# Waves in a conductor

1. Oct 25, 2012

### aaaa202

I don't really understand how electromagnetic waves in wires are created. Sure you can see from the Maxwell equations that the fields satisfy the wave equation. But if you plug some cables onto a battery isn't the situation more or less static. I mean the electric field from the battery has existed since t=-∞ so I don't see why it should take time for the electric field to reach the other end of the cables.
Also it seems that there in general two ways to reach the equations for transmission in a cable. the telegraph equations. One goes by simply solving maxwell equations and applying the boundary conditions that a linear media gives. Another seems to be to view a cable as a sum of small capacitors and conductors. Either way you find precisely the same speed for the signal. Why is that? Surely Maxwells equations don't incorporate anything about the capacitance or inductance of the linear media.

2. Oct 25, 2012

### tiny-tim

hi aaaa202!
no, when you first connected the battery to the circuit, it took a finite time for the charge to get round
Maxwell's equations include the ampere-maxwell-law …

curlB = µ jfree + µε ∂E/∂t

which include µ and ε, the permeability and permittivity, which could be (but aren't) called "inductivity" and "capacitivity" respectively

3. Oct 25, 2012

### Studiot

Electromagnetic waves are not created in conductors they are created on and around conductors.
The conductance of conductors is so high that any waves attenuate very very rapidly.

This is known as the skin effect.

4. Oct 25, 2012

### aaaa202

studiot: okay that makes sense, although I don't see what the telegraph equations describe then. Is the potential in it the potential around the conducting wires?

Tim: I do realize that it takes time for the charges to move. But we are conserned about how the field from our battery drives the current around right? And that field has existed always so I don't understand how it should take time for the field to propagate information around. Or what field are we really looking it when we study this apparant wavel like behaviour?

5. Oct 25, 2012

### Studiot

We do not usually calculate this way but the wave dues to electric power at 50/60 Hz have wavelengths of thousands of kilometers in air but nanometres in copper.

The telegraph equations and the transmission line equations describe waves in a transmission medium between two conductors, not in the conductors.

The waves, of course propagate at the local speed of light, which is close to c in air but much slower in copper.

I do not have more time now.

6. Oct 26, 2012

### aaaa202

To continue this discussion: What is it that is actually responsible for the wave motion of the field. At first I thought that it's the electrons bouncing into each other but that would be a transversal wave and it doesn't seem right since electromagnetic waves don't need a medium.

7. Oct 26, 2012

### Studiot

This question caused controversy for a century or so.

In terms of wave motion one way to think of it is to consider the electromagnetic wave as carrying its medium with it, in a manner that feeds on itself.

It is a fundamental experimental observation that a changing electric field gives rise a magnetic one and a changing magnetic field gives rise to an electric one. There is no theoretical requirement for this in classical physics but it is observed to be so.

Last edited: Oct 26, 2012
8. Oct 26, 2012

### AJ Bentley

Several people have already said this but you seem to be missing it.

A transmission line is not a wire. It's (usually) a coaxial cable used to transmit AC signals.

The theory is about EM waves travelling down such a cable - not about electricity from a battery running down a wire.

9. Oct 26, 2012

### PhilDSP

Hi Studiot,

You bring up an important observation but it's resolution may not be what you believe it to be. We should remember that none of the Maxwellians (and Faraday and Maxwell himself) assumed that the cause of a changing magnetic field is a changing electric field (and vice versa). That seems to be a 20th century bit of confusion. Please see Jefimenko's clear analysis of the actual causal relationships (or a secondary source such as Jackson's textbook).

http://en.wikipedia.org/wiki/Jefimenko's_equations

In regard to the OP, in Maxwell theory there are 2 kinds of current:

conduction current - the movement of electrons or other charged particles
displacement current - the movement of energy whose characteristics are described by 'fields'

The Maxwell equations give us the rules for determining how both types of current affect each other.

10. Oct 26, 2012

### Studiot

Hello Philip.

Look at the title of this thread.
How much displacement current exists in a conductor?

Does it not imply that it is theoretically impossible to avoid having a magnetic field without an electric one and vice versa?

Yet the conventional view is that it is change of one that gives rise to the other.
If this is true it, does it not preclude the possibility of a steady uniform field throught time and space?

11. Oct 26, 2012

### PhilDSP

The displacement current is perfectly described by wave equations.

Yes, I think you are pointing out that the observation of the changing of an electric field is very often linked to the changing of a magnetic field. So the difference between correspondence and a causal relationship is a bit subtle. But there are cases where one or both of the fields propagate as evanescent waves. That is, they aren't traveling waves and their changing values don't continuously propagate. In those situations you may find exceptions.

P. S. The Poynting theorem is good to look at in conjunction with these questions. It shows the movement of energy (which is also described by wave equations related to field fluctuations). But the Poynting theorem shows that the energy moves at right angles to the flow of electrons in the wire - from outside the wire into it.

Last edited: Oct 26, 2012
12. Oct 26, 2012

### Studiot

I think all this is a digression. Do gauge theories (on which I am not an expert) have any place in the classical physics section?

Further I don't see the connection between mass and conductivity in either view. Conductors are conductors because of their electron arrangement, not because of their proton arrangement. Neutron matter offers an enormous density but does it have high conductivity?

13. Oct 26, 2012

### PhilDSP

The movement of charges in the wire induces a displacement current outside of the wire (and to some extent inside the wire). The displacement current propagates as an evanescent wave according to the Maxwell equations.

14. Oct 26, 2012

### Studiot

Which is the point we have been trying to get over to aaaa2002.

15. Oct 26, 2012

### ZapperZ

Staff Emeritus
I don't understand this at all. How does the "large amount of mass concentrated in a small volume" have anything to do with non-wave behavior?

I can look at the current in an AC circuit, and I definitely see electron current being described as a wave.

And if you are arguing about the actual physical behavior, then there's the circuit equivalent of the 2-slit experiment, such as in SQUIDs. Those are certainly wave-like description to me.

Zz.

16. Oct 26, 2012

### PhilDSP

Yes, I was thinking about what you said as I wrote it. The mass only changes the velocity that the free electron moves at (slower than c of course). The free electron should still move in the same manner as a wave more or less. But the Lorentz force law probably best describes its potential movement.

17. Oct 26, 2012

### ZapperZ

Staff Emeritus
That still doesn't explain anything.

A buckyball is many orders of magnitude more massive than an electron. No one can say now that a buckyball doesn't exhibit wave-like behavior after we've show that it can produce 2-slit interference pattern!

But this is neither here nor there. The very fact that we have experimental observation of wavelike behavior of conduction electrons should be enough to falsify what you said. So if you disagree with this, you need to address directly these experimental facts, not some other conjectures.

Zz.

18. Oct 26, 2012

### Studiot

This does not address my objection to the same quote.

I do not know Jackson, but Griffiths is oft quoted here.

Section 9.4 of Griffiths is entitled Electromagnetic Waves in Conductors and follows the conventional path I described.

P396 has a particularly good sketch of the rapid attenuation of an EM wave attempting to propagate in a conductor.
Beneath is a good question
"Find the skin depth in a good conductor in nanometers.......

Chapter 10 of Griffiths relates to your earlier intervention, although I cannot find any dependence on mass in any of the equations presented, there is certainly none in Maxwell.

However I think Plonus has a more comprehensive treatment of the subject (EM waves) in his chapter 13. It includes many practical examples, facts and figures.

19. Oct 26, 2012

### PhilDSP

Yes, I agree. The word wave-like fits very well. What I really meant was that momentum needs to be factored into the equations of motion for the electron in addition to the wave equations.

20. Oct 26, 2012

### ZapperZ

Staff Emeritus
That is a very strange statement. I can have momentum in a classical wave! The classical treatment of EM wave certainly has momentum in it, and this is without having to resort to having any mass either!

Zz.

21. Oct 26, 2012

### PhilDSP

Studiot, It wasn't clear what your objection was. If it is this:

In a medium, the conductivity tensor is often determined by the amount of (spatial density of) each species of free charged particle. The species of particle gives both mass and charge: positive or negative as well as number of charges if the particle is composite (an atom, ion or molecule). That's often simplified by ignoring the contribution of heavier particles since electrons move much farther and faster in response to changing fields. The movement of any charged particle induces additional field fluctuations of course.

Neutronic matter in free particles will affect the conductivity because it increases the mass of ions, atoms and molecules. But only meagerly. I mean it affects conductivity meagerly.

Last edited: Oct 26, 2012
22. Oct 26, 2012

### PhilDSP

Yes, of course! The beauty of the wave equations associated with the Maxwell equations for fields is that they are transparent to momentum of the fields. If you are aware of any literature studying why that can be so, I'd be very interested in considering it.

23. Oct 26, 2012

### ZapperZ

Staff Emeritus
What does it mean to be "transparent to momentum of the fields"? I wish you'd state your case more clearly here, because you seem to be backpeddling with each post, but at the same time, still hanging on your original statement.

If the wave description has the ability to include momentum, then what you said is no longer true. In fact, I find it difficult to find what part of your argument remains true.

Zz.

24. Oct 26, 2012

### Studiot

But you have suggested that the conductivity is is some way related to the mass.

If the mass of say an electron were suddenly quintupled or that of the proton divided by 1000 what difference would that make to your calculations?

I also asked another two questions (post# 10) about your equations, you have yet to comment on.

Surely if you are prepared to post theory you must be prepared to offer the calculated consequences as I have done?

25. Oct 26, 2012

### PhilDSP

To ZapperZ: I mean that no explicit factor for momentum needs to introduced into the wave equation in order to account for its effects (for fields)

But the exact same equation doesn't work for massive particles. The particle velocity factor will no longer be c. As far as I have seen, rigorous equations of motion for charged particles such as a free electron require the use of either the constitutive relations or the Lorentz force law or both.