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Divergence and curl of vector field defined by \vec A = f(r)vec r

  1. Oct 11, 2008 #1
    1. The problem statement, all variables and given/known data
    A vector field is defined by A=f(r)r
    a) show that f(r) = constant/r^3 if [tex]\nabla[/tex]. A = 0
    b) show that [tex]\nabla[/tex]. A is always equal to zero


    2. Relevant equations
    divergence and curl relations


    3. The attempt at a solution
    I tried using spherical co-ordinates to solve this. But I am not sure if i am totally right.
     
    Last edited: Oct 11, 2008
  2. jcsd
  3. Oct 11, 2008 #2

    G01

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    I would also use spherical coordinates on this problem.

    I cannot say if you are right since I haven't seen any of your work. I'll need to see some work if you want more specific advice.
     
  4. Oct 11, 2008 #3

    olgranpappy

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    Certainly parts (a) and (b) can not be consistent as you have written them... i suppose part (b) should be a curl not a divergence?
     
  5. Oct 11, 2008 #4
    here is what i've done and where i got stuck:

    [tex]\nabla[/tex]A = [tex]\frac{1}{r^{2}}[/tex] [tex]\frac{\partial}{\partial r} (r^{2}f(r))[/tex]

    on simplifying this i got:

    f(r) = [tex]\frac{-1}{2} r f'(r)[/tex]

    then i integrated with a hope to get an expression for f(r) but did not end up with what i needed i.e. f(r) = constant/ r^{3}
     
  6. Oct 12, 2008 #5

    nicksauce

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    Well the way you interpreted it, you have [tex]\mathbf{A}=f(r)\hat{r}[/tex], but the way the question is stated it is [tex]\mathbf{A}=f(r)\mathbf{r}=[/tex], or [tex]\mathbf{A}=f(r)r\hat{r}[/tex]. This extra factor of r should give the answer required.
     
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