# Q & A: waves of electricity

Q:
Hello! I have a question about distribution of charges on a conductor. I have come to the conclusion, that the only way it can possibly happen so fast (e.g on the entire 100 m wire), is that it must be a wave propagating. Charging one end with plusses (taking some of the electrons away), creates a wave of neighboring electron to accelerate towards the "positive "spot"". What I am not quite grasping (even though I can prove this without any dificulty using for example Gauss' law) is who it can become an equipotential surface. I am clearly missing something, because so far I can only imagine the wave propagation back and forth. What is the part I am ignoring?
- Ott (age 25)
Rapla, Estonia
A:
I think that when you write that  you "can only imagine the wave propagation back and forth" the part that you're missing is that the electrons can dump some of their energy into other modes. That allows the wave to settle down into the equipotential form, like water waves settling down and leaving a level surface.

In an ordinary conductor, the energy is mainly dumped in the form of phonons (sound waves) in the material. These rapidly form a thermal distribution- the material heats up a little bit. A simple picture of an electron fluid interacting with a static set of atoms would lack the processes (called inelastic scattering) by which energy gets transferred to those sound waves. Perhaps that's the sort of picture you had in mind.

Mike W.

(published on 11/21/2011)

## Follow-Up #1: electronic conduction

Q:
Thank you for your answer! Howeven, what I mean is that I quite do not understand or imagine how that wave settles down into equipotential and charges almost instantly cover the entire surface. I mean, if the wire is hundreds of feet long, how does this happen so fast, what is the mechanism, how does it come by? That is an aspect i never have grasped entirely
- Ott
A:
The electrons accelerate in the electric field, but reach an average velocity after about a scattering time. The characteristic scattering times for electrons in a piece of metal are typically very short, say in the neighborhood of 10-14 sec. That means that a current density proportional to electric field gets established very quickly.

In the sort of situation you're asking about, that current then shifts charge so as to reduce the electric field, so the current density falls proportionately. You can actually convert the conductivity of the metal into a characteristic decay rate for the electric field. In fact, in the CGS system of units the conductivity is given directly in inverse seconds, so no conversion is needed in those units. The conversion factor from conventional units is about 1 mho/cm -> 1012/s. For a typical piece of good conductor, with conductivity of about 105 mho/cm, that would give relaxation times faster even than the short time it takes for the field to set up the current density, a result which isn't physical. In practice, that means that within about one electron scattering time, the little shift in positions of the electrons is enough to make the potential about uniform. Depending on geometry, magnetic fields produced by the current flow can slow down the process. Those magnetic induction effects become very important for applied ac fields.

In case I've still missed the core question, please follow up. Meanwhile, you might look for a copy of Purcell's book on Electricity and Magnetism, which does a great job of conveying a feel for how these things work.

Mike W.

(published on 11/21/2011)

## Follow-Up #2: field propagation

Q:
Thank you for your answer and literature suggestion. I am not exactly sure whetter I got the answer to the core of the question, maybe I should put it this way: Why does the charge spread out on a conductor so quickly and how exaclty does it happen? If I have two conducting spheres that are connected with a wire, and when I charge up one sphere say with plusses, then how does it happen, that the wire and the other sphere is almost instantly charged too. Why and how does it happen and why does it happen so fast (as I understand at the speed of light almost). Therefore: my question does not involve so much of a current, rather electrostatics.
- Ott
A:
Aha, now I see your point.  You see how once there's an electrical field charges in conductors will respond to it very quickly, flowing to rapidly reduce the electric field. The question is how the field got set up so quickly to begin with.

The propagation of electromagnetic fields is described by Maxwell's equations. The electric fields can come from charges or from changes in the magnetic field. The magnetic field can come from currents or from changes in the electric field. When Maxwell first looked at the equations he realized that they give solutions which propagate through space. The terms that determine the speed of the propagation are standard constants easily measurable in a lab. The resulting speed is the speed of light.

So by the time you've set up the charges on those spheres, all the nearby region will have long since picked up the fields coming from those charges. Then, as we argued above, the electrons will very rapidly move around under the influence of those fields.

Mike  W.

(published on 11/22/2011)