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Electrostatics: work moving point dipole

  1. Oct 7, 2009 #1
    1. The problem statement, all variables and given/known data

    Two electrical dipoles with dipole moments [tex]\vec{p}_0=p_0\hat{y}[/tex] and [tex]\vec{p}_1=p_1\hat{y}[/tex] are located in the xy-plane. [tex]\vec{p}_0[/tex] i located at the origin and [tex]\vec{p}_1[/tex] is initially in (x,y)=(a,0). What work is required to move [tex]\vec{p}_1[/tex] (with unhanged directon) to the point (x,y)=(0,a)?

    2. Relevant equations

    Dipole potential: [tex]\phi_D(\vec{r})=\frac{\vec{p}\cdot\hat{r}}{4\pi\epsilon_0r^2}[/tex]
    Work: [tex]W=Q\phi[/tex]

    3. The attempt at a solution

    Taking the difference of the potential in the two cases:

    [tex]\left(\phi_{D, p_o}+\frac{p_1\hat{y}\cdot a\hat{x}}{4\pi\epsilon_0a^2}\right) - \left(\phi_{D,p_0}+\frac{p_1\hat{y}\cdot a\hat{y}}{4\pi\epsilon_0a^2}\right)=-\frac{p_1}{4\pi\epsilon_0a}[/tex]

    Here comes the step that I'm not sure about. The work is the potential times the point charge - how is it when we have a point dipole? Attempt:

    [tex]W=Q\phi = -\frac{p_0}{a}\cdot\frac{p_1}{4\pi\epsilon_0a}=-\frac{p_0p_1}{4\pi\epsilon_0a^2}[/tex]

    Does this make sense? Any help appreciated!

    Thanks!
     
  2. jcsd
  3. Oct 7, 2009 #2

    kuruman

    User Avatar
    Homework Helper
    Gold Member

    Start with the expression for the potential energy of one dipole in the external electric field of the other dipole

    [tex]U=-\vec{p}_{1}\cdot\vec{E}_{2} [/tex]

    where

    [tex]E_{2}=\frac{1}{4 \pi\epsilon_{0}}\large(\frac{(3\vec{p}_{2} \cdot \vec{r})\vec{r}}{r^{5}}-\frac{\vec{p}_2}{r^3} \large) [/tex]

    ** Addition on edit **
    Here, vector r is the position of dipole 1 relative to dipole 2, i.e. dipole 2 is at the origin.
     
    Last edited: Oct 7, 2009
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