Classical Electromagnetism: Question on Ampere's Law and Displacement Current

jaseh86

Hi, just out of curiosity...

Ampere's Law describes that an electric current produces a magnetic field. When corrected with Maxwell's displacement current, it describes that a magnetic field is also created by a time-varying electric field.

Does this mean that an electric current produces the magnetic field purely BECAUSE of the changes in electric field associated with moving charges. Or is electric current one of two distinct ways (the other being changing electric field) to produce magnetic field.

A little confused on the matter.

Thanks.

Defennder

Ampere's circuital law does not say that an electric current produces a magnetic field. That is the Biot-Savart law. I don't see how a changing electric field (without considering the current flow) can induce a B-field.

jaseh86

Thanks for the reply.

But doesn't Maxwell's Theory of Light explain that changing electric fields induce magnetic fields due to Ampere's Law / Biot-Savart Law. In vacuum for instance, there is no electric current, so why is it that a changing electric field without current flow can't produce magnetic field?

cabraham

Good question. All we really can say from Maxwell's equations is that under time varying conditions, electric and magnetic fields cannot exist independently. It is futile to even attempt to ascertain which one comes first. Which is the cause and which is the effect is an endless vicious circle. In his 1905 paper "On The Electrodynamics Of Moving Bodies", Einstein states that "questions as to which one (electric or magnetic field) is the "seat" (root, fundamental, canonical, principle, basis) no longer have any point." The parentheses are mine.

Defennder

Hmm, actually having taken a closer look at your thread title, I realise that I might have misunderstood your question. The displacement current, which is associated with a changing E-field does itself induce a B-field. The only question that remains if every time-varying E-field is associated with a displacement current.

nicksauce

To which I would definitely say no. The easiest case is solving Maxwell's equation in the absence of sources, and getting a wave equation for E (and B), which is certainly a time-varying field.

rbj

hunh?? that's what one of them 4 Maxwell's Equations tells us. of course a changing electric field induces a magnetic field.

imagine a simple two parallel plate capacitor with axial leads and a steady (DC) current flowing in one lead (and out the other lead). between the plates of the capacitor there is no current, but there is a magnetic field induced.

Defennder

Solving which equation?

That is because of the displacement current. The question I posed was whether every time-varying E-field would be associated with a displacement current. nicksauce said no, so how about those cases whereby we have a time-varying field but no displacement current or current flow? Are there such cases, and is any B-field induced?

cmos

Can you defennd this statement a bit (mind the pun )? Ampere's law clearly states that the presence of electric current induces a curling magnetic field. Does this not imply that electric current produces a magnetic field?

The laws of Ampere and Biot and Savart both say that current induces the magnetic field; I would call it a matter of convenience in choosing which law to invoke for a specific problem. Note that you may consider either law as an axiom to EM theory from which you can derive the other.

cmos

The term known as the displacement current (dD/dt) is called that because it is correct to think of it as a current in the traditional (or at least semi-traditional) sense; that is as the movement of charge. You have to remember that the electric displacement (D-field) contains in it two facets of information: 1) the electric field (E-field) and 2) the electric polarization (which is essentially a sum over electric dipole moments).

It is the time-varying realignment of dipole moments, when immersed in a time-varying electric field, that gives rise to the displacement current. Thus we can see the displacement current as a movement of real charge; the charge, however, is a bound charge as opposed to a free charge.

Now in freespace, the polarization vector is identically zero, thus (taking the freespace permittivity to be unity) D=E. Clearly then, what is called the displacement current has nothing to do with an actual current, but is a pure time-varying electric field. But seeing that we have already named Maxwell's correction to the Ampere law as the displacement current, we might as well keep calling it that (or so some would argue). Really, its a matter of taste.

Defennder

I have already answered this part in my reply to rbj with my questions below. I can't edit my old posts so I'll appreciate if you'll read both my later replies as well as the earlier posts. As for the fact that Ampere's and Biot-Savart's law follows from one another, I am aware of that. I was only saying that the Biot-Savart law gives us a more explicit vector equation of B due to I, in comparison to Ampere's.

How does this answer the questions
1. Is every time-varying E-field would be associated with a displacement current D?
2. Are there any cases whereby you have a varying E-field but no associated displacement current?
3. If the answer to 2 is yes, then would there be a B field induced?

These are just yes/no questions. I don't see how you have answered them.

Defennder

Anyway, here's what Wikipedia says:
But since it's Wikipedia, I'll leave it to the experts to verify it. Otherwise it answers all of my questions.

cmos

I believe I answered all these questions in my previous post. When asking a question, you learn nothing by just getting a yes or no response; hence my lengthy response. But, to be explicit:

1) Well first off D is not the displacement current, it is the electric displacement (sometimes called the flux density). The time-derivative of D is the displacement current. In principle, there is nothing wrong with talking about the D-field in freespace, but since it is equivalent to the E-field ( [tex]D=\epsilon_0E[/tex] ), there really is no point.

As I said before, whether you wish to call the time-derivative of the electric field in freespace a displacement current or not is really just a matter of taste.

2) See above.

3) As you Wikipedia article states, a time-varying electric field induces a time-varying magnetic field and vice-versa. This is the mechanism by which light propagates.

Defennder

Thanks, got it. Should have read your post more closely. I was just trying to be sure I understood it correctly.

cmos

I think it just may have been your choice of wording in your first post that may lead some to confusion. To make it clear, Ampere's law does, in fact, state that a current induces a magnetic field. In principle, it is a fundamental law of nature. In practice, however, Ampere's law is not always so easy to play with and there are time when deference to the Biot-Savart law gives a simpler way to reach a solution for the magnetic field.

jaseh86

Hi everyone, thanks for all the input, even though it has kind of deviated from the intention of my original question.

Perhaps the questions I really should have asked are :

1. Does the movement of charges directly result in a displacement current (varying E-field)?
3. Can an electric current induce a magnetic field if there was NO consideration/existence of a displacement current?

Ultimately, from cabraham's post, I assume it becomes pointless to question whether moving charges induce a time-varying magnetic field which then induces a time varying E field (Faraday's), or moving charges have a time-varying electric field which then induce a B field (Maxwell-Ampere).

Thanks.

cmos

Well now your line of questioning is getting a bit more complex. It comes from the upper-level idea that "there is really no such thing as magnetostatics." What this means is that what is usually taught as "magnetostatics" should strictly be called magneto-quasi-statics. This, as you have said, is because a steady current is really a movement of charge which follows from it time-variances in the fields. But, we can often neglect these effects since they would add only the smallest of corrections (which is good because it is so much easier to write "magnetostatics").

To answer your questions:

1) Yes, the movement of charge implies modification in the electric field, but for a steady current, this can often be neglected. Note, however, that an AC current will induces an AC B-field by Ampere's law which, in turn, will induce an AC E-field by Faraday's which, in-turn, will induce an AC B-field by Maxwell's correction to Ampere's law which, in-turn, will ................

3) Sure. This is just Ampere's law without Maxwell's correction.