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Permanent Dipole - Permanent Dipole Interaction Derivation

  1. Jul 4, 2017 #1
    1. The problem statement, all variables and given/known data
    I am trying to derive the dipole-dipole interaction derivation, which is:

    U=(-p1p2/4πϵ_0) (1/z^3) ((2cosθ_1cosθ_2)− (sinθ_1sinθ_2cosζ))

    Where p1 and p21 are the two dipole moments, r is the distance between two dipoles on the y axis, θ_1 and θ_2 are the angles between the z axis and dipoles, and ζ is the dihedral angle

    I am using U = -p2 * E_1 to derive the equation, where E_1 is the electric field of p1 and * is the dot product

    So far, I have deduced E-1 in spherical coordinates as:
    E_1= (1/4πϵ_0)(1/r^3) (2cosθ r + sinθ θ)
    Where the bolded r and θ are the unit vectors in spherical coordinates.

    Now from this, I have problems:
    1. How do I formulate p2 in spherical coordinates
    2. How do I dot product two vectors of spherical coordinates

    2. Relevant equations
    See above

    3. The attempt at a solution
    I have tried to convert E_1 into Cartesian, but it won't solve it if I don't know how to formulate p2
  2. jcsd
  3. Jul 4, 2017 #2


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    Did you mean to say that θ1 and θ2 are angles measured relative to the y axis?

    You can rotate your xyz axes about the y axis so that p1 lies in the yz plane.

    Can you draw the vectors r and θ at the location of p2 in the above diagram?

    Then try to find expressions for the r and θ components of p2.

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