Energy of an electric quadrupole in an Electric Field

  1. Hello, I have tried to look for related threads but could not find any. Please excuse me if this is a repetition. I was curious about the energy of an electric quadrupole moment in an electro-magnetic field.

    Basically, i am trying to follow first order perturbation theory and derive the matrix element of an atom interacting with a field in Bohm's Quantum Theory book - upon expansion, we get terms that "look" like an electric dipole, magnetic dipole and so on.

    For example, the energy of an electric dipole in an electric field is E dotted with d (dipole moment). Likewise, that of a magnetic dipole would mu (magnetic moment) dotted with B. I am looking for a similar expression for the quadrupole.

    Any help, as well as references are greatly appreciated. Thanks again!
  2. jcsd
  3. hmm....thanks for the link friend, but it doesnt give me enough detail unfortunately :(
  4. clem

    clem 1,276
    Science Advisor

    [tex]U_Q = -\frac{1}{3}\bf[Q]:[{\nabla} E][/tex].
  5. thanks a lot clem, that really helps you have reference by any chance?
  6. clem

    clem 1,276
    Science Advisor

    Section 2.4 of Franklin's "Classical Electromagnetism" discusses electric quadrupoles.
  7. What does the operation : signify? I assume it yields a scalar here, since the expression is for energy, but how is it defined? Thank you
  8. What clem said about Franklin's "Classical Electromagnetism" are wise words in this context :P

  9. Meir Achuz

    Meir Achuz 2,076
    Science Advisor
    Homework Helper
    Gold Member

    That is a misprint.
    The correct equation is
    [tex]U_Q = -\frac{1}{3}\bf[Q]:[{\nabla}\nabla E][/tex]
    [tex]=-\frac{1}{3}[(\bf[Q]\cdot\nabla)\cdot\nabla] E[/tex].
    Last edited: Dec 12, 2012
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