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Prove ∇ × J = 0 means B=0

  1. Mar 27, 2016 #1
    1. The problem statement, all variables and given/known data

    Prove that a current density J(r, t) such that ∇ × J = 0 implies the magnetic field B = 0.

    2. Relevant equations

    Maxwell's equations, vector calculus

    3. The attempt at a solution

    I've played around with Maxwell's equations and with the properties of vector calculus but I can't reach the necessary conclusion. Any hints would be greatly appreciated.
  2. jcsd
  3. Mar 27, 2016 #2


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    I think the problem may have been mis-stated because, as stated, it looks false.
    Consider a point outside a long, straight wire carrying a steady, direct current. ∇ × J=0 at that point because J=0 in an open neighbourhood of that point. But B is not zero. It is a stable, nonzero field that runs around the wire.
    The proposition fails inside the wire too: see these calcs.
  4. Mar 27, 2016 #3
    Thank you very much, I had the feeling something was wrong when the math just didn't agree with the statement.
    Do you know of any property that is similar to the one I was trying to prove? I mean, if the problem is mis-stated, any ideas as to what the correct statement is?
  5. Mar 27, 2016 #4


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    Given those conditions, if we also have ##\frac{\partial\mathbf{E}}{dt}=0##, which will for instance be correct if the current is steady, then we can deduce that ##\nabla^2\mathbf{B}=0##. Perhaps they meant that.
  6. Mar 27, 2016 #5
    Ok thanks, I really appreciate your help.
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