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Homework Help: Please explain this step

  1. Apr 7, 2008 #1
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Apr 7, 2008 #2

    HallsofIvy

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    Science Advisor

    It's a little hard to distinguish between [itex]\nabla[/itex] and V!!
    What you are asking is how they went from
    [tex]\int\int\int (\nabla\cdot E)V d\tau[/tex]
    to
    [tex]\int\int\int E\cdot\nabla V d\tau+ \int\int V (E\cdot dA)[/tex]
    (some people prefer [itex]E\cdot n dA[/itex] rather than [itex]E\cdot dA[/itex] where "n" is the unit normal to the surface.) V here is the scalar potential and E is a vector function.

    There are actually two steps in there. First they are using the "product" rule:
    [tex]\nabla\cdot(VE)= V\nabla\cdot E+ (\nabla V)\cdot E[/itex]
    where [itex]\nabla\cdot E[/itex] and [itex]\nabla\cdot (VE)[/itex] are the divergence (div) of the vectors and [itex]\nabla V[/itex] is grad V.

    so
    [tex]\int\int\int\nabla\cdot(VE)d\tau= \int\int\int V\nabla\cdot Ed\tau +\int\int\int(\nabla V)\cdot E d\tau[/itex]

    Now use the divergence theorem to convert that first integral on the right to
    [tex]\int\int VE\cdot dA[/itex]

    But I'm not at all clear why the [itex]\epsilon_0/2[/itex] only multiplies the first integral!
     
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