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Vector Potential

  1. Nov 7, 2009 #1
    Hello!
    Can you tell me the sufficient condition for the existence of the vector potential?
    Thank you very much!
     
  2. jcsd
  3. Nov 7, 2009 #2
    when a field is divergence free it has a vector potential.
     
  4. Nov 7, 2009 #3
    That is a necessary condition, but I do not think is enough!
    Are you sure?
    Thank you.
     
  5. Nov 7, 2009 #4
    It is necessary and sufficient in Euclidean space.

    For instance, in magnetostatics a magnetic field always has a vector potential.

    If the field is B = (u,v,w) consider the 2 form udy^dz -vdx^dz + wdx^dy

    Since B has zero divergence, the 2 form is closed. By Poincare's Lemma it is therefore the exterior derivative of a 1 form adx + bdy + cdx. The curl of (a,b,c) equals B.
     
  6. Nov 7, 2009 #5
    Thank you very much!
     
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