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Length contraction on charged wire

  1. Mar 10, 2013 #1
    Lets say we have an infinite charged wire with a line charge [itex] \lambda [/itex]
    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. jcsd
  3. Mar 10, 2013 #2


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    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.
  4. Mar 10, 2013 #3


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    Staff Emeritus
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    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.
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