what is a criterion(s) to define some quantity as a vector or scalar?

[steveB]

I don't have too much to say about that. One thing I can say is that one truck is more like one electron and traffic is more like a continuous flow of electrons or a current. When a road is in place the traffic and the cars individually might seem to be describable by the same vector. But the point I'm making about a properly defined current in general is that it will not always be directed along with the local current density and hence direction has no place in the description of current.

A better analogy in the general case is a bunch of dune buggies driving across the desert with no roads. Now imagine that I erect a large arch for buggies to drive through and I define the current to be the rate of buggies going through the arch. The arch does not define a direction and the buggies can go through the arch at any angle. Hence a whole bunch of buggies going through at various angles do not define a clear direction. Basically the definition of current does not consider direction, but if you build a road or a wire, the road or wire can define a direction. In the analogy, forcing cars to drive on the road gives the illusion that the current has direction, but it's a false vision. It is the current density that has the direction and it just so happens that, in this special case, all current density streams are going in the same general direction. So, the illusion works in practice for many special cases.

 

[MrAl]

Hi again Steve,


Well my argument was that the root cause for any effect was the electrons themselves, or at least we can track what is happening by looking at the electrons. And sometimes when the electrons change direction we see a different effect for each direction, and sometimes we dont.

For example, two wires that meet at one point and join to a larger single wire. If we have 1 amp in wire 1 and 2 amps in wire 2, the larger wire has 3 amps (DC). And it doesnt matter if we bend the angles of the two wires away from each other by 10 degrees, 20 degrees, 50 degrees, etc., the addition is still the same, 1 plus 2 equals 3. Here the direction of the current is not relevant to the total current.

But take that same set up (with a battery to provide power of course) and look at the magnetic field that arises from the two smaller wires and the larger wire. Outside the larger wire we see the same field, but outside the two smaller wires we see a different field. Now "Classical" (note the quotes) theory would tell us that the path of the electrons in the two smaller wires change direction as we bend the two wires at different angles to each other. But my point was that the electrons are really what cause (or react to) the magnetic field effects here. At least in classical theory the electrons are causing the field, and their direction has an immense effect on the outcome of the field, yet we tend to want to describe this as the path rather than the electron itself.

 

[steveB]

Hi again Steve,


Well my argument was that the root cause for any effect was the electrons themselves, or at least we can track what is happening by looking at the electrons.

OK, your description above is clearer to me without the analogy, and I don't have any problem with what you say here. But, I've lost track of what this is an "argument" for. Do we disagree about anything? Is there something I've said that is incorrect that needs to be corrected?

It seems that both you and Rachit feel that current must "always" have a vector nature to it and that "sometimes" we just ignore the direction and use it as a scalar. Rachit even provided a reference that seemed to imply this point of view. However, this is a misleading viewpoint. There is no doubt that scalar charge density (actually a 3-form treated as a scalar in vector analysis) and vector current density (actually a 2-form treated as a vector in vector analysis) are the sources of electromagnetic fields. Hence, the directions and vector nature of moving charges are indeed important. However, the vector nature of the charge flow and the critical information is encoded in what we call "current density" for continuos charges and qv (charge times velocity) for discrete charges.

The "current" I'm talking about and claiming can't contain directional information (and is therefore a scalar) is the current "I" that appears in Ampere's law, as I quoted above. This is the accepted scalar definition of current. It does not have any direction to it from it's mathematical form, and it can't have a direction because the Law of Nature implied by Ampere's Law says that we define the current relative to a closed path (which we choose, by the way) that we integrate the magnetic field over. We can attach ANY open surface to the chosen closed loop and we can integrate charge density over that open surface to calculate the current "I". The beauty of Ampere's Law is that ANY surface gives the same answer for current. If any surface gives the same value then direction can't have any meaning in this definition.

I suspect that when Ratchit returns from vacation, he will disagree with this, but do you disagree with this? This is the ONLY point I was trying to make about Ratchit's statement that

"Any current, whether electrical or not, is a spacial vector quantity. Current has magnitude and direction."

His statement is inaccurate. It's more than the fact that this statement ignores an accepted definition. The statement steps on and squashes one of the most beautiful Law's of Nature in our chosen field of study.

 

[MrAl]

Hello again Steve,

Well all i can say at this point is this deserves more thought. What clouds the issue is that the electron(s) is(are) intrinsically linked to the path (and the motion along that path), and without either there is no electromagnetic field. My point was that the current definitions "work" so we use them, but the heart of the matter is what is really causing the phenomenon. We can get away with calling the current a scalar so we do it that way and accepted theory is built on that premise. So i have to agree that for the most part we'll have to accept it as a scalar for many problems and accept the path as the vector.

 

[steveB]

More thought is generally a good thing, but what issue is clouded? The current definition "I" in Ampere's law is just a useful invariant scalar value that can be associated with any closed loop path in 3D space. That seems to me to be useful and illustrative of what Ampere's law is saying.

This is the very concept that triggered Maxwell to invent the displacement current dD/dt. Previously, Ampere, Maxwell and other's at that time recognized a flaw in the existing Ampere's law. For example, in a circuit with a capacitor, one can evaluate the integral of H around the closed path and get the correct current I. Then, one can choose a surface through the conductor and integrate J over the surface to again get the correct current I. But if you choose a surface that goes though the gap between the capacitor plates, you don't get the right value unless you include dD/dt as the displacement current density.

This was Maxwell's unique contribution to Maxwell's equations, - add dD/dt to the current density J. All other EM concepts came from others like Ampere, Faraday, Orsted etc. Once Maxwell discovered the universality of the correct form of Ampere's law, he showed that EM waves were possible, and then showed that the propagation speed was that of light. That's quite a discovery to emerge from trying to define currents in a consistent and universal way.

Speaking about "more thought", think about all the thought these great men put in over many decades of work. These guys are the ones that unclouded the physics for us. But, I agree that this is not an easy subject, so much thought is required on our part.

 

[MrAl]

Hello again Steve,

In terms of classical theory you dont have to clarify anything it's already clear. You missed the point because you are stuck in classical theory because that is your framework of knowledge. Move outside of that and it all changes. Think outside the box. I am bringing up a question that lies outside of classical theory so dont bother to repeat it over and over again it wont help here. Do you understand my intent now?

 

[steveB]

Hello again Steve,

In terms of classical theory you dont have to clarify anything it's already clear. You missed the point because you are stuck in classical theory because that is your framework of knowledge. Move outside of that and it all changes. Think outside the box. I am bringing up a question that lies outside of classical theory so dont bother to repeat it over and over again it wont help here. Do you understand my intent now?

Hi MrAl,

Honestly, I don't understand your intent. I agree, I must have missed your point. Can you restate the question you are bringing up? So far, I don't disagree with the points you are bringing up, but I must have missed the question.

I don't think I'm stuck in classical theory, but it just so happens that classical theory has the exception to the "rule" that Rachit stated. So, that's why I bring it up. I'm glad this exception and the classical theory it comes from is clear, however.

Thanks,
Steve

 

[MrAl]

Hi there Steve,


Try to just forget about Classical Theory for a minute. We want to take a fresh look at this without any previous bias of how to attempt to solve a problem involving current flow.
Think about a computer that has Classical Theory built into it's bios, but then we swap the bios chip for some other chip where we need to establish the program. Classical Theory is no longer available.

Now look at an electron moving. The electron moving along a path creates a magnetic field we can measure. But what causes that field, it's the movement of the electron not really the path. If we try to remove either the field goes away. They are linked through nature, so that both are required. If the electron is moving then the field is created.
Interesting though is that once the electron passes, the field goes away even though the path is still there.
So what causes the field, the electron or the path? We see a field when we have both, but take away the electron and the path is still there yet there is no field.
So it seems more correct to say that the electron caused the field.

Perhaps i am just getting to nit picky on this but it is interesting to think about. The whole thing stems from assigning the vector property to the path not the electron, but if the electron did not follow that path then all bets are off.

It is also interesting i think that current in two close wires that are parallel, if the current is in the same direction in both wires the force is such that the wires move together, while if one current is reversed the two wires repel. In this case we change nothing but the direction of the current.

We are probably getting a little off topic here too because the main theory requires a scalar for equations like Biot Savart.

 

[steveB]

Hi MrAl,

So, if I completely forget about classical theory and just think along the lines you mentioned, I find nothing wrong with what you are saying. I'm in complete agreement that the electron (or charges in general) cause the fields, and I completely agree with everything you just said there.

However, I can't help but wonder if I'm missing an important point (whether subtle or obvious, I'm not sure). I say this because, if I reinstall my "classical theory BIOS chip", I find that I'm still in complete agreement with everything you said. I find it is correct; and yes, I find that it is interesting to think about too.

So, is there some reason why I should not agree, or should I encounter a sticking point about what you said, if I think in terms of classical theory?

 

[MrAl]

Hello again,


Yes i'll have to give this some more thought. It appears that making the path the vector works good enough so why change that right?

 

[steveB]

Hello again,


Yes i'll have to give this some more thought. It appears that making the path the vector works good enough so why change that right?

My thought on this is that your viewpoint is correct. Hence, the version of the Biot Savart law that uses the current density vector J (rather than scalar current I) is the correct and more fundamental relation. A real wire has physical extent and current is spread over the cross sectional area of the wire. Solving for the real distribution of J over the cross section is a pain, especially since the real distribution is not significant to the answer if you have thin wires. One could assume a uniform distribution over the actual cross section of the wire, but this would lead to an integral that is hard to solve mathematically.

However, for mathematical convenience, when we have simple wires that can be approximated as infinitely thin (filamentary wires), we convert the proper Biot Savart Law into a simpler one useful in the special case of wires. Essentially, J is treated as a two-dimensional impulse function over the cross section of the wire.

Now, here is where we have a dilemma. We can either try to invent a new vector current (call it I'), or we can use the already existing scalar current (call it I) and use the vector paths that are typical in vector calculus. If we use I', then we've confused everyone by inventing a new entity that might get confused with I. The new I' is only meaningful in the special case of thin wires, because there I' is replacing J which truly is a vector.

In some sense, you might even say this is all just a mathematical trick. Perhaps it's not quite right to say that it's a trick, but this idea is good enough to let my mind get past the issue you are bringing up.