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Divergence in spherical coordinates

  1. Sep 12, 2013 #1
    Problem:

    For the vector function [tex]\vec{F}(\vec{r})=\frac{r\hat{r}}{(r^2+{\epsilon}^2)^{3/2}}[/tex]

    a. Calculate the divergence of ##\vec{F}(\vec{r})##, and sketch a plot of the divergence as a function ##r##, for ##\epsilon##<<1, ##\epsilon##≈1 , and ##\epsilon##>>1.

    b. Calculate the flux of ##\vec{F}## outward through a sphere of radius R centered at the origin, and verify that it is equal to the integral of the divergence inside the sphere.

    c. Show that the flux is ##4 \pi## (independent of R) in the limit ##\epsilon \rightarrow 0##.

    Solution:

    a. Calculating the divergence is simply a matter of plugging the function correctly into the spherical coordinates divergence formula. I get [tex]F_r=\frac{r}{(r^2+{\epsilon}^2)^{3/2}} \Rightarrow \frac{\partial F}{\partial r}=3r^2(r^2+{\epsilon}^2)^{-3/2}-3r^4(r^2+{\epsilon}^2)^{-5/2} [/tex]

    Therefore,

    [tex]\nabla \cdot \vec{F}= 3(r^2+{\epsilon}^2)^{-3/2}-3r^2(r^2+{\epsilon}^2)^{-5/2} [/tex]

    So for the different values of ##\epsilon## do I just substitute?
     
  2. jcsd
  3. Sep 12, 2013 #2
    Also can someone point me in the right direction for parts b. and c.? Thanks.
     
  4. Sep 12, 2013 #3
    Anyone?
     
  5. Sep 13, 2013 #4

    ehild

    User Avatar
    Homework Helper
    Gold Member

    Check your ∂F/∂r.

    Anyway, the end result is correct.

    As for b), what is flux?

    ehild
     
    Last edited: Sep 13, 2013
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