Conductor Charge/Electric Fields

[Cyrus]

Hi, I have two questions.

The first is on the charge at the surface of a conductor. Let's say we have a charge distribution at the surface of the conductor, but there is a spot on the conductor that is not at the same potential. The result will be that there is a motion of this charge disribution towards the lowest potential energy possible. But once it gets to that point, it won't stop, because its gained kinetic energy in the process. Therefore, it will over shoot, until it looses all its kinetic energy, and repeats the process over again. Obviously, this does not happen, because eventually all the charge does come to rest. What is the reason behind this? Is the loss due to the charge at the surface colliding with atoms as they move through simple harmonic motion, heating up the condutor until they have given up all of their energy and reached equilibrium at the lowest possible potential energy?


My second question is about the electric field inside a wire. If a wire is hooked up to a battery, then there is a voltage, or a potential difference at the two ends of the wire. I know that the potential difference is a result of an electric field being present. But how come the electric field is always inside the wire, and always points in the same direction everywhere inside the wire.

I don't see why the electric field in a wire would always be perpendicular to the cross sectional area of the wire.

 

[Tide]

A couple of points:

In a conductor, the number of charge carriers (electrons) is enormous as are the fixed charges (ions). When an electric field (potential difference) is applied, electrons don't just go streaming down the pipe and overshoot. The electric field is balanced by the current flow and is felt very quickly at the other end. Electrons making up the current collide with the ions which impedes their motion.

Under a fixed potential difference the electrons will drift along at a speed given by

$$v_d = \frac {\sigma \Delta V}{ne}$$

where $\sigma$ is the electrical conductivity, n is the number density of conduction electrons and $\Delta V$ is the potential difference. (I'm using cgs units so you may have to adjust for your favorite system)

You will find that for typical conductors (copper, in particular) that the drift speed is small and that it would take a considerable time (minutes to hours IIRC) for an electron at the switch to make it up the circuit to your lightbulb.

As to why there are no radial fields, you will only get those when the currents are sufficiently high and you can get a pinching effect from the intense and collapsing magnetic fields. Otherwise, the conditions we spoke about before prevail for the radial part.

 

[Cyrus]

Im not understanding your anweser tide. IM asking about a sphere that is charged, not a light wire conected to a light blub. I am saying if this sphere has charge on it, and there is a point with potential difference, then the charges will move. But once the get to the lower potential, they will have kinetic energy and over shoot. But having a static charge at the surface implies that the surface is at equipotential. So in the end they all have to be at the same potential levels. But once they move and get to that potential spot, they over shoot and return to the equipotential and so on and so on exhibiting simple harmonic motion. So something has to give friction so that over time the harmonic motion dampens out and becomes stable at the same potential everywhere.

 

[Tide]

I was addressing your question about electric fields in a wire.

 

[reilly]

As Tide points out, moving charges (electrons mostly) collide with the nucleii of the metal, and also collide with each other. So we are talking about electrical resistance, which will damp any current flow on the surface of an isolated, charged sphere.

The surfaces of an ideal conductor are equipotentials, thus there are no radial electric fields generated by an ideal conducting wire. In practice, currents will damp with distance from the surface, so most of the current is very close to the surface.

Actually this is a well studied area of E&M, and, consequently, is discussed in many textbooks.

Regards,
Reilly Atkinson