Distinction free and bound current (or charge) [very confused]

In summary: When we talk about conductors, however, we no longer have a perfect insulator separating us from the free charges. Instead, the free charges are constantly bombarding the conductor, and it's only because we have a perfectly permeable material that the free charges are able to flow slowly through the material. In general, then, the free current in a conductor is proportional to the electric field. However, because we cannot independently control the flow of charge, the bound current also exists. This is why the magnetic permeability of a conductor is not equal to the material's permeability for electric currents, because there is a component of bound current flowing through the material.
  • #1
nonequilibrium
1,439
2
Hello,

I'm reading Griffiths' Introduction to Electrodynamics and I got quite confused in 9.4.1 page 392 (but the question is general, for anyone who does not have that book):

It's about EM-waves in conductors. I will quote a paragraph:

In Sect. 9.3 I stipulated that the free charge density [tex]\rho_f[/tex] and the free current density [tex]\vec J_f[/tex] are zero, and everything that followed was predicated on that assumption. Such a restriction is perfectly reasonable when you're talking about wave propagation through a vacuum or through insulating materials such as glass or (pure) water. But in the case of conductors we do not independently control the flow of charge, and in general [tex]\vec J_f[/tex] is certainly not zero. In fact, according to Ohm's law, the (free) current density in a conductor is proportional to the electric field: [tex]\vec J_f = \sigma \vec E.[/tex]

What is the definition of bound (or free) current? I can't find a decent one online, and the book (chapter 6) seems to keep it to something like "bound current is the current you can't directly control", but that is not only too vague for me (there are situations where I'm caught in a limbo), the above paragraph seems to contradict that! I say this because of how he blames the existence of the free current on the fact that you cannot independently control the flow of (all) charge, yet that was his definition of bound current, so this should not concern free current.

Even more so: as he states that E is proportional to the free current, he implies that the bound current is zero (because E is proportional to all the current, in an Ohmic material). But that seems impossible, because if so, then the magnetic permeability of conductors would have to be equal to [tex]\mu_0[/tex], because there's nothing they're keeping track off (e.g. the magnetic permeability of an isolator is simply a way to easily add the (simplified) contribution due to bound current, but if the bound current is always zero, there is no need for an adapted constant, so [tex]\mu_{\textrm{conductor}} = \mu_0[/tex]).

I'm hoping I'm just misunderstanding the concepts, cause at the moment I'm not getting it. If I put an EM wave in a (uncharged) conductor, I would think the magnetic permeability and electric permittivity of the conductor take care of any subsequently induced currents, so that there is no talk of [tex]\rho_f[/tex] or [tex]\vec J_f[/tex]. Where am I wrong?
 
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  • #2
I think it's most clear to go back to the definition of polarization and bound charge. P=[tex]\epsilon[/tex]E; as you know a static electric field induces a static shift of charge. If there are free charges, they can move forever in a static field, at the drift velocity. Bound charges can only shift while remaining attached to an atom, etc.

Because [tex]\nabla[/tex].P=bound charge (sorry, having trouble with the math text), if you take the time derivitive of each side, by comparison to the continuity equation, you see that the time derivitive of polarization "acts like" a current. But we know that a static electric field only causes a static polarization. So the bound current only occurs when the electric field is changing in time.

This is probably what Griffiths means when he says we don't control the bound current. The idea is that it is easy for us to apply a static field of any magnitude in order to directly control the magnitude of free current; whereas it is not so simple to control the time derivative of the field.

I agree that his wording is poor when he says that we don't directly control the flow of charge in a conductor.
 
  • #3
Hm, I'm sorry if I'm being daft, but I don't get how your reply answers my questions?

You would call the derivative of the polarization a free current, am I right? Why? I would think a bound current.
 
  • #4
mr. vodka said:
Hm, I'm sorry if I'm being daft, but I don't get how your reply answers my questions?

You would call the derivative of the polarization a free current, am I right? Why? I would think a bound current.

No, the derivative of the polarization is the bound current, since it is related by the continuity equation to the flux of bound charge.

What I said was intended to answer your question about the definition of bound current. I wasn't directly answering all the points about "controlling" currents because that seems inherently ambiguous and I didn't want to try to read Griffiths' mind. But here goes:

When he is talking about insulators, he says that Jf and pf are zero because there are no free charges in a perfect insulator, and hence no free currents. Since every charge carrier is fixed to an atom and immobile, the only free currents that could exist are currents that we "put there", for example by conceptually adding free charges to the system. In this way, we "control" the free currents and charges, and hence are able to make the statement that they are zero.

However, the bound charges can still move in an applied dynamic field, via a time varying polarization. In this case I think the idea that bound current is the current we "don't control" is related to the fact that we can arbitrarily set the free current by applying a suitable static field via Ohm's law. We can't control the bound current because we can't set the value of dP/dt in anything other than a transient way.


But again, my advice is not to worry too much about these notions of "control", just keep going back to the constitutive relations and make sure all of those make sense.

Even more so: as he states that E is proportional to the free current, he implies that the bound current is zero (because E is proportional to all the current, in an Ohmic material). But that seems impossible, because if so, then the magnetic permeability of conductors would have to be equal to , because there's nothing they're keeping track off (e.g. the magnetic permeability of an isolator is simply a way to easily add the (simplified) contribution due to bound current, but if the bound current is always zero, there is no need for an adapted constant, so ).

You're correct that in a static field the bound current is zero, but there is still a static polarization in the material. This is the information that the material's permittivity gives you.

I'm hoping I'm just misunderstanding the concepts, cause at the moment I'm not getting it. If I put an EM wave in a (uncharged) conductor, I would think the magnetic permeability and electric permittivity of the conductor take care of any subsequently induced currents, so that there is no talk of or . Where am I wrong?

The permittivity of a material just allows you to treat the material as a homogeneous medium, as if you were dealing with free space. What you are stating here is like saying that the permittivity of free space would "take care of" any subsequently induced currents, and there would be no free current. But of course if we place isolated charges in free space and apply a field, there will be a free current. It's the same thing in a conductor. The conductor has some polarizability due to its bound charges, and this defines a permittivity. But it also has free charges which determine the free current response.
 
  • #5
Thank you for your detailed response, I'm starting to get what you mean; I have two specific points I'd like to reply with, in an attempt to understand it better, if you have the time (and the points are separate from each other). I feel like I'm getting closer to understanding.

1) If I read your last post well, you're taking the meaning of "free charge" to be a charge that is not attached to something (like a free electron in a metal), and a "bound charge" to be a charge that is (like an electron in an insulator, attached to the atom). Fair enough, the names would match, but I don't think these are the definitions of free and bound charge? Bound charge is simply the charge due to polarization, and the free charge is the other (i.e. non-bound) charge. That might seem to be the same to you, but I don't think so: take a perfect conductor and slide it in between a conductor, there will be a perfect polarization that cancels the whole electric field, and this polarization is not only due to the bound electrons in the metal atoms, but mostly due to the (in your sense of the word) free electrons in the metal (otherwise the electric permittivitiy would never be infinity). In my Maxwell equations (with D instead of E), I will have no free charge (because the permittivity of the conductors takes care of the polarization), although you would if you were to follow your definition of "charge that is not attached to an atom". Can you tell me if I'm taking your words the wrong way, and if so, how do you separate bound from free charges?

2) Quoting you:
You're correct that in a static field the bound current is zero
as a response to me stating that Griffiths' equation [tex]\vec J_f = \sigma \vec E[/tex] implies that [tex]\vec J_b = 0[/tex]. Fair enough, but Griffiths even uses this equation for time-varying E-fields (as the section I'm quoting from is about EM-fields in conductors). So then according to Griffiths' interpretation, [tex]\vec J_b = 0[/tex] always in conductors?...
 
  • #6
mr. vodka said:
take a perfect conductor and slide it in between a conductor, there will be a perfect polarization that cancels the whole electric field, and this polarization is not only due to the bound electrons in the metal atoms, but mostly due to the (in your sense of the word) free electrons in the metal (otherwise the electric permittivitiy would never be infinity). In my Maxwell equations (with D instead of E), I will have no free charge (because the permittivity of the conductors takes care of the polarization), although you would if you were to follow your definition of "charge that is not attached to an atom.

I'm not sure if I understand what you mean about sliding a conductor in between a conductor, but I think you're getting at the fact that in a conductor, the free electrons completely screen any external field such that the internal field is zero.

Why are you attributing this effect to a polarization? According to my definition, it's not a polarization specifically because the charges responsible for the screening are not bound charges, and polarization only refers to the distribution of bound charge.

You said you have no free charge in this situation, but I don't know where you draw that conclusion. A conductor always has free charge; there may or may not be a current, but that charge is in the conductor somewhere. In the bulk of the conductor the free charge will be uniformly distributed, and this satisfies D=0 from Gauss's law no matter what value of free charge density is present.

as a response to me stating that Griffiths' equation [tex]\vec J_f = \sigma \vec E[/tex] implies that [tex]\vec J_b = 0[/tex]. Fair enough, but Griffiths even uses this equation for time-varying E-fields (as the section I'm quoting from is about EM-fields in conductors). So then according to Griffiths' interpretation, [tex]\vec J_b = 0[/tex] always in conductors?...

I think I might need to take a look at the section you are talking about. I have a copy of Griffiths if you can reference.

I'm not entirely sure on this point but I think this really depends on the definition of conductivity. To be completely general, it is frequency dependent and complex (and a tensor). As far as I know, Griffiths is referring to an (isotropic) real, DC conductivity. In that case you can define it as the proportionality between free current and electric field. If the frequencies of the time varying field are sufficiently low, it is acceptable to use the DC conductivity for time varying fields.

In general though, I think you're right that you wouldn't be able to separate the free charge and polarization response so easily.
 
  • #7
Thanks for your swift reply!

Why are you attributing this effect to a polarization? According to my definition, it's not a polarization specifically
Well, I can't find an exact definition of polarization, but I think everybody will agree with something of the following:
the net charge is zero, the dipole moment is nonzero, and the other terms in the multipole expansion are negligible in comparison to the dipole moment
Why this definition? Because these three conditions are sufficient to describe the resulting electric field in function of the dipole moment in the form of something like [tex]\int \frac{ \vec r \cdot \vec P}{r^2} \mathrm d \vec r[/tex] (with some constants left out), with P being the dipole moment (density). (this is discussed in chapter 3.4 page 146 of Griffiths).

I think you'll agree the three conditions are fullfilled in the case of a slab of conductor in between two conductor plates.

In chapter 4.2 (page 166), then, Griffiths uses that dipole moment density P to define the bound charge (top of p168) as [tex]\rho_b = -\vec \nabla \cdot \vec P[/tex].

Another way of making my point (if you're not yet convinced) that the metal polarizes: say it wasn't polarization, then D, being equal to [tex]\epsilon_0 \vec E + \vec P = \epsilon \vec E[/tex], is simply [tex]\epsilon_0 E + 0[/tex], such that [tex]\epsilon_0 = \epsilon[/tex] for the piece of metal, while metals do have their own electric permittivity. Proof by contradiction.

You said you have no free charge in this situation, but I don't know where you draw that conclusion. A conductor always has free charge;
I hope it's now clear why we differ on this matter: when you say "free charge", you say "not bound to an atom", while I'm saying "charge that is not due to the polarization", which is not the same, as I hope I just explained my point of view well.

I think I might need to take a look at the section you are talking about. I have a copy of Griffiths if you can reference.
The section that inspired this confused mess is 9.4 starting on page 392 :)

As for your second to last paragraph, I'm not sure if I understand, but I think I'll wait for you to read 9.4, if you're still up for it.

NB: Of course one has the choice to not work with the distinction of bound and free charges, in the sense that one can choose not to hide the induced/bound/etc currents & charges in the modified constants [tex]\epsilon_0 \to \epsilon[/tex] and [tex]\mu_0 \to \mu[/tex], and then everything is a free charge or current (by the earlier definition that I'm following, not yours, obviously), and then I'd agree with everything Griffiths says in 9.4, except for the fact he uses [tex]\mu[/tex] and [tex]\epsilon[/tex] where I'd put the vacuum constants (because in my view then there is nothing extra/induced that the constants should account for, as I'm putting it into the equations myself under the name of [tex]\rho_f[/tex] and [tex]J_f[/tex], unless there are other effects I'm forgetting)
 
  • #8
I didn't mean that the conductor doesn't polarize at all; this is still consistent with my definitions since of course any metal has a huge number of valence band or core electrons. The only thing I took issue with is the fact that you said there is a "perfect polarization" which cancels the field; the polarization will of course exist in accordance with the permittivity, but free charges (under my definition) are needed to fully screen a field in general, in my understanding this is the reason the internal field is fully screened in a conductor but not a dielectric.

Anyway I'm going to have to take a look at those Griffiths sections and get back to you. At least one of us is going to learn something from all this! :biggrin:
 
  • #9
Hm, so the disagreement is that I also call the movements of the free electrons ("free" in the sense of "not tied to an atom", I will call this "metal-free" from now on) polarization, resulting in the fact that I call the metal-free electrons bound charges (so here "bound" in the sense of [tex]\rho_b = - \vec \nabla \cdot \vec P[/tex], and thus being absorbed in the permittivity constant of that specific metal).

A result of our disagreement -unless it's an empty disagreement- is that you would put a non-zero [tex]\rho_f[/tex] into your Maxwell equations (the equations for matter, that is), whereas I would put it equal to zero. Correct?

And I'll eagerly await your reading of Griffiths :) (or the reply thereof, I suppose)
 
  • #10
Well, most of my response is probably going to be less rigorous than you would like, but I'll say what I can.

mr. vodka said:
Fair enough, but Griffiths even uses this equation for time-varying E-fields (as the section I'm quoting from is about EM-fields in conductors). So then according to Griffiths' interpretation, always in conductors?...

I think the footnote on p393 is relevant here. Griffiths says that Jf = [tex]\sigma[/tex]E, and you were arguing that the total current density should be in that expression. In the footnote he notes that Ohm's law breaks down on time scales shorter than the collision time. For the DC case, conductivity is related to the net motion of charge in response to a field, but any individual electron will still be moving in a quasi-random direction (if I can be semiclassical here). On very short time scales you can not treat the problem as a collective drift of charge, and you would have to consider the complex dynamics of electrons which will depend on the band structure, etc. By the time that things have equilibrated enough that you can use Ohm's law, the bound currents are not significant. On that time scale you can treat the polarization of a material as an instantaneous effect.


mr. vodka said:
the net charge is zero, the dipole moment is nonzero, and the other terms in the multipole expansion are negligible in comparison to the dipole moment.

I think you'll agree the three conditions are fullfilled in the case of a slab of conductor in between two conductor plates.

Another way of making my point (if you're not yet convinced) that the metal polarizes: say it wasn't polarization, then D, being equal to [tex]\epsilon_0 \vec E + \vec P = \epsilon \vec E[/tex], is simply [tex]\epsilon_0 E + 0[/tex], such that [tex]\epsilon_0 = \epsilon[/tex] for the piece of metal, while metals do have their own electric permittivity. Proof by contradiction.

The free charges in a conductor can screen any electric field of frequency below the plasma frequency; at a distance into the conductor of a few skin depths, there is no electric field present at all. Then the polarization can be zero without contradicting the measurable permittivity of the metal. So yes, near the surface the metal polarizes, but in situations where high frequency fields or very thin metallic layers aren't explicitly involved, people often ignore this.

In terms of DC permittivity of a metal, I'm not really sure, and I would probably want to know how those values are experimentally measured in order to understand what it means physically.


mr. vodka said:
NB: Of course one has the choice to not work with the distinction of bound and free charges, in the sense that one can choose not to hide the induced/bound/etc currents & charges in the modified constants [tex]\epsilon_0 \to \epsilon[/tex] and [tex]\mu_0 \to \mu[/tex], and then everything is a free charge or current (by the earlier definition that I'm following, not yours, obviously), and then I'd agree with everything Griffiths says in 9.4, except for the fact he uses [tex]\mu[/tex] and [tex]\epsilon[/tex] where I'd put the vacuum constants (because in my view then there is nothing extra/induced that the constants should account for, as I'm putting it into the equations myself under the name of [tex]\rho_f[/tex] and [tex]J_f[/tex], unless there are other effects I'm forgetting)

A result of our disagreement -unless it's an empty disagreement- is that you would put a non-zero into your Maxwell equations (the equations for matter, that is), whereas I would put it equal to zero. Correct?

Well, if I understand you right you are saying that if you incorporate the bound charges and currents into pf and Jf, you will use the vacuum constants, and I agree with that.

He also mentions on 393 that the dissipation of the charge density pf is an expression of the empirical fact that charge on a conductor flows to the edges. This is another piece of evidence that all of this treatment is ignoring the polarizations and charge distributions at the surface; of course in reality any metal may be terminated with charge from various sources.
 

What is the difference between free and bound current?

Free current refers to the flow of charged particles that are not bound to specific atoms or molecules. These particles are able to move freely in a material, such as in a conductor. On the other hand, bound current refers to the flow of charged particles that are bound to specific atoms or molecules and cannot move freely. These particles contribute to the overall current, but their movement is restricted.

What is the distinction between free and bound charge?

Free charge refers to the charged particles that are not bound to specific atoms or molecules and are able to move freely in a material. Bound charge, on the other hand, refers to charged particles that are bound to specific atoms or molecules and cannot move freely. These particles contribute to the overall charge of a material, but their movement is restricted.

How are free and bound current related?

Free and bound current are related in that they both contribute to the overall current in a material. Free current is typically associated with the movement of electrons, while bound current is associated with the movement of ions or other charged particles that are bound to specific atoms or molecules. The sum of these currents is known as the total current.

What is the importance of distinguishing between free and bound current/charge?

Distinguishing between free and bound current/charge is important in understanding the behavior of materials and their electrical properties. For example, in a conductor, the presence of free charge allows for the flow of electricity, while in an insulator, the lack of free charge restricts the flow of electricity. Understanding the distinction between these types of current and charge can also help in the design and development of electronic devices.

How is the distinction between free and bound current/charge applied in practical applications?

The distinction between free and bound current/charge is applied in various practical applications, such as in the design of electronic circuits and devices. For example, in designing a capacitor, the presence of bound charge on the plates allows for the storage of electrical energy, while the flow of free current through the circuit is what allows the capacitor to discharge. This distinction is also important in understanding the behavior of materials in different electrical fields and in the study of electromagnetism.

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