Does Kirchhoff's Law Hold? Disagreeing with a Master

[Sredni]

The dependence on Path2 comes from the fact that bsfeechannel defined the strength of the field being as strong to generate 1V of EMF for the entire loop. So given this being a uniform field means that a larger surface area always means more of the field is enclosed in the loop, this means the field has to be weaker to still produce 1V. Hence why making the loop area around Path2 larger causes the voltage on Path1 appear smaller because the field is still in the same spot but its weaker.

This post has been shortened and cleansed to avoid upsetting other children.
Whatever was written here can be found in one or more of the following books (in no particular order, and without mentioning the usual suspects Feynman, Purcell, Griffiths, Ohanian, Jackson):

Panofsky, Phillips
Classical Electricity and Magnetism 2nd ed

John Kraus
Electromagnetism 2nd to 4th ed

Ramo, Whinnery, VanDuzer
Fields and Waves in Communication Electronics 2nd or 3rd ed

Bleaney
Electricity and Magnetism 3rd ed

Nayfeh, Brussel
Electricity and Magnetism

Kip
Fundamentals of Electricity and Magnetism 2nd ed

Lorrain, Courson
Electromagnetic Fields and Waves 2nd ed

[Berni]

And what about path #1?

Sorry when i headed off for the night i remembered that i forgot to consider the electrostatic effect of the wire on the other path. So what i calculated in the first example is the voltage across Path2 when the wire along Path1 is not there.

Going back to the first example with numbers the 866.6 mV is just the magnetic EMF, but in the wire charge separation equivalent to 133.4 mV has occurred in order to null out the sum E field. This field is obviusly not confined to the the wire so it also creates a electrostatic field of 133.4 mV on the outside of the wire and this includes Path2 so the two have to add up 133.4 mV + 866.6 mV = 1V.

So in the more recent example yeah Path1 is 1V now.

[Berni]

RESPONSE

This makes sense, but, if the field is uniform - and you made no assumptions on what has generated it (remember the boundary conditions set by the shape of the primary coil?) how do you decide how to split the area? You assume the point with the B is the "origin" of the B  field, and this term sounds a bit strange to me.
Are you assuming a circular generating coil so that the induced E field is directed along concentric circles centered at that point?

All magnetic fields have an origin no matter how they are generated, this is something i realized along the way of this thread.

One way you could generate this uniform field is a coil going around the whole page outside of view, origin is the center of this coil. Alternatively you could have the whole page set inside of a airgap of a giant magnetic core. In that case whatever source somewhere on the core is generating the field (Be it a coil or a moving magnet) and the core is bringing the field trough the page, in this case the geometric center of the core poles above and below the circuit determine the field origin. Another way is to have a flexible core with a fixed permanent magnet somewhere in it and then you change the surface area of the cores poles above and below the circuit to concentrate the field down into a smaller area and so make it stronger as the shape of the core moves inwards, in this case the center point of this uniform scale transformation applied to the core is the magnetic origin.

No matter how it is created this is a point where the magnetic filed lines stay put while other lines crowd up or disperse as the field changed strength.

But in a lot of cases this origin can be ignored because the effects of changing where it is null out.

Oh, before I forget, I finally managed to scan my drawings about your 'magnetically shielded' circuit. Let me see if I can show you that your lumped circuit with four lumped coils is a different system from the unlumpable Romer-Lewin circuit.

Let's say we have a hypermu material that captures all magnetic field lines and lets nothing out. We put primary and secondary coil inside it in this way:


Fig. the hypermu shield

Nice perspective drawing there by the way.

Now, let's look it as a cross section showing the B field lines entering and exiting the page. Since the material is magical, no field lines escape and I can exaggerate the holes needed to put our probes in. Circuit A1 is closest to the physical system, while circuit A2 is the same, but with the arc exaggerated.


Fig circuits A1 and A2

Do you recognize circuit (a) of my previous post? Here KVL works on both inner and outer circuit paths.
(Edit: with 'external circuit path' I mean the external voltmeter you placed to show 0V, as long as you enclose zero net (varying) magnetic field it behaves according to KVL)
(Edit: it seems I've used letters consistent with my previous post, after all)

Now, do you realize that systems A1 and A2 are different from system B, the Romer-Lewin ring?


Fig circuit A

Here KVL DIES on both inner and outer circuit paths. (the field lines close at infinity).


EDIT: added the missing "zero" before net magnetic field. Time to go to bed.

Well the way i see it is that Fig3 is sort of a zoomed in Fig2.

The field still returns back the other way to close itself, except that it happens a lot farther away (infinitely far). So anything the happened inside the hollow part of the toroid is now happening across the whole page.

The equivalent for going in reverse to get Fig3 to be like Fig2 is to take everything on the page in Fig3 and stuffing it inside that hollow HyperMu™ toroid. So the Romer-Lewin cirucit inside can see no difference between Fig2 and Fig3 since its inside in both cases. But if we move anything on the outside of the ring in Fig2 inside the toroid it suddenly becomes a transformer secondary as well, hence why i have modeled the voltmeter probe wires as coupled inductors. The only way the probes could avoid being affected by the field is having them run outside of the toroid in your so called "KVL safe area". Wires outside the toroid still have inductance, but it never shows any voltage across it (No intersecting field to generate it and no current flow trough the voltmeter to generate its own magnetic field) so we can just pretend the inductors are not there as it doesn't affect the circuit in any way.

Since as you pointed out this "KVL safe area" does not exist in Fig3 means that we must consider all wires passing around the field to be magnetically coupled to it.

[bsfeechannel]

And what about path #1?

[snip]
So in the more recent example yeah Path1 is 1V now.

Thank you. We are almost there. Please bare with me. So far, your answers for the last example are checking OK against Feynman's lectures. But we need to check a few more concepts.

Now can you explain why exactly 1V for path #1?

[Berni]

Because the total EMF around the loop is 1V.

So the voltage contribution from the EMF field down Path1 is 133.4 mV while the charge separation in the ends of the wire on Path2 cause an additional  866.6 mV of electrostatic field. So then 133.4 mV + 866.6 mV = 1V

Or another way is to just says that Faradays law says its 1V because that's how much you need to get to 1V for the integral around the loop. But that explanation is a bit underwhelming because it doesn't really show where the voltage comes from.

[bsfeechannel]

Alright. The important thing here is to notice that the EMF is 1V with or without the wire. The wire seems to be irrelevant for the generation of that EMF, albeit it causes a distortion of the electric field along paths #1 and #2 (reducing #2 to zero and enhancing #1 in compensation).

Now comes the crucial question. What is generating this 1V EMF? In other words, what is the fundamental cause behind it?

[Berni]

That depends how low do you define as being fundamental, so i'm going to go lower and lower in steps to hopefuly cover the level you are after.

Faraday law:
EMF Voltages comes from a loop area enclosing a changing magnetic field

Maxwell:
EMF Voltage comes from an the electric fields relation to the vector curl of the magnetic field.

Special relativity:
EMF Voltage comes from an apparent electric field generated by warping of spacetime due to the relative motion of charged particles.

Quantum electrodynamics:
EMF Voltage is the result of a moving observers interaction with a electromagnetic field trough the exchange of particles. (Please don't ask too many questions about this one because i don't really understand much about it)

[Sredni]

Now, do you realize that systems A1 and A2 are different from system B, the Romer-Lewin ring?

Well the way i see it is that Fig3 is sort of a zoomed in Fig2.

I've corrected my captions, apologies for that. I believe you mean circuit (B) is different equivalent to circuit (A2)

It is not.

This post has been shortened and cleansed to avoid upsetting other children.
Whatever was written here can be found in one or more of the following books (in no particular order, and without mentioning the usual suspects Feynman, Purcell, Griffiths, Ohanian, Jackson):

Kip
Fundamentals of Electricity and Magnetism 2nd ed

Lorrain, Courson
Electromagnetic Fields and Waves 2nd ed

John Kraus
Electromagnetism 2nd to 4th ed

Ramo, Whinnery, VanDuzer
Fields and Waves in Communication Electronics 2nd or 3rd ed

Bleaney
Electricity and Magnetism 3rd ed

Nayfeh, Brussel
Electricity and Magnetism

Panofsky, Phillips
Classical Electricity and Magnetism 2nd ed

[bsfeechannel]

That depends how low do you define as being fundamental, so i'm going to go lower and lower in steps to hopefuly cover the level you are after.

Faraday law:
EMF Voltages comes from a loop area enclosing a changing magnetic field

Maxwell:
EMF Voltage comes from an the electric fields relation to the vector curl of the magnetic field.

Special relativity:
EMF Voltage comes from an apparent electric field generated by warping of spacetime due to the relative motion of charged particles.

Quantum electrodynamics:
EMF Voltage is the result of a moving observers interaction with a electromagnetic field trough the exchange of particles. (Please don't ask too many questions about this one because i don't really understand much about it)

What about somewhere between Faraday and Maxwell?

For these two the ultimate cause of the EMF is the varying magnetic field. But OK, this means we can proceed.

Now let me introduce you to the resistor. Differently from our ideal wire, resistors do allow the existence of an electric field inside them. When connected to a circuit, they will let a current flow that is proportional to the line integral of the electric field along their paths divided by their resistance.

Now let's connect this resistor to our wire . A current will obviously flow from A to B. This will generate a magnetic field with a direction that will tend to oppose the magnetic field that generated the E.M.F. However, let's suppose that the resistance is sufficiently high, the current is too low, and consequently this opposing magnetic field is so weak that we can consider it negligible.

Let's suppose that R = 1k ohms (producing a 1mA current).

Using my drawing below, i.e. respecting the geometry of the paths, how would you please model this circuit using your circuit analysis technique?

[Doctorandus_P]

I could only watch a part of the (always respected and liked) big brow guy.

When you draw a schematic, it is only a model, a representation of the outside world.
If there is some external magnetic field which induces a current in your schematic, you have to put a representation of that induced current it in your model.
So then it becomes a schematic with 2 resistors and a current source.
Problem solved.

Lewin is making assumptions from an incomplete / incorrect model by ignoring the current source while drawing his model for the Kirchof "law".
There really is no need to go any deeper into this.

[LukeW]

Lewin draws a schematic diagram that will be familiar to any freshman engineer - a battery, a couple of resistors. Let's analyze it using Ohm's and Kirchoff's laws.

It's tempting to immediately put an end to this "paradox" by specifying a more rigorous assumption for Kirchoff analysis - that Kirchoff's voltage law can't be applied to a loop with a magnetic flux applied through it. And KCL can't be applied to systems that emit or capture external electrons into that node.

Asking a question about the effect of an external magnetic flux on the circuit really makes this an electromagnetic compatibility question - and you're not going to understand EMC problems with lumped models and ideal wires.

Kirchoff's and Ohm's laws are not Maxwell's equations. They are deliberately simplified, and very useful tools within a certain context - but they're not really the right tools for an electrodynamics problem. They're not designed to do an electrodynamics job.

It's easy to see, naively, that applying magnetic flux may "break" Kirchoff's voltage law, in its traditional first-year-undergrad expression.

Similarly, we can imagine scenarios that "break" Kirchoff's current law - a thermionic tube, a cathode-ray tube, a Faraday cup, a radiation detector like an ion chamber. Any system that involves the emission or capture of electrons from outside, or to outside, the simplified system boundary will appear to "break" Kirchoff's current conservation at that circuit node.

But the problem here, the real paradox, is not the use of Kirchoff's laws.

Kirchoff's voltage law is, basically, a statement of conservation of energy (simplified within a certain context.) Kirchoff's current law is basically a statement of conservation of charge. These are really solid, foundational principles of physics - it's hard to imagine genuine violation of energy or charge conservation.

Kirchoff's laws work. Something is wrong with the circuit model.

The "paradox" here is not the use of lumped circuit elements, either.
We can make a reasonable model of the circuit with lumped elements.

The problem here is that we draw the schematic symbol for a battery, and the schematic symbol for a couple of resistors, and then we draw these lines between them. What are those lines on the blackboard, the lines between the battery and the resistor?
This is Lewin's great "paradox" in a nutshell.
Nobody teaching basic electronics ever talks about the lines, and we need to talk about the lines.

These lines are "ideal wire".

Ideal wire has no resistance, no capacitance to the groundplane, no mass, no cost, infinite tensile strength, infinite flexibility, infinite resistance to corrosion or insulation degradation, no resistive heating, no skin effect, no crosstalk, no limit to its current-carrying capacity before the insulation melts off, no voltage drop etc. It's really easy to solder, strip and terminate.

It's a spherical frictionless cow in a vacuum.

We take a coil of Ideal Wire, put it between the poles of a magnet, with a commutator, and spin it around. What voltage is observed?

No EMF generated? Nothing?

Frustrated, you check the Ideal Wire datasheet again.
Inductance: 0 nH/m.

Hmmm.

Ideal Wire can't couple to a magnetic (or electric) field.

In the freshman physics class, we don't really tease out the inductance of the wire loop as an important quantity that can be measured - it's not used the same way engineers use it.

But the treatment of electromagnetic systems with Maxwell's equations and integrals of the B vector dot product with the area vector and all that sort of thing can be used to shake out the fact that inductance is an intrinsic physical property of the wire loop, and it is not zero.

You can include the effect of magnetic flux coupling into the loop by drawing in an inductor as a lumped circuit element and identifying the appropriate voltage across it. You can keep it as a lumped element connected by ideal wires.

There's a lesson here for the students.
All the interesting things in EE, all the complicated things of practical importance - electromagnetic compatibility, signal integrity in high-speed digital systems, antennas, transmission lines, shielding, RF design, EMI, crosstalk, power transmission - all have at their core an understanding of Real Wires.

Cable assembly, manufacturing, labor, economics, testing at scale, reliability in demanding environments such as automotive or aerospace - Real Wires (and their connectors and terminations, which are a key part of real wires) are crucial here too.

When students are frustrated with breadboards, or when student projects don't work in the lab, it's never because their stock of resistors or transistors or opamps are faulty. It's almost always because of wires.

[bsfeechannel]

Lewin is making assumptions from an incomplete / incorrect model by ignoring the current source while drawing his model for the Kirchof "law".
There really is no need to go any deeper into this.

Those who say Lewin is wrong have a tendency to not want to go deeper in the subject. If Lewin was really wrong, they would not demonstrate this consistent fear.

[Sredni]

We take a coil of Ideal Wire, put it between the poles of a magnet, with a commutator, and spin it around. What voltage is observed?
No EMF generated? Nothing?
Frustrated, you check the Ideal Wire datasheet again.
Inductance: 0 nH/m.
Hmmm.
Ideal Wire can't couple to a magnetic (or electric) field.

What. Are. You. Talking. About.

<sigh>

Edit: added context.

[Berni]

What about somewhere between Faraday and Maxwell?

For these two the ultimate cause of the EMF is the varying magnetic field. But OK, this means we can proceed.

Now let me introduce you to the resistor. Differently from our ideal wire, resistors do allow the existence of an electric field inside them. When connected to a circuit, they will let a current flow that is proportional to the line integral of the electric field along their paths divided by their resistance.

Now let's connect this resistor to our wire . A current will obviously flow from A to B. This will generate a magnetic field with a direction that will tend to oppose the magnetic field that generated the E.M.F. However, let's suppose that the resistance is sufficiently high, the current is too low, and consequently this opposing magnetic field is so weak that we can consider it negligible.

Let's suppose that R = 1k ohms (producing a 1mA current).

Using my drawing below, i.e. respecting the geometry of the paths, how would you please model this circuit using your circuit analysis technique?

The circuit is simple enough that very little circuit analysis is actually needed on it.

We know the voltage around it is 1V, We know the total loop resistance is 1 KOhm so trough I=U/R=1V / 1KOhm = 1mA.
At this point the voltages and currents across components are known so the circuit is solved.

Since you are looking for the textbook definition the voltage between points AB is both 0V and 1V.

If you are after the charge density at the points then it involves a bit more work as the resistive bar must be modeled as a resistor in series with an inductor and then it does start to matter where in space the points are. But since you are not interested in this unscientific made up voltage il save myself some work and leave it there.

Right. The whole point is that when the lines close at infinity or, for what matters outside your lab, you have no way to access the circuit without having a net varying B field enclosed by the circuit's path. And this is what makes KVL fail.

Exactly because i don't have access to a wire that goes outside of it i simply model the wire as part of the cirucit. If i know the exact voltage across the wire means i can use math to figure out what voltage is on the other end when i know the voltage on this end(Where its shown by the voltmeter)

Instead of avoiding the field, its simply compensated for

Look better: in one case you cannot get rid of the varying B field inside the circuit path. The reason is that the 'returning lines' are way out of your reach. So you cannot 'hide' the whole field inside the components, hence you cannot have lumped components.
If you cannot see this, I don't know how to help you.


As for the rest, man, are you sure you do not have a reset button somewhere? I really mean no disrespect (I believe we've been able to argue without resorting to insults and keeping it quite polite) but... the way you try to compute 'partial emfs' based on areas makes me think of cargo cult science.

Using the area ratios works as long as the symmetry of the system allows it such as in the case of primary and secondary circular coils on the same axis. When they are off center, or have a different shapes, things are not so easy. So, while it is true that you get different results according to where the circuit is placed inside the field (let's focus on the field generated by a circular primary coil), you need to take the shape of the induced E field into account if you want to find the path integral on a portion of the closed curve (or the distribution of surface charge on the conductor).
(EDIT: come to think of it again, it seems that all is required is that the primary coil be circular as that seems the only way to get a uniformly distributed magnetic field inside. Someone has proof of that?)

And regarding the 'origin of the field' as the point where field lines stay put when you change the intensity... It looks intriguing, but does it have any physical motivation? To me is just a graphical representation, nothing more. You seem to think that when the field diminishes in strength the 'number of points' where you can find the B field diminished as well...


Edit: corrected duplicate "positioning" with "shapes"
Edit: grammar
EDIT: different --> equivalent. Man, I really need to sleep better. That fucking neighbor's dog, I can hear him with all windows closed. Maybe a 400W ultrasound whistle...

It works in 3D space just as well, except that the origin is not a point anymore but instead a line.

So far the way i worked with partial wire segments seamed to align with experimental results. The goal of this is circuit modeling to consistently construct a mesh model that behaves like the circuit in question. This model then allows to to reliably apply many circuit analysis tools(Including KVL) that don't always work when just slapped directly on a real circuit. Its not only KVL that sometimes breaks when directly slapped onto a real circuit, many other tools can misbehave just as badly if not worse. Hence why i never claimed that KVL always works. Circuit meshes are very bad at explaining what fundamentally happens to the electrons, they are good at explaining behavior of circuits in a high level way. This helps in making sense of large circuits containing 100s of components. Much like C++ is nice at explaining large complex programs, but glances over details that one has to deal in Assembler.

If you think the methods i have used to circuit model Dr. Lewins experimental circuit are flawed you are welcome to suggest a circuit that would create a incorrectly behaving circuit mesh when modeled with the same methods. Provided its easy to experimentally recreate i can built it later and compare the test results to the calculations.

I'm not trying to say that Mawell or Faraday, or the definition of voltage is wrong. I was just trying to show a way of applying KVL in a way that works without adding any magical voltages to make it work for just this special case, the same modeling methods can be applied to other circuits just as well.

I suppose if you want a physical representation of field lines in 3D you could imagine each field line being a long stringy bar magnet. All of those long bar magnets have the same magnetic flux and they contain all of this flux inside of them. So if you ware to recreate the field in a loop of wire using these stringy bar magnets as the current is ramping up you would find more and more of these magnets appearing at the edge of the loop and moving towards the center. If we now look at the 2D cross section of the area inside the loop it would look like a area of uniformly distributed dots that is "compressing together" around the center point, putting more and more of them into view as they get closer together. This is sort of analogues to a magnetic field moving inwards so a conductor going around this center point feels the same effect as if it was moving trough a magnetic field. This lets you think about it analogous to the explanation of a comutated DC motor/generator, we also have an equation for calculating the size of this voltage its U_emf=v*B*l .Similarly the force generated by a current carrying wire in a magnetic field can be calculated, putting both together gives you a connection between mechanical power and electrical power in the wire. None of this requires a defined loop area, just a section of wire. Look it up yourself instead of just reading my words.

Oh and by the way a ultrasonic blaster just makes them bark more. The trick is to blast it for a few seconds right after they bark so that they learn what awaits them the next time they do so.

[rfeecs]

I suppose if you want a physical representation of field lines in 3D you could imagine each field line being a long stringy bar magnet. All of those long bar magnets have the same magnetic flux and they contain all of this flux inside of them. So if you ware to recreate the field in a loop of wire using these stringy bar magnets as the current is ramping up you would find more and more of these magnets appearing at the edge of the loop and moving towards the center. If we now look at the 2D cross section of the area inside the loop it would look like a area of uniformly distributed dots that is "compressing together" around the center point, putting more and more of them into view as they get closer together. This is sort of analogues to a magnetic field moving inwards so a conductor going around this center point feels the same effect as if it was moving trough a magnetic field. This lets you think about it analogous to the explanation of a comutated DC motor/generator, we also have an equation for calculating the size of this voltage its U_emf=v*B*l .Similarly the force generated by a current carrying wire in a magnetic field can be calculated, putting both together gives you a connection between mechanical power and electrical power in the wire. None of this requires a defined loop area, just a section of wire. Look it up yourself instead of just reading my words.

This is where thinking about "field lines" causes a problem.  In this case of a changing magnetic flux, nothing is moving.  The geometry of the magnetic field does not move.  By that I mean that at every point, the direction of the magnetic field vector doesn't change.  Only the amplitude changes.

The magnetic field is not creating a force to move the charges in the wire.  That force would be perpendicular to the wire, so it would have no effect on the current.

Faraday's law says that a magnetic field that is changing with time creates an electric field that rotates around the magnetic flux.  It is the electric field, not the magnetic field that moves the charge and creates the current.

This has nothing to do with special relativity.  It is just Faraday's law.

[rfeecs]

(EDIT: come to think of it again, it seems that all is required is that the primary coil be circular as that seems the only way to get a uniformly distributed magnetic field inside. Someone has proof of that?)

If the solenoid is infinite or very long such that the field lines are parallel and the field outside can be neglected, then the field inside will be uniform.  This can be proven by Ampere's law.  Choose a rectangular path parallel and perpendicular to the field and surrounding some turns of the solenoid.  Vary the position of the side of the path that is inside the solenoid.  Then since the current surrounded by the path is always the same, the internal field must also always be the same, no matter what position.  The solenoid doesn't have to be circular.  It should work for any shape as long as the other conditions are met.

[Sredni]

(EDIT: come to think of it again, it seems that all is required is that the primary coil be circular as that seems the only way to get a uniformly distributed magnetic field inside. Someone has proof of that?)

If the solenoid is infinite or very long such that the field lines are parallel and the field outside can be neglected, then the field inside will be uniform.  This can be proven by Ampere's law.  Choose a rectangular path parallel and perpendicular to the field and surrounding some turns of the solenoid.  Vary the position of the side of the path that is inside the solenoid.  Then since the current surrounded by the path is always the same, the internal field must also always be the same, no matter what position.

Yep, that's what I was looking for. Of course it was in Ramo Whinnery VanDuzer as well.

The solenoid doesn't have to be circular.  It should work for any shape as long as the other conditions are met.

So it seems. And yet there has to be some difference in the induced electric field that is generated when the field varies, but I get into nasty integrals as soon as I try to change shape of the coil.
The reason I'm trying to find the induced E field expression for shapes other than a circle is that I believe that the area partitioning Berni proposes can cease to be useful when there is not a 'center' for the induced E field lines and even if there is one (because even if the field follows the profile of the coil close to the coil, it will smooth out farther away), the lines from such a center to the extremes of the segment we want to compute the emf contribute are not perpendicular to the field.

The circular E-field is particular in that it is perpendicular to the radial segments joining the center to the extremes of the open curve we want to find the emf contribute.
But I need to sleep on this. I'm just adding a draft of what I mean here



In the first part the values 12, 4, 4 12 are the line integrals computed along the square of side 4 immersed in the circular induced field whose magnitude grows with the distance from the center C. These were actually computed values since I had the expression for the field. So the emf of the whole square is 36 units. Sure enough, the ratio of the areas (A1/Atot) is the same as the ratio of the line integrals (on AB / on full perimeter).

In the second part I arbitrarily added two values to the radial paths, without changing the other values. This is not legit, but to push my hypothesis: if and I mean IF there is a net nonzero contribute to the emf on this radial paths, the ratio of the areas might not mirror the ratio of the path integrals.
I need to find the analytical expression of the E field for a non-circular generating coil to test it. It might as well be that, when the field is uniform, the net contribute of the path integrals along the two radial paths be zero.

Time to sleep.

[Berni]

This is where thinking about "field lines" causes a problem.  In this case of a changing magnetic flux, nothing is moving.  The geometry of the magnetic field does not move.  By that I mean that at every point, the direction of the magnetic field vector doesn't change.  Only the amplitude changes.

The magnetic field is not creating a force to move the charges in the wire.  That force would be perpendicular to the wire, so it would have no effect on the current.

Faraday's law says that a magnetic field that is changing with time creates an electric field that rotates around the magnetic flux.  It is the electric field, not the magnetic field that moves the charge and creates the current.

This has nothing to do with special relativity.  It is just Faraday's law.

I was not trying to say that the field actually moves that way, but it just looks like its moving in such a way from the perspective of the wire in it.

As for the force on the wire that only happens when a current is flowing trough it where the force comes from the usual two magnets interacting with each other. My point was that there are formula for working with sections of wire rather than just loops. Tho usually they will still be in some way connected to Faradays law as in the wire is enclosing a different amount of field.

[Berni]

The question of is the field inside a square coil still uniform got me thinking quite a bit too and i couldn't come up with a reasonable answer.

The math to do it does indeed get quite complicated so i decided to cheat my way around it a bit by creating a C# app that calculates it. It sidesteps the integrals by applying the vector math to every point and then just adding everything together. It mostly makes use of vector curl combined with the formula for finding the field around a wire that is based on Ampere's law. This then gets slapped onto every point in the array of vectors to make the magic happen. However this is NOT a proper EM simulation by far. It just assumes that the input is steadily ramping so it doesn't need to have timesteps and it works somewhere between 2D and 3D. It works on a 2D array of 3D vectors (Mostly because my bad unomptimised code already takes a few seconds to calculate this)

The Cyan arrows are the E field forced forced upon the world. This is essentially the path of a wire connected to a power source.
The Red/Green shading indicates the magnetic flux in the Z direction (This is generated by the Cyan E field)
The White arrows are the E field that is generated by the magnetic flux (This is actually the part that takes by far the longest to compute in my app)

[Berni]

So here are the actual interesting results with complete closed loops.

As expected all of the field suddenly happens on the inside while the circulating E field spreads out to infinity. But a bit more surprisingly the inside of rectangular shapes indeed appear completely uniform, even with a long aspect ratio rectangle. The point where the EMF circulates around also ends up in the middle of the shape.

I have also included a square with a open side to show that this is indeed computed and not just a hand made diagram. Some similarities can be seen to the wire screenshots from above.





For curiosity i have also built a shielded box in it. As you can see it is perfectly functional too as there is no field visible outside while the field inside behaves as normal. Tho the way the box is created in the simulation is a bit of a cheat. There is no support for varying magnetic permeability (And it probably needs to be 3D to make sense) so instead the box is actually another coil with a current set to create the same size of flux in order cancel out the inner coil. This could be analogus to a superconducting box where this current would appear in the form of eddy currents.

[rfeecs]

The Cyan arrows are the E field forced forced upon the world. This is essentially the path of a wire connected to a power source.

That's a little confusing.  I guess it would be equivalent to a displacement current, or a real current?  I'm not sure how the wire segments or open ended solenoid could exist in the real world, though.  I'm assuming "somewhere between 2D and 3D" means this is showing a cross section of an infinite/long solenoid.

The results look reasonable.  Very amazing you can create this so quickly with C#, graphics and all.  Great.

[Sredni]

@Berni

Can you create a big trapezoidal (the more asymmetric the better) primary coil and then compute the line integrals on the sides of a square coil closed path placed off-center (where 'center' is where the E field appears to go around), like the square I posted?
In my case, the side was 4 units, and the center C was placed at (3,1) (when the square center was at (2,2)).

Alternatively, instead of color-coding amplitudes, draw arrows with lenght proportional to the amplitude (but this might lead to too long and too short arrows...). If you can create a big enough picture, with enough resolution and linear scale for the arrows, we can compute the path integrals by adding the arrows' projections.

[Berni]

Yeah there is no such thing as "current" in the E field, its just representing the voltage placed on the coil. If you look closely you can see the white arrows always point opposite of the cyan arrows, this is the magnetic field trying to resist a current being put trough it by generating EMF in the opposite direction.

The part about it being sort of 2.5D is that it acts much like the diagrams we have been drawing in this thread where everything happens in a flat plane. For a physical analogy you can imagine this flat plane sandwiched between two infinite permeability plates. So the fields that come up "out of the page" simply get eaten up by the plate and returned back around infinitely far. This is the reason why you don't see the returning field come back in around a wire loop, it doesn't have enough room to bend back around the outside like it does with a real like solenoid in 3D space. To see that happen in here i likely need to have a 3D array of vectors, but that would likely turn the 3 second computation time into a 15 to 60 minutes. Real EM simulators likely have a whole bunch of math tricks and optimizations to make it calculate faster but my app is just quickly thrown together (I dont even use multiple CPU cores).

This thing did still take a good few hours to program, tho most of the time was spent sorting out the math to work correctly. Dealing with all these 3D vectors, angles, rotations etc... can easily go wrong when two things are the wrong way around or a - mistakenly becomes a +,

@Berni

Can you create a trapezoidal (the more asymmetric the better) primary coil and then compute the line integrals on the sides of a square coil placed off-center (where 'center' is where the E field appears to go around), like the square I posted?

Alternatively, instead of color-coding amplitudes, draw arrows with lenght proportional to the amplitude (but this might lead to too long and too short arrows...).

I have been experimenting with weird T or C shaped coils to try to break it and it was still surprisingly uniform. Il make some more screenshots tomorow

I do have a feature for scaling the arrow length, but during testing i found that making stronger arrows be brighter was easier to see (Short arrows loose ability to show direction as they become a point, or long arrows get excessively long). I don't have a feature for taking an integral around a path yet, but i could see that being very useful in simulating a secondary coil.

Tho a bit of a problem is that the EMF generated by the magnetic field appears to be 6 times larger than the E field that original caused the magnetic field. From my understanding the two should be almost the same. My guess is that i perhaps forgot a 2*PI somewhere or that my cheat of using 2.5D rather than full on 3D is the cause of some extra troubles. But i can fix this by simply overlaying another secondary coil over the primary and measure the ratio instead. Apart from that scaling problem all the rest does appear to act as it should according to Maxwell.

[Sredni]

If by uniform you mean the magnetic field, it has to be. Rfeecs has posted the demonstration it has to be - as long as there is nothing outside the coil.

What I am interested in is the shape of the curly E field.

[Berni]

Here are some more shapes.

To visualize the strength of the E field better i just put the other field into the heatmap. Its always green because vector magnitude can never go negative.

Some interesting things do happen. The trapezoid shape appears to have some returning field on the outside of the loop, not sure why but my guess is that not having proper 3D that is at fault here.

The dark spot in the middle of the E field seems to sit at the magnetic fields origin point as expected. But it does become a bit more complicated once this origin is located outside of the loop as it is in the C shape. You can still see the dark spot in there, but there are additional dark spots on the edges of the shape. Also a slight non uniformity in the field can be spotted on the inside corners of the C shape, but that's probably just a math hiccup due to too small step size.







EDIT: Fixed spelling