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Griffith's 3.33

  1. Oct 1, 2015 #1
    I am "continuing this thread" in hopes of asking questions that deal with the meaning of the question. https://www.physicsforums.com/threa...dipole-moment-in-coordinate-free-form.359973/
    1. The problem statement, all variables and given/known data

    Show that the electric field of a "pure" dipole can be written in the coordinate-free form
    $$E_{dip}(r)=\frac{1}{4\pi\epsilon_0}\frac{1}{r^3}[3(\vec p\cdot \hat r)\hat r-\vec p].$$
    2. Relevant equations
    $$E_{dip}(r)=\frac{p}{4\pi\epsilon_0r^3}(2\cos \hat r+\sin\theta \hat \theta)$$

    3. The attempt at a solution
    I am trying to understand what "coordinate free" means. If the answer is in terms of r hat and theta hat, doesnt that contradict "coordinate free"? AND i would get $$ p = pcos(\theta) \hat r - psin(\theta) \hat \theta $$. Why doesn't p depend on PHI? If it's coordinate free why are we restricting our coordinates to r and theta??
     
    Last edited: Oct 1, 2015
  2. jcsd
  3. Oct 2, 2015 #2

    blue_leaf77

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    "Coordinate free" means you don't need to define the coordinate system to write your equation. ##\hat{r}## is a unit vector from the center of the dipole to the observation point, so given the orientation of ##\mathbf{p}## in space, the relative direction of ##\hat{r}## with respect to ##\mathbf{p}## will automatically follow.
     
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