Electrons in a Moving Wire: The Lorentz Force

[scariari]

Consider an infinite current carrying wire along the z-axis. Let I be the current in the wire and suppose that flowing electrons produce this current. The wire has no net charge density because the density of positive ions is assumed to compensate the density of flowing electrons.

Let the density of electrons per meter in the wire be p (phi). What is the typical velocity of the current electrons expressed in terms of p and I?

velocity density and current in one equation... J=pv should be ok, since it is for a 3D region and we are talking about the density INSDIE a wire.

Give the Lorentz Transformation that describes the transformation into the electrons rest frame

hmmm, help?

What is the density of electrons and ions in this frame? Is the wire also uncharged in the rest frame?

density is always the least in the frame where charge is at rest. if it were uncharged wouldn't it mean that there is no current? my only other explanation is that is is uncharged because the electrons and ions cancel, but they dont, or else they would cancel in the normal frame as well...

What is the total current in this frame?

there is a current, but it doesn't take into account the electrons because they are at rest. how to find the current? the current can't be calculated by part (a) because it doesn't make a distinction between electrons and ions. could it be that the current is the same?

Is there a frame conceivable in which the total current vanishes?

in the in the normal frame, electrons are moving and ions are at rest. at a rest frame, electrons are at rest and ions are moving. so is there a frame where the ions and electrons cancel?

 

[arcnets]

The point of this problem is to show that Lorentz force is a relativistic effect. You got to be familiar with special relativity to answer this. If you aren't, I think there's no point in trying to answer this.
Has your prof. made clear that you'd need SR to participate in this course?

 

[scariari]

my class is really strange to be honest. I agree i should know more about special relativity to do this, but that's what I am doing this problem for. I want to learn about it through doing this problem... so i am all ears to any suggestions/ hints/ keywords.

the lorentz transformation that describes going to the rest-frame is the reverse of the famous Lorentz transformation:

i') x=(gamma)(x' + vt')
ii') y=y'
iii') z=z'
iv') t=(gamma)(t' + (v/c^2)x')

 

[arcnets]

Originally posted by scariari
I want to learn about it through doing this problem...

OK, fine. Then please let me tell you what the basic idea, IMO, is.
The problem states your wire has zero net charge because the ions compensate for the electrons.
Relativistically, this is only true if the electrons are at rest.
But if the electrons move, they appear length-contracted. That is, the entire electron-gas appears length-contracted. And thus more dense. And thus, you get a net negative charge on the wire. Creating an electrical field.
If you work out the math, you find that the resulting electrical force is just the same as what we call 'Lorentz force'.