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Electric Dipole - Calulating the electic field from Potential

  1. Feb 18, 2013 #1
    1. The problem statement, all variables and given/known data
    Calculate the Electric field of a Dipole from its Potential.

    [tex] \vec E=-\operatorname{grad}(\Phi_D)[/tex]

    2. Relevant equations
    [tex]\Phi_D(\vec R)=\frac{Q}{4\pi\epsilon_0} \cdot \frac{\vec d \vec R}{R^3}[/tex]



    3. The attempt at a solution
    Hi all!

    I am trying to calculate the electric Field of a Dipole from its Potential.

    Specifically, I am trying to understand the following equation:


    [tex]\operatorname{grad}(\Phi_D)=\frac{Q}{4\pi\epsilon_0R^3}\cdot((\vec d\cdot \vec R )\cdot \operatorname{grad}(\frac{1}{R^3})+\frac{1}{R^3} \cdot\operatorname{grad}(\vec d \cdot \vec R))[/tex]

    I tried re-writing the equation according to the Einstein Notation:
    [tex]\operatorname{grad}(\Phi_D)=\frac{Q}{4\pi\epsilon_0}\cdot\frac{\partial}{\partial R_j}\frac{d_k R_l}{(R_i R_i)^{\frac{3}{2}}}[/tex]

    Is that correct? If so, how will I go about proving the above identity?

    Thank you all for your help!
     
  2. jcsd
  3. Feb 18, 2013 #2
    It is fairly straightforward to show, using the definition of gradient and partial derivatives, that gradient satisfies the Leibniz rule, [tex] \operatorname{grad}(fg) = f \operatorname{grad}(g) + g \operatorname{grad}(f) [/tex]
    where f and g are scalars
     
  4. Feb 18, 2013 #3
    But they are not scalar, are they?

    [tex] \vec d \text{ and } \vec R [/tex] are vectors...

    The rule for vectors is, in my opinion, [tex]

    \operatorname{grad}(\vec a \cdot \vec b)=(\vec a \cdot \nabla) \cdot \vec b+(\vec b \cdot \nabla) \cdot \vec a+\vec a \times (\nabla \times\vec b)+\vec b \times (\nabla \times\vec a)[/tex]
     
    Last edited: Feb 18, 2013
  5. Feb 18, 2013 #4
     
  6. Feb 18, 2013 #5
    Oh ok.

    I actually tried that already, but the result was 0 for some reason...
    But it's good to know I had the right idea and obviously just got lost in the math on the way.
    I guess I'll simply try it again :)

    Thanks for your help!
     
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