Ampere's Law: Questions from E.M. Purcell's Textbook

[exmarine]

I posted questions about this subject some time back. Didn't get them answered to my satisfaction, but did learn of an EXCELLENT textbook - E.M.Purcell's Electricity and Magnetism - thanks to you all that responded! Now I have two questions about some of his material.

(1) I waded through his derivations of Ampere's law for PARALLEL currents, and thought I understood them. Once current is flowing in the wire, it sheds electrons until it is neutral again, IN THE LAB FRAME. But then, IN THE ELECTRONS' FRAME, the protons appear Lorentz contracted, so the wire appears positive to parallel moving charges outside the wire, etc.

Sounds good, until you get to a later chapter, where he derives the E-field around a MOVING charge: E'(perpendicular to v) = gamma * E(perp), and E'(parallel to v) = E(para). Obviously the electrons are moving with their drift velocity in the lab frame, so wouldn't the electrons' perpendicular E-field exceed the perpendicular E-field of the stationary protons, and the wire appear to be positive rather than neutral, even in the lab frame?

(2) Then in his derivation of Ampere's law for PERPENDICULAR currents (see page 198), Purcell invokes Gauss' theorem to show that the E-field perpendicular to the wire must remain zero - because there are equal numbers of positive and negative charges per unit length of wire. What?! Isn't Gauss' theorem limited to static charges? Obviously, a wire carrying a current cannot have all the positive and negative charges stationary in ANY reference frame. So is Purcell being careless, or am I missing another subtle point?

Thanks,
BB

 

[Doc Al]

Isn't Gauss' theorem limited to static charges?

No it's not.

 

[Naty1]

I'm not sure I understand the conditions you are setting for your inquiry, but the typical drift velocity of electons in a wire are on the order of only a few meters per second...for all practical purposes, if considering relativity effects, nearly at rest in the lab frame.

You can increase the current flow, make the electrons move faster, by using extremely high voltages, and maybe would be able to measure some relativistic effects...

 

[Doc Al]

I'm not sure I understand the conditions you are setting for your inquiry, but the typical drift velocity of electons in a wire are on the order of only a few meters per second...for all practical purposes, if considering relativity effects, nearly at rest in the lab frame.

You can increase the current flow, make the electrons move faster, by using extremely high voltages, and maybe would be able to measure some relativistic effects...

Even ordinary drift speeds are enough to invoke relativistic effects.

 

[atyy]

Sounds good, until you get to a later chapter, where he derives the E-field around a MOVING charge: E'(perpendicular to v) = gamma * E(perp), and E'(parallel to v) = E(para). Obviously the electrons are moving with their drift velocity in the lab frame, so wouldn't the electrons' perpendicular E-field exceed the perpendicular E-field of the stationary protons, and the wire appear to be positive rather than neutral, even in the lab frame?

Purcell's argument is correct to some order of approximation only. I don't actually know if the effect you mention is smaller than the Lorentz contraction effect. Another book which might be useful is Ohanian's "Classical Electrodynamics".

 

[exmarine]

#1 I meant wouldn't the wire appear NEGATIVE of course.
#2 How can we prove that? I am surprised that Purcell didn't even assert it, at least that I can find. The integral looks pretty tricky. I'll try it, at least numerically. Ain't computers wonderful! Just imagine what Newton, Maxwell, and Einstein could have done with a PC!
#3 & #4. Somebody always says that. But the big "simplification" by Einstein's SRT is supposed to be that one can DERIVE Maxwell from SRT. I always have to prove things for myself, so that's what I am trying to do.
#5 Approximate? Purcell's solution matches Ampere's, so I doubt if it is approximate...