Harold Ensle:

Note that the current vector is not zero (since its magnitude is not zero).

Current is not a vector. I = \integral J . ds

[OC]
In general electric current is not a scalar.

[Bilge]
In general, the current is a scalar. A charged particle beam,
for example is not the same thing as a current in a wire. In
particular, a charged particle beam has an electric field.

[OC]
And the electrons in a wire do not produce an electric field?
What evidence do you have for this?

[Bilge]
The fact that the electric field outside a wire carrying a current is
zero. Should I assume that you weren't aware of this fact?

[OC]
The wire has zero NET charge (remember Gauss' law?).
And strictly speaking the electric field outside the wire is not
exactly zero. That's because when there is a current flowing, there is
a voltage gradient along the wire (remember Ohm's law?). This voltage
gradient produces an electric field outside the wire. But in most
cases the field can be neglected.



[...]
[Bilge]
What you are calling a ``current'' is not a current.

[OC]
Physicists all over the world don't agree with you.

[Bilge]
Except for at least two physicists your survey missed. Me and appar-
ently, jackson, who states on pg 175 [2nd edition], ``Since the surface
integral of the current density is the total current I [denoted as
a scalar] passing through the closed curve C...''

[OC]
Let me appeal to Feynman's Lectures, vol. 2.
In the chapter on magnetostatics, after the definition of current as
flow of current density through a surface, we find the current I as
vector.
He explicitly speaks of "vector current I".
Maybe you should read Jackson's book more carefully and go beyond page
175.

By the way, in order to use the definition of current with the
integral, you have give to give a specific surface.
What happens if you want to treat the current without choosing a
specific surface (which is not unusual)?

[OC]
Please give a proper definition of current or current density.

[Bilge]
What you mean is, give a definition that is ammenable to your argument,
but has nothing to do with E&M. Your argument seems to hinge on my
agreement to ignore E&M, guess your personal definition of current
and agree that it's a vector despite the fact that it isn't.

[OC]
No. So far you have not given a definition of current density at all.
You have not written j = ...
You only tried to side-step the question by using Ampere's law.

[OC]
(Hint: you cannot use Ampere's law or Maxwell's equations.)

[Bilge]
Oh, yeah. I forgot. Current has nothing to do with maxwell's equations.
So, ok, use this definition:
_
j^v = u(q\gamma^v)u

where u is the free particle wavefunction for a fermion, q is the
electric charge and \gamma^u are the dirac matrices. How's that?

[OC]
After several posts you finally have given a definition of current
density!
So, you agree that current is defined in terms of charged particles.
Now, how do you apply this definition to classcial Electromagnetism?


[...]
[Bilge]
Is your experience with currents limited to describing them in
terms of a lab frame which is an absolute rest frame? Does the
current change sign if I walk faster than the drift velocity of
the charges in the wire?

[OC]
Another attempt to divert the attention from the topic?
Is it really so hard for you to stay on topic?

[Bilge]
Is it really so hard to figure out that current has something to
do with electromagnetism?

[OC]
I never said that electric current has nothing to do with
Electromagnetism.
I was saying that current is a vector.


[...]
[OC]
The integral version of Maxwell's equations do not require necessarily
charge densities and current densities.

[Bilge]
I asked you to show that, not assert it, so please do so for an
electron using the integral version of gauss' law. Tell me what
electrostatic energy you get for an electron which has an experimentally
determined radius of less than 5 x 10^-17 meters. Compare that with the
rest mass energy.

[OC]
This is another attempt to divert the attention from the topic.

[Bilge]
In other words, I just contradicted your assertion regarding maxwell's
equations, so this is no longer on topic.

[OC]
Are you confusing me with somebody else (again)?
The question was never a classical description of the electron.
The question was about electric current being a vector or a scalar.

About the integral version of Maxwell's equations, you really should
read more carefully your books. Usually it is explained.



[OC]
We already know that a classical description of the electron does not
work.

[Bilge]
Then why did you argue with me in the first place?

[OC]
See above.
I was arguing about your claim that electric current is a scalar.

[OC]
How do you define electric current or current density, this is the
question.

[Bilge]
Yes, I know. But from the rules you've imposed, apparently the only
definition that you'll accept is your personal definition, not one that
relates to electromagnetic phenomena.

[OC]
No, I did not give any rules.
It is just the normal way Electromagnetism is explained.
And you should know if you read your books.



[...]
[Bilge]
But it is not a vector and maxwell's equations do not describe currents
as moving charges. Current densities are time dependent charge densities.
In magnetostatics, the charge density is obviously not time dependent,
since there is no time varying electric field measured due to a current.

[OC]
Maxwell's equations to not define currents. They link electric and
magnetic fields, charge and currents, but do not define any of these.

[Bilge]
OK, then to be perfectly rigorous, we can start without any charge
or current at all and just derive it all from fundamental principles.

Start with the dirac lagrangian for a free particle. Impose local U(1)
gauge invariance and use noether's theorem to obtain the gauge covariant
derivative, D_u = du + ieA_u. Now to find the conserved current, find the
commutator of the covariant derivative with itself, i.e., (1/ie)[D^u, D^v],
and take the divergence twice. The first gives you j^v and the second
gives you the continuity equation, d_v j^v = 0. Any guess as to what that
commutator is or what taking that first divergence gives you?

[OC]
Why are you trying to make things more complicated than necessary?
Are you trying to show off your "knowledge", or is it just another
distraction tactic?

I settle for a definition of current density within classical
Electromagnetism.
Please show us that how do apply what you exposed above within a
classical framework.

[OC]
Consider a wire carrying a non-zero but steady current.

[Bilge]
I did. You just want to ignore the fact that for a steady current
in a wire, div J = 0 which is what we call ``magnetostatics''.

[OC]
I did not ignore it.
I pointed out that it is irrelevant, because it does not define
current density.

[OC]
Using your "definition" ("Thee current is the integral of the scalar
product of the current density and the vector normnal to the surface
bounding the current density"), what do you get?

[Bilge]
A scalar known as the current, I. Integrate it yourself and see.

[OC]
In this situation, the flux of the current density (what you call
"current") would be zero for any closed surface (for any volume of the
wire, there is as much current going in as going out).

[Bilge]
Your point being what? Since div J = 0 for magnetostatics and charge
conservation requires that d(\rho)/dt + div J = 0, what else did you
expect?

[OC]
For any closed surface you choose, the integral is zero.
But there is a NON-zero current flowing in the circuit.

Since you find that the integral is zero for any closed surface, would
you say that the current is zero?

You can argue that, although the integral is always zero, there is a
steady non-zero current density.
But if you do not know a priori that the current density is non-zero,
you cannot know it from the current if you use the integral.


[...]
[Bilge]
What charges do you see moving in curl B = J? Which of maxwell's
equations explain the current in a wire as moving charge? If it isn't
obvious, maxwell's equations cannot explain currents in wires as moving
charges.

[OC]
Once more.
Maxwell's equations do not define electric and magnetic field or
charge or electric current. We can still use them, even if we do not
know that particles carry a discrete charge.

[Bilge]
Yawn... Sure, but you are trying to use maxwell's equations incorrectly.

[OC]
Very cute.
Maxwell's equations do not define current or current density. But you
are trying to use them for that (otherwise we would not be arguing),
which is incorrect.

[OC]
Your insistence on bringing up Maxwell's equations or Ampere's law
only shows your ignorance.

[Bilge]
You are the one insisting on a definition of current that doesn't
follow from electromagnetism, not me.

[OC]
Are you saying that without Maxwell's equation there is no
Electromagnetism?

What about Coulomb's law, Ampere's law, Faraday's law, Ohm's law...?
Maxwell's equations summarize the relationships between electric
fields, magnetic fields, charges and currents. But all these entities
are defined by other means.


[...]
[Bilge]
Then to what was ampere referring in ampere's law?

[OC]
Answer the question: How is a current density defined?

[Bilge]
OK. Here it is again:
_
j^v = u(q\gamma^v)u

[OC]
Apply this to classical Electromagnetism.


[OC]
Do not try to divert the attention to something irrelevant.

[Bilge]
I'm sorry, but I consider E&M relevant to the definition of current.

[OC]
You consider Ampere's law relevant to the definition of current.
This is simply wrong, accept it.

[Bilge]
That's news to me. I've always regarded d_u F^uv = j^v to be the
definition of the electromagnetic current.

[OC]
And you were wrong!

[Bilge]
Naturally and you seem to be unable to do anything but require me
to use your personal definition of a current to validate your personal
definition of a current.

[OC]
It is not my personal definition.
Read your books more carefully.

[OC]
You obviously did not pay attention when you read that.
Where does j^v come from?

[Bilge]
From the requirement that the dirac lagrangian be locally gauge
invariant. Where did you think?

[OC]
Now apply it to classical Electromagnetism.

[Bilge]
At first I thought you simply didn't know any better. Now it appears
you are a full fledged crackpot. So be it. You're just one of the
many who assert much, yet support nothing and who won't accept any
explanation that contains any physics.

[OC]
If you just read the book you refer to...
You are convinced that the definition of current using the integral is
the one and only used in physics.
You are sadly mistaken. And by insisting on it, you only show that you
do not even understand the books you quote.

OC

OC:
Bilge:
The fact that the electric field outside a wire carrying a current is
zero. Should I assume that you weren't aware of this fact?

The wire has zero NET charge (remember Gauss' law?).

Why do you think I pointed that out?

And strictly speaking the electric field outside the wire is not
exactly zero.

Apply gauss' law to a wire with no net charge.

That's because when there is a current flowing, there is
a voltage gradient along the wire (remember Ohm's law?).

You are digressing and in any case, ohm's law only validates
my point, since V = IR is a relation between scalars and in
vector form, ohm's law reads, J = \sigma E, or if you want to
be fancy, J = \sigma . E, where \sigma == \sigma_ij is a 3 x 3
tensor to account for anisotropic media. To which version of
ohm's law were you referring and what about it invalidates anything
I said?

This voltage gradient produces an electric field outside the wire.
But in most cases the field can be neglected.

No. Try proving that.

Except for at least two physicists your survey missed. Me and appar-
ently, jackson, who states on pg 175 [2nd edition], ``Since the surface
integral of the current density is the total current I [denoted as
a scalar] passing through the closed curve C...''

Let me appeal to Feynman's Lectures, vol. 2.
In the chapter on magnetostatics, after the definition of current as
flow of current density through a surface, we find the current I as
vector.
He explicitly speaks of "vector current I".
Maybe you should read Jackson's book more carefully and go beyond page
175.

Perhaps you should tell me what's wrong with the material on page 175
and where jackson retracts it later on.

By the way, in order to use the definition of current with the
integral, you have give to give a specific surface.

Your point being what?

What happens if you want to treat the current without choosing a
specific surface (which is not unusual)?

I note you have provided no example of this ``not unusual'' situation.

Oh, yeah. I forgot. Current has nothing to do with maxwell's equations.
So, ok, use this definition:
_
j^v = u(q\gamma^v)u

where u is the free particle wavefunction for a fermion, q is the
electric charge and \gamma^u are the dirac matrices. How's that?

After several posts you finally have given a definition of current
density!
So, you agree that current is defined in terms of charged particles.

No, because the charge doesn't exist independently of the current.
j^u is a four-vector and the charge density is only the time component
of that four vector. In case you haven't learned anything about electro-
magnetic radiation, light has no charge, yet it's source is a transverse
current density.

Now, how do you apply this definition to classcial Electromagnetism?

The same way I've been applying it. The definition was for your
benefit, not mine.
[...]

Is it really so hard to figure out that current has something to
do with electromagnetism?

I never said that electric current has nothing to do with Electromagnetism.
I was saying that current is a vector.

Except that you insist that I define current without using
electromagnetism, which makes no sense.

[...]

In other words, I just contradicted your assertion regarding maxwell's
equations, so this is no longer on topic.

Are you confusing me with somebody else (again)?
The question was never a classical description of the electron.

What classical moving charges exist?

The question was about electric current being a vector or a scalar.

Then stop insisting that current be defined independently of
electromagnetism.

[...]

Yes, I know. But from the rules you've imposed, apparently the only
definition that you'll accept is your personal definition, not one that
relates to electromagnetic phenomena.

No, I did not give any rules.

Yes, you did. You said that I can't use ampere's law.

It is just the normal way Electromagnetism is explained.
And you should know if you read your books.

And here I thought that classical E&M was explained using maxwell's
equations. Could you tell me which equation of classical E&M other
than ampere's law contains any reference to a current? To the best
I can determine, that's the only one of maxwell's equations containing
anything at all related to current and if maxwell's equations aren't
classical E&M, I'm not sure what equations I should be using.

[...]
(1/ie)[D^u, D^v], and take the divergence twice. The first gives you
j^v and the second gives you the continuity equation, d_v j^v = 0.
Any guess as to what that commutator is or what taking that first
divergence gives you?

Why are you trying to make things more complicated than necessary?

No, but you insisted on a definition that did not start with ampere's law.
I would have preferred to make this simplest by not having it come up.

Are you trying to show off your "knowledge", or is it just another
distraction tactic?

I don't suppose you are going to do as I requested and take the divergence
of that commutator, are you?

I settle for a definition of current density within classical
Electromagnetism. Please show us that how do apply what you exposed
above within a classical framework.

As soon as you tell me what you consider the equations of classical
E&M. since ampere's law doesn't seem to one of the ones you accept.

I did. You just want to ignore the fact that for a steady current
in a wire, div J = 0 which is what we call ``magnetostatics''.

I did not ignore it. I pointed out that it is irrelevant, because
it does not define current density.

Your point being what? Since div J = 0 for magnetostatics and charge
conservation requires that d(\rho)/dt + div J = 0, what else did you
expect?

For any closed surface you choose, the integral is zero.
But there is a NON-zero current flowing in the circuit.

Yep. I = \integral J . dS

Since you find that the integral is zero for any closed surface, would
you say that the current is zero?

You can argue that, although the integral is always zero, there is a
steady non-zero current density.

The integral isn't zero. That is the point. The integral is a scalar
known as the current.

But if you do not know a priori that the current density is non-zero,
you cannot know it from the current if you use the integral.

I didn't say the current density is zero. I said the divergence of
the current density is zero.
[...]

Very cute.
Maxwell's equations do not define current or current density. But you
are trying to use them for that (otherwise we would not be arguing),
which is incorrect.

Yawn... If maxwell's equations don't define the current (and charge)
density, then exactly what relates the current density to the charge
density? After all, the _only_ relationship between charges and currents
is that those two things provide a conservation law for electromagnetism.
Outside of that, those two things are meaningless.

You are the one insisting on a definition of current that doesn't
follow from electromagnetism, not me.

Are you saying that without Maxwell's equation there is no
Electromagnetism?

What about Coulomb's law, Ampere's law, Faraday's law, Ohm's law...?

Maxwell's equations

div E = \rho gauss' law
div B = 0 gauss' law for magnetic charge
curl E = -dB/dt faraday's law
curl B = J + dE/dt ampere's law (with maxwell's addition)

Ohm's law is nothing but an empirical relationship derived from
curve fitting and has no explanation outside of quantum mechanics.
It also doesn't help your argument, as I've already demonstrated
above. But, for the sake of completeness, you'll note that I can
get a magnetic field when the resistance is zero.

Maxwell's equations summarize the relationships between electric
fields, magnetic fields, charges and currents. But all these entities
are defined by other means.

Such as?

OK. Here it is again:
_
j^v = u(q\gamma^v)u

[OC]
Apply this to classical Electromagnetism.

[...]
[OC]
Now apply it to classical Electromagnetism.

Rather than keep repeating yourself, _you_ apply it.

[Bilge]
The fact that the electric field outside a wire carrying a current is
zero. Should I assume that you weren't aware of this fact?

[OC]
The wire has zero NET charge (remember Gauss' law?).

[Bilge]
Why do you think I pointed that out?

[OC]
If you can explain why, be my guest.

[OC]
And strictly speaking the electric field outside the wire is not
exactly zero.

[Bilge]
Apply gauss' law to a wire with no net charge.

[OC]
That's because when there is a current flowing, there is
a voltage gradient along the wire (remember Ohm's law?).

[Bilge]
You are digressing and in any case, ohm's law only validates
my point, since V = IR is a relation between scalars and in
vector form, ohm's law reads, J = \sigma E, or if you want to
be fancy, J = \sigma . E, where \sigma == \sigma_ij is a 3 x 3
tensor to account for anisotropic media. To which version of
ohm's law were you referring and what about it invalidates anything
I said?

[OC]
You brought up the electric field of a current-carrying wire.

By the way, if you did your homework, you would see that more often
then not, current is treated as vector, even with ohm's law.


[OC]
This voltage gradient produces an electric field outside the wire.
But in most cases the field can be neglected.

[Bilge]
No. Try proving that.

[OC]
No what?

No, the voltage gradient does not produce an electric field, or
No, in most cases the field can be neglected?

About the first: "voltage gradient" is not accurate, I should have
said "potential gradient"; and a potential gradient IS an electric
field.

About the second: in most cases it is neglected, as you prove when you
said that the electric field outside a wire is zero.

By the way, look up "energy density" and "energy density flow" of
electric fields, and you will see if the electric field outside a wire
is zero or not.


[Bilge]
Except for at least two physicists your survey missed. Me and appar-
ently, jackson, who states on pg 175 [2nd edition], ``Since the surface
integral of the current density is the total current I [denoted as
a scalar] passing through the closed curve C...''

[OC]
Let me appeal to Feynman's Lectures, vol. 2.
In the chapter on magnetostatics, after the definition of current as
flow of current density through a surface, we find the current I as
vector.
He explicitly speaks of "vector current I".
Maybe you should read Jackson's book more carefully and go beyond page
175.

[Bilge]
Perhaps you should tell me what's wrong with the material on page 175
and where jackson retracts it later on.

[OC]
Please pay more attention.
I never said that the definition of current with the integral is
wrong, neither did I claim that it is retracted.

I just gave you an example of current used as a vector.

You seem to be convinced that defining electric current as flux of
current density through a closed surface is the one and only possible
and acceptable definition.
I brought up exmples to show that you are wrong.
If you do not accept it, there is not much I can do about it.


[OC]
By the way, in order to use the definition of current with the
integral, you have give to give a specific surface.

[Bilge]
Your point being what?

[OC]
What happens if you want to treat the current without choosing a
specific surface (which is not unusual)?

[Bilge]
I note you have provided no example of this ``not unusual'' situation.

[OC]
Current in a straight wire.
Uncountable situations where the conductors involved are uniform or
symmetric.

Have you ever done homework exercises?

[Bilge]
Oh, yeah. I forgot. Current has nothing to do with maxwell's equations.
So, ok, use this definition:
_
j^v = u(q\gamma^v)u

where u is the free particle wavefunction for a fermion, q is the
electric charge and \gamma^u are the dirac matrices. How's that?

[OC]
After several posts you finally have given a definition of current
density!
So, you agree that current is defined in terms of charged particles.

[Bilge]
No, because the charge doesn't exist independently of the current.
j^u is a four-vector and the charge density is only the time component
of that four vector. In case you haven't learned anything about electro-
magnetic radiation, light has no charge, yet it's source is a transverse
current density.

[OC]
Are you talking about electromagnetic waves? Where there are
time-dependent electric fields that can move charges?

Above you defined current density in term of moving charged particles
(something you would not admit in previous posts).

Then there is this exchange:
OC:"So, you agree that current is defined in terms of charged
particles."
Bilge:"No, because the charge doesn't exist independently of the
current."

Do you mean that if there is a current, then there are charged
particles moving?
Do you mean that if the current density is zero, then there is no
charge?


[OC]
Now, how do you apply this definition to classcial Electromagnetism?

[Bilge]
The same way I've been applying it. The definition was for your
benefit, not mine.

[OC]
We were talking about classical Electromagnetism, weren't we?
Why can't you simply show how you apply your definition to it?



[...]
[OC]
Is it really so hard to figure out that current has something to
do with electromagnetism?

[Bilge]
I never said that electric current has nothing to do with Electromagnetism.
I was saying that current is a vector.

[Bilge]
Except that you insist that I define current without using
electromagnetism, which makes no sense.

[OC]
Apparently you think that only Maxwell's equations "are"
Electromagnetism.
This is simply wrong. (See below)



[...]
[Bilge]
In other words, I just contradicted your assertion regarding maxwell's
equations, so this is no longer on topic.

[OC]
Are you confusing me with somebody else (again)?
The question was never a classical description of the electron.

[Bilge]
What classical moving charges exist?

[OC]
Please explain how this relates to the problem of current being a
vector.

[OC]
The question was about electric current being a vector or a scalar.

[Bilge]
Then stop insisting that current be defined independently of
electromagnetism.

[OC]
I never said such thing.
If we talk about charge and electric current, we are dealing with
Electromagnetism.
Only because charge or electric current do not require Maxwell's
equations to be defined, it does not mean that they are outside the
domain of Electromagnetism.


[...]
[Bilge]
Yes, I know. But from the rules you've imposed, apparently the only
definition that you'll accept is your personal definition, not one that
relates to electromagnetic phenomena.

[OC]
No, I did not give any rules.

[Bilge]
Yes, you did. You said that I can't use ampere's law.

[OC]
Because Ampere's law does NOT define electric current or current
density.
It just links magnetic fields to electric currents, but defines
neither of them.


[OC]
It is just the normal way Electromagnetism is explained.
And you should know if you read your books.

[Bilge]
And here I thought that classical E&M was explained using maxwell's
equations. Could you tell me which equation of classical E&M other
than ampere's law contains any reference to a current? To the best
I can determine, that's the only one of maxwell's equations containing
anything at all related to current and if maxwell's equations aren't
classical E&M, I'm not sure what equations I should be using.

[OC]
Show me how electric fields, magnetic fields and current are defined
through Maxwell's equations.

I bet that you end up with circular reasoning, because Maxwell's
equations are not enough.



[...]
[Bilge]
(1/ie)[D^u, D^v], and take the divergence twice. The first gives you
j^v and the second gives you the continuity equation, d_v j^v = 0.
Any guess as to what that commutator is or what taking that first
divergence gives you?

[OC]
Why are you trying to make things more complicated than necessary?

[Bilge]
No, but you insisted on a definition that did not start with ampere's law.
I would have preferred to make this simplest by not having it come up.

[OC]
Ampere's law does NOT define electric current or current density.
I you don't get it, you shouldn't even be talking about
Electromagnetism.


[OC]
Are you trying to show off your "knowledge", or is it just another
distraction tactic?

[Bilge]
I don't suppose you are going to do as I requested and take the divergence
of that commutator, are you?

[OC]
I am still waiting for your definition of current density within
classical Electromagnetism.


[OC]
I settle for a definition of current density within classical
Electromagnetism. Please show us that how do apply what you exposed
above within a classical framework.

[Bilge]
As soon as you tell me what you consider the equations of classical
E&M. since ampere's law doesn't seem to one of the ones you accept.

[OC]
Ampere's law does NOT define electric current or current density.
I you don't get it, you shouldn't even be talking about
Electromagnetism.


[Bilge]
I did. You just want to ignore the fact that for a steady current
in a wire, div J = 0 which is what we call ``magnetostatics''.

[OC]
I did not ignore it. I pointed out that it is irrelevant, because
it does not define current density.

[Bilge]
Your point being what? Since div J = 0 for magnetostatics and charge
conservation requires that d(\rho)/dt + div J = 0, what else did you
expect?

[OC]
For any closed surface you choose, the integral is zero.
But there is a NON-zero current flowing in the circuit.

[Bilge]
Yep. I = \integral J . dS

[OC]
Since you find that the integral is zero for any closed surface, would
you say that the current is zero?

You can argue that, although the integral is always zero, there is a
steady non-zero current density.

[Bilge]
The integral isn't zero. That is the point. The integral is a scalar
known as the current.

[OC]
I have to spell it out for you, haven't I?

I = \integral(S) J.dS = \integral(V) div J dV

if div J = 0 then the integral I is zero, no matter what surface S you
choose!

Bilge, were you bluffing all this time? (This is a rhetorical
question.)



[OC]
But if you do not know a priori that the current density is non-zero,
you cannot know it from the current if you use the integral.

[Bilge]
I didn't say the current density is zero. I said the divergence of
the current density is zero.

[OC]
Which was not the point.
I said that using the definition of current with the integral, you do
not get information on the current density (besides the divergence).
You cannot even distinguish betweeen zero current density or non-zero
steady current density.



[...]
[OC]
Very cute.
Maxwell's equations do not define current or current density. But you
are trying to use them for that (otherwise we would not be arguing),
which is incorrect.

[Bilge]
Yawn... If maxwell's equations don't define the current (and charge)
density, then exactly what relates the current density to the charge
density? After all, the _only_ relationship between charges and currents
is that those two things provide a conservation law for electromagnetism.
Outside of that, those two things are meaningless.

[OC]
An equation giving a relation between two entities, does not
necessarily define them.
Oh, by the way, the conservation law known as continuity equation is
NOT one of Maxwell's equation (as you acknowledge in your list below).


[Bilge]
You are the one insisting on a definition of current that doesn't
follow from electromagnetism, not me.

[OC]
Are you saying that without Maxwell's equation there is no
Electromagnetism?

What about Coulomb's law, Ampere's law, Faraday's law, Ohm's law...?

[Bilge]
Maxwell's equations

div E = \rho gauss' law
div B = 0 gauss' law for magnetic charge
curl E = -dB/dt faraday's law
curl B = J + dE/dt ampere's law (with maxwell's addition)

Ohm's law is nothing but an empirical relationship derived from
curve fitting and has no explanation outside of quantum mechanics.
It also doesn't help your argument, as I've already demonstrated
above. But, for the sake of completeness, you'll note that I can
get a magnetic field when the resistance is zero.

[OC]
Please show us how E, B, \rho, and J are defined.

[OC]
Maxwell's equations summarize the relationships between electric
fields, magnetic fields, charges and currents. But all these entities
are defined by other means.

[Bilge]
Such as?

[OC]
Why are you so keen on showing off your ignorance?

The textbooks you quoted have plenty of explanations, why don't you
read them?


[Bilge]
OK. Here it is again:
_
j^v = u(q\gamma^v)u

[OC]
Apply this to classical Electromagnetism.

[...]
[OC]
Now apply it to classical Electromagnetism.

[Bilge]
Rather than keep repeating yourself, _you_ apply it.

[OC]
You brought it up, you apply it.

OC

OC:
[Bilge]
The fact that the electric field outside a wire carrying a current is
zero. Should I assume that you weren't aware of this fact?

[OC]
The wire has zero NET charge (remember Gauss' law?).

[Bilge]
Why do you think I pointed that out?

[OC]
If you can explain why, be my guest.


Buy an E&M textbook. You evade every point. Crackpot.

plonk


[OC]
And strictly speaking the electric field outside the wire is not
exactly zero.

[Bilge]
Apply gauss' law to a wire with no net charge.

[OC]
That's because when there is a current flowing, there is
a voltage gradient along the wire (remember Ohm's law?).

[Bilge]
You are digressing and in any case, ohm's law only validates
my point, since V = IR is a relation between scalars and in
vector form, ohm's law reads, J = \sigma E, or if you want to
be fancy, J = \sigma . E, where \sigma == \sigma_ij is a 3 x 3
tensor to account for anisotropic media. To which version of
ohm's law were you referring and what about it invalidates anything
I said?

[OC]
You brought up the electric field of a current-carrying wire.

By the way, if you did your homework, you would see that more often
then not, current is treated as vector, even with ohm's law.


[OC]
This voltage gradient produces an electric field outside the wire.
But in most cases the field can be neglected.

[Bilge]
No. Try proving that.

[OC]
No what?

No, the voltage gradient does not produce an electric field, or
No, in most cases the field can be neglected?

About the first: "voltage gradient" is not accurate, I should have
said "potential gradient"; and a potential gradient IS an electric
field.

About the second: in most cases it is neglected, as you prove when you
said that the electric field outside a wire is zero.

By the way, look up "energy density" and "energy density flow" of
electric fields, and you will see if the electric field outside a wire
is zero or not.


[Bilge]
Except for at least two physicists your survey missed. Me and appar-
ently, jackson, who states on pg 175 [2nd edition], ``Since the surface
integral of the current density is the total current I [denoted as
a scalar] passing through the closed curve C...''

[OC]
Let me appeal to Feynman's Lectures, vol. 2.
In the chapter on magnetostatics, after the definition of current as
flow of current density through a surface, we find the current I as
vector.
He explicitly speaks of "vector current I".
Maybe you should read Jackson's book more carefully and go beyond page
175.

[Bilge]
Perhaps you should tell me what's wrong with the material on page 175
and where jackson retracts it later on.

[OC]
Please pay more attention.
I never said that the definition of current with the integral is
wrong, neither did I claim that it is retracted.

I just gave you an example of current used as a vector.

You seem to be convinced that defining electric current as flux of
current density through a closed surface is the one and only possible
and acceptable definition.
I brought up exmples to show that you are wrong.
If you do not accept it, there is not much I can do about it.


[OC]
By the way, in order to use the definition of current with the
integral, you have give to give a specific surface.

[Bilge]
Your point being what?

[OC]
What happens if you want to treat the current without choosing a
specific surface (which is not unusual)?

[Bilge]
I note you have provided no example of this ``not unusual'' situation.

[OC]
Current in a straight wire.
Uncountable situations where the conductors involved are uniform or
symmetric.

Have you ever done homework exercises?

[Bilge]
Oh, yeah. I forgot. Current has nothing to do with maxwell's equations.
So, ok, use this definition:
_
j^v = u(q\gamma^v)u

where u is the free particle wavefunction for a fermion, q is the
electric charge and \gamma^u are the dirac matrices. How's that?

[OC]
After several posts you finally have given a definition of current
density!
So, you agree that current is defined in terms of charged particles.

[Bilge]
No, because the charge doesn't exist independently of the current.
j^u is a four-vector and the charge density is only the time component
of that four vector. In case you haven't learned anything about electro-
magnetic radiation, light has no charge, yet it's source is a transverse
current density.

[OC]
Are you talking about electromagnetic waves? Where there are
time-dependent electric fields that can move charges?

Above you defined current density in term of moving charged particles
(something you would not admit in previous posts).

Then there is this exchange:
OC:"So, you agree that current is defined in terms of charged
particles."
Bilge:"No, because the charge doesn't exist independently of the
current."

Do you mean that if there is a current, then there are charged
particles moving?
Do you mean that if the current density is zero, then there is no
charge?


[OC]
Now, how do you apply this definition to classcial Electromagnetism?

[Bilge]
The same way I've been applying it. The definition was for your
benefit, not mine.

[OC]
We were talking about classical Electromagnetism, weren't we?
Why can't you simply show how you apply your definition to it?



[...]
[OC]
Is it really so hard to figure out that current has something to
do with electromagnetism?

[Bilge]
I never said that electric current has nothing to do with Electromagnetism.
I was saying that current is a vector.

[Bilge]
Except that you insist that I define current without using
electromagnetism, which makes no sense.

[OC]
Apparently you think that only Maxwell's equations "are"
Electromagnetism.
This is simply wrong. (See below)



[...]
[Bilge]
In other words, I just contradicted your assertion regarding maxwell's
equations, so this is no longer on topic.

[OC]
Are you confusing me with somebody else (again)?
The question was never a classical description of the electron.

[Bilge]
What classical moving charges exist?

[OC]
Please explain how this relates to the problem of current being a
vector.

[OC]
The question was about electric current being a vector or a scalar.

[Bilge]
Then stop insisting that current be defined independently of
electromagnetism.

[OC]
I never said such thing.
If we talk about charge and electric current, we are dealing with
Electromagnetism.
Only because charge or electric current do not require Maxwell's
equations to be defined, it does not mean that they are outside the
domain of Electromagnetism.


[...]
[Bilge]
Yes, I know. But from the rules you've imposed, apparently the only
definition that you'll accept is your personal definition, not one that
relates to electromagnetic phenomena.

[OC]
No, I did not give any rules.

[Bilge]
Yes, you did. You said that I can't use ampere's law.

[OC]
Because Ampere's law does NOT define electric current or current
density.
It just links magnetic fields to electric currents, but defines
neither of them.


[OC]
It is just the normal way Electromagnetism is explained.
And you should know if you read your books.

[Bilge]
And here I thought that classical E&M was explained using maxwell's
equations. Could you tell me which equation of classical E&M other
than ampere's law contains any reference to a current? To the best
I can determine, that's the only one of maxwell's equations containing
anything at all related to current and if maxwell's equations aren't
classical E&M, I'm not sure what equations I should be using.

[OC]
Show me how electric fields, magnetic fields and current are defined
through Maxwell's equations.

I bet that you end up with circular reasoning, because Maxwell's
equations are not enough.



[...]
[Bilge]
(1/ie)[D^u, D^v], and take the divergence twice. The first gives you
j^v and the second gives you the continuity equation, d_v j^v = 0.
Any guess as to what that commutator is or what taking that first
divergence gives you?

[OC]
Why are you trying to make things more complicated than necessary?

[Bilge]
No, but you insisted on a definition that did not start with ampere's law.
I would have preferred to make this simplest by not having it come up.

[OC]
Ampere's law does NOT define electric current or current density.
I you don't get it, you shouldn't even be talking about
Electromagnetism.


[OC]
Are you trying to show off your "knowledge", or is it just another
distraction tactic?

[Bilge]
I don't suppose you are going to do as I requested and take the divergence
of that commutator, are you?

[OC]
I am still waiting for your definition of current density within
classical Electromagnetism.


[OC]
I settle for a definition of current density within classical
Electromagnetism. Please show us that how do apply what you exposed
above within a classical framework.

[Bilge]
As soon as you tell me what you consider the equations of classical
E&M. since ampere's law doesn't seem to one of the ones you accept.

[OC]
Ampere's law does NOT define electric current or current density.
I you don't get it, you shouldn't even be talking about
Electromagnetism.


[Bilge]
I did. You just want to ignore the fact that for a steady current
in a wire, div J = 0 which is what we call ``magnetostatics''.

[OC]
I did not ignore it. I pointed out that it is irrelevant, because
it does not define current density.

[Bilge]
Your point being what? Since div J = 0 for magnetostatics and charge
conservation requires that d(\rho)/dt + div J = 0, what else did you
expect?

[OC]
For any closed surface you choose, the integral is zero.
But there is a NON-zero current flowing in the circuit.

[Bilge]
Yep. I = \integral J . dS

[OC]
Since you find that the integral is zero for any closed surface, would
you say that the current is zero?

You can argue that, although the integral is always zero, there is a
steady non-zero current density.

[Bilge]
The integral isn't zero. That is the point. The integral is a scalar
known as the current.

[OC]
I have to spell it out for you, haven't I?

I = \integral(S) J.dS = \integral(V) div J dV

if div J = 0 then the integral I is zero, no matter what surface S you
choose!

Bilge, were you bluffing all this time? (This is a rhetorical
question.)



[OC]
But if you do not know a priori that the current density is non-zero,
you cannot know it from the current if you use the integral.

[Bilge]
I didn't say the current density is zero. I said the divergence of
the current density is zero.

[OC]
Which was not the point.
I said that using the definition of current with the integral, you do
not get information on the current density (besides the divergence).
You cannot even distinguish betweeen zero current density or non-zero
steady current density.



[...]
[OC]
Very cute.
Maxwell's equations do not define current or current density. But you
are trying to use them for that (otherwise we would not be arguing),
which is incorrect.

[Bilge]
Yawn... If maxwell's equations don't define the current (and charge)
density, then exactly what relates the current density to the charge
density? After all, the _only_ relationship between charges and currents
is that those two things provide a conservation law for electromagnetism.
Outside of that, those two things are meaningless.

[OC]
An equation giving a relation between two entities, does not
necessarily define them.
Oh, by the way, the conservation law known as continuity equation is
NOT one of Maxwell's equation (as you acknowledge in your list below).


[Bilge]
You are the one insisting on a definition of current that doesn't
follow from electromagnetism, not me.

[OC]
Are you saying that without Maxwell's equation there is no
Electromagnetism?

What about Coulomb's law, Ampere's law, Faraday's law, Ohm's law...?

[Bilge]
Maxwell's equations

div E = \rho gauss' law
div B = 0 gauss' law for magnetic charge
curl E = -dB/dt faraday's law
curl B = J + dE/dt ampere's law (with maxwell's addition)

Ohm's law is nothing but an empirical relationship derived from
curve fitting and has no explanation outside of quantum mechanics.
It also doesn't help your argument, as I've already demonstrated
above. But, for the sake of completeness, you'll note that I can
get a magnetic field when the resistance is zero.

[OC]
Please show us how E, B, \rho, and J are defined.

[OC]
Maxwell's equations summarize the relationships between electric
fields, magnetic fields, charges and currents. But all these entities
are defined by other means.

[Bilge]
Such as?

[OC]
Why are you so keen on showing off your ignorance?

The textbooks you quoted have plenty of explanations, why don't you
read them?


[Bilge]
OK. Here it is again:
_
j^v = u(q\gamma^v)u

[OC]
Apply this to classical Electromagnetism.

[...]
[OC]
Now apply it to classical Electromagnetism.

[Bilge]
Rather than keep repeating yourself, _you_ apply it.

[OC]
You brought it up, you apply it.

OC

"Bilge" wrote in message
...
OC:
[Bilge]
The fact that the electric field outside a wire carrying a
current is
zero. Should I assume that you weren't aware of this fact?

[OC]
The wire has zero NET charge (remember Gauss' law?).

[Bilge]
Why do you think I pointed that out?

[OC]
If you can explain why, be my guest.


Buy an E&M textbook. You evade every point. Crackpot.

plonk

Hahahahaha...........and bilge falls apart.

I see that events on this thread have unfolded exactly as I predicted.

H.Ellis Ensle

[Bilge]
The fact that the electric field outside a wire carrying a current is
zero. Should I assume that you weren't aware of this fact?

[OC]
The wire has zero NET charge (remember Gauss' law?).

[Bilge]
Why do you think I pointed that out?

[OC]
If you can explain why, be my guest.

[Bilge]
Buy an E&M textbook. You evade every point. Crackpot.

plonk


Are you running away now? What happened?

Have you realized that you cannot support your position in a point by
point discussion?

Have you realized that your attempts to divert the attention away from
the topic do not work?

Are you afraid to expose your ignorance in public? (It's a bit late
for that.)

Or have you finally understood that you are wrong and you cannot admit
it?

You say "Buy an E&M textbook".
My advice: don't buy textbooks, read them, carefully.

OC