# Length contraction on charged wire

1. Mar 10, 2013

### 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?

2. Mar 10, 2013

### Staff: Mentor

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.

3. Mar 10, 2013

### bcrowell

Staff Emeritus
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.