Length contraction on charged wire

[port31]

Lets say we have an infinite charged wire with a line charge $\lambda$
on it. Now when I move with respect to this wire the E field will increase do to length contraction. And there will also be a B field that we could calculate with ampere's law.
But the increased E would make it seem that there is more total charge.
Because the E field exists every where in space at a stronger strength.
Is this only because we have an infinite wire that this is happening?

 

[Nugatory]

Lets say we have an infinite charged wire with a line charge $\lambda$
on it. Now when I move with respect to this wire the E field will increase do to length contraction. And there will also be a B field that we could calculate with ampere's law.
But the increased E would make it seem that there is more total charge.
Because the E field exists every where in space at a stronger strength.
Is this only because we have an infinite wire that this is happening?

No, an infinite length is not required.

Also check out the description of this problem in the FAQ at http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html#Length_Contraction... You don't need Ampere's Law to calculate the B field at all; it turns out that the velocity-dependent contraction of the E field produces exactly the same forces as the classically computed B field.

 

[bcrowell]

But the increased E would make it seem that there is more total charge. Because the E field exists every where in space at a stronger strength. Is this only because we have an infinite wire that this is happening?

Here's an analysis with a loop rather than an infinite wire: https://www.physicsforums.com/showthread.php?t=631446 The finite total charge on the loop is the same in both frames, because charge is a relativistic scalar.

The seeming paradox in the case of the infinite wire doesn't seem to me to be specifically about relativity or E&M. I think it's really just a paradox about infinity of the same general flavor as Hilbert's hotel paradox: http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

The relativistic analysis of the infinite wire is a classic way of introducing magnetism. This pedagogy originated with Purcell. This WP article discusses it in some detail: http://en.wikipedia.org/wiki/Relativistic_electromagnetism There are other seeming paradoxes that can come up when you do this approach. See, e.g., the discussion question at the end of section 23.2 of this book: http://www.lightandmatter.com/lm/ . The resolution is that the paradox (not yours, but the one stated there) is stated in a way that incorrectly assumes simultaneity to be frame-independent.