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Irrotational and divergenceless?

  1. Feb 3, 2004 #1
    I'm not really sure what I'm being asked to do with the following questions. Will someone help me?

    How can you show that E(r)=-gradV(r) has zero curl an is irrotational, i.e. the quantity (grad x E) =0


    How can you show that the magnetic field ,B(r)=grad x A(r), is divergenceless i.e. that the quantity grad dot B(r) =0?
     
  2. jcsd
  3. Feb 3, 2004 #2

    HallsofIvy

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    Staff Emeritus
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    Basically, you just go ahead and do it!

    That is, start by assuming that you have V(x,y,z) so that E(r)=-gradV(r)= -Vxi- Vy-Vz and calculate curl(E)= curl(grad V). See what happens!
     
  4. Feb 4, 2004 #3
    if B=curl A, then div B = div (curl A), the divergence of a curl is always zero. Actually this proof is usually done in the opposite order since the div B=0 is one of Maxwell's equations, valid even for time dependent phenomenon (the old there are no magnetic monopoles theorem). If div B=0 then B can be expressed as the curl of another vector, which gets labeled A and we call it the magentic vector potential.
    Hope this helps.
    Cheers,
    Norm
     
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