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Flux from divergence

  1. Apr 29, 2009 #1
    1. The problem statement, all variables and given/known data

    Suppose [tex] \vec{G} [/tex] is a vector field with the property that [tex] div\vec{G} = 5 [/tex] for [tex] 2 \leq ||\vec{r}|| \leq 14 [/tex] and that the flux of [tex] \vec{G} [/tex] through the sphere of radius 4 centered at the origin is [tex] 20\pi [/tex]. Find the flux of through the sphere of radius 12 centered at the origin.


    2. Relevant equations



    3. The attempt at a solution

    what I tried so far is

    [tex] 20\pi \int_0^{2\pi} \int_0^{\pi} \int_0^{12} \rho^2 sin(\phi)d\rho d\phi d\theta [/tex]

    is this wrong
     
  2. jcsd
  3. Apr 29, 2009 #2

    dx

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    No integration is needed for this problem. The flux through the region 2 < R < 4 is just

    5(volume of region 2 < R < 4)

    by the divergence theorem. Use this, together with the fact that the flux through 0 < R < 4 is 20π, to find the flux through the region 0 < R < 2.

    Can you see how to take it from here?
     
  4. Apr 29, 2009 #3
    ok so after I got the flux through region 2< r< 4and through 0 < r< 4 I just substract it right?
     
  5. Apr 29, 2009 #4

    dx

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    Yes, but in which order are you going to do the subtraction?
     
  6. Apr 29, 2009 #5
    it's the 0<r<4 - 2<r<4 correct?
     
  7. Apr 29, 2009 #6

    dx

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    Correct.
     
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