there be a difference between the potential at a given point and the potential at the same point?

rbj
oh, and i never said anything about you "making progress" (which, if i did, would sound sorta patronizing, which is something i really do not intend to do). i only said that when you said

"then the potential is multivalued and nonphysical..."

that was a meaningful statement that we could both understand (and that i disagreed with it). this is in contrast to saying

"The loop rules apply to an instant of time."

which was sorta a straw-man. it was non-sequitur and i didn't disagree with it (assuming "loop rules" meant KVL) nor depend on the contrary to either state or defend the point i was making (about KVL not being exactly accurate when there is a changing magnetic field).

rbj
we had posts that were "crossing in the mail".

Tide said:
Specifically, you're interested in the potential difference from one point to the next and, more to the point, how can the there be a difference between the potential at a given point and the potential at the same point?
this deviation from KVL (which happens only in the presence of a changing magnetic field, did i say that before? - it seems that this important qualification is going unnoticed) is about the sum of a bunch of voltages (call them "potential differences" if you like, but this concept of potential is a bit problematic in this case of changing magnetic fields because there ain't any single valued potentials, so then what the hell do they mean?) around a closed loop that you would think would add to zero, but they don't.

even though there is no single-valued potentials that are simple properties of each point in space, there is still a concept of how much work (per unit charge) it takes to drag a test charge through some electrical component from one terminal to another. that is a potential difference and it is reasonably well defined since the path is also well defined. so you add up these well defined potential differences and you don't get exactly zero.

stated another way, there is a well defined amount of work (per unit charge) that it takes to drag this test charge around the loop from point A through this specific route and back to point A. if the field is conservative (that is there are no changing magnetic fields and the right-hand side of Faraday's Law is zero), then that well defined amount of work is zero. but, in the other case, with a changing magnetic field, that well defined amount of work (per unit charge) is equal to the right hand side of Faraday's Law.

clear as mud?