Flux: Surface integral of a sphere.

  1. 1. The problem statement, all variables and given/known data
    Find the surface integral of [tex]\vec{r}[/tex] over a surface of a sphere of radius a and center at the origin. Also find the volume integral of [tex]\nabla[/tex] [tex]\bullet[/tex] [tex]\vec{r}[/tex].

    2. Relevant equations
    Divergence theorem.

    3. The attempt at a solution
    First I did the volume integral part of the divergence theorem. I obtained [tex]\nabla[/tex] [tex]\bullet[/tex] [tex]\vec{r}[/tex] = 1 + 1 + 1 = 3. So I figured, the answer must be 3*volume = 4[tex]\pi[/tex]r[tex]^{3}[/tex] (I don't know why the pi looks like an exponent, but it's 4 pi r^3)

    This answer seems like a correct one.

    Now the surface integral I'm having trouble with. Knowing that the equation of the sphere is
    x[tex]^{2}[/tex]+y[tex]^{2}[/tex]+z[tex]^{2}[/tex]=a[tex]^{2}[/tex], I found [tex]\nabla[/tex] [tex]\bullet[/tex] (x[tex]^{2}[/tex]+y[tex]^{2}[/tex]+z[tex]^{2}[/tex]) to obtain the normal. The [tex]\vec{r}[/tex] [tex]\bullet[/tex] [tex]\vec{n}[/tex] = 2x[tex]^{2}[/tex] + 2y[tex]^{2}[/tex] + 2z[tex]^{2}[/tex].

    So I would integrate this over the surface in Cartesian coordinates, or convert to spherical and integrate? Is the normal suppose to be the normal unit vector? I appear to be obtaining the wrong answer no matter which way I am doing this. What exactly would the integral in cartesian coordinates contain for boundaries?

    Thanks. Sorry if the latex syntax is not perfect.
  2. jcsd
  3. LCKurtz

    LCKurtz 8,454
    Homework Helper
    Gold Member

    For the first integral your answer should be 4 pi a^3, not 4 pi r^3.

    You probably aren't asked to find "the surface integral of [tex]\vec{r}[/tex] over a surface of a sphere". I'm guessing you are asked to find the flux integral for the vector field [tex]\vec{r}[/tex]. In other words you are to calculate

    [tex]\int\int_S \vec r \cdot \hat n\, dS[/tex]

    where [tex]\hat n[/tex] is the unit outward normal. In the case of your sphere your unit outward normal is:

    [tex] \hat n = \frac {\vec r}{|\vec r |}[/tex]

    Now the natural way to do such an integral would be spherical coordinates. But when you evaluate [tex] \vec r \cdot \hat n[/tex] on the surface of the sphere you should see a shortcut.
  4. Thanks!
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