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Homework Help: Potential energy of a dipole point and point charge

  1. Oct 18, 2012 #1
    1. The problem statement, all variables and given/known data
    A point dipole p is at the origin of coordinates, while a point charge q is at position r. Calculate the potential energy U of this system in two ways (a) by treating the dipole as a source and q as a test charge in the dipole field (b) by treating the dipole as a test-object in the Coulomb electric field of the point-charge, now treated as a source


    2. Relevant equations
    U=-pE
    V(r)=Integral of Edr

    3. The attempt at a solution
    I found the potential

    V(r)=p.r/(4πε0r^3)=p.r(hat)/(4πε0r^2)=pcosθ/(4πε0r^2)

    Then I differentiated since E(r)=d/dr(V(r))

    E(r)=-d/dr(V(r))=-d/dr(pcosθ/(4πε0r^2))=-(-3pcosθ/(4πε0r^3))=3pcosθ/(4πε0r^3)

    So my U would be equal

    U=-pE=-p(3pcosθ/(4πε0r^3))=-3p^2cosθ/(4πε0r^3) ????

    It doesn't make sense
     
  2. jcsd
  3. Oct 18, 2012 #2
    First, [itex]\vec{E}= -\nabla \Phi \ne -\frac{\partial \Phi}{\partial r}[/itex].
    Second, the energy of a charge q in potential V is [itex]U=qV[/itex].
    Third, the energy of a dipole p in field E is [itex]U=-\vec{p}\cdot\vec{E}[/itex].
    For part a, you're supposed to use a charge q in a dipole field.
    For part b, you're supposed to use a dipole [itex]\vec{p}[/itex] in point charge field.
     
  4. Oct 25, 2012 #3
    Thank you!
     
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