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Electric Field of a Pure Dipole(Math Stuff)

  1. Mar 3, 2005 #1
    Working on Griffiths 3.33. I'm supposed to show that the Electric Field of a pure dipole can be written in the following coordinate free form:
    [tex]
    \vec{E}(\vec{r}) = \frac{1}{4 \pi \epsilon_0 r^3} [3(\vec{p} \cdot \hat{r})\hat{r} - \vec{p}]
    [/tex]
    Where p is the dipole. I know that the potential is equal to
    [tex]
    V(r,\theta) = \frac{\hat{r} \cdot \vec{p}}{4 \pi \epsilon_0 r^2}
    [/tex]
    and I tried to take the negative gradient of that, but got lost in the math. If you assume the dipole points along the z-axis you get the solution
    [itex]
    \vec{E}(r,\theta) = \frac{p(2cos \theta \hat{r} + sin \theta \hat{\theta})}{4 \pi \epsilon_0 r^3}
    [/itex]
    And I tried to work with that a little to no avail. Any ideas how to solve this beast?
     
  2. jcsd
  3. Mar 3, 2005 #2

    dextercioby

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    Yes,there's one advice for each part:don't get lost in maths.And use spherical components for the 2 vectors involved in the scalar product of the second...

    Daniel.
     
  4. Mar 3, 2005 #3
    I'm still rather confused....should I calculate the dot product by components or say it equals [tex] pcos \acute{\theta} [/tex] and then try to find some weird relation between theta prime and theta? Or perhaps I'm missing something? Cuz either way I keep getting lost. :yuck:
     
  5. Mar 4, 2005 #4

    dextercioby

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    What primes are u talking about...?There is no prime in your equations.

    As for incapacity of differentiation,well,that's simply bad.

    Daniel.
     
  6. Mar 4, 2005 #5
    Nevermind I got it...I was trying to use spherical coordinates all the way through instead of the spherical components of the rectangular coordinates
     
  7. Mar 5, 2005 #6
    In such questions, it is almost always good to define your own coordinate systems. Here, the vecotors p, p (cross) r and (p cross (p cross r)) can serve as the ideal coordinate axes since they are mutually perpendicular and of course, you know the component of E along p and along p cross r.
     
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