# Energy of an electric quadrupole in an Electric Field

by karanmohan
 P: 9 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!
 P: 9 hmm....thanks for the link friend, but it doesnt give me enough detail unfortunately :(
P: 1,233

## Energy of an electric quadrupole in an Electric Field

$$U_Q = -\frac{1}{3}\bf[Q]:[{\nabla} E]$$.
 P: 9 thanks a lot clem, that really helps alot...do you have reference by any chance?
 Sci Advisor P: 1,233 Section 2.4 of Franklin's "Classical Electromagnetism" discusses electric quadrupoles.
 P: 14 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
P: 5
 Quote by JJfortherear 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
What clem said about Franklin's "Classical Electromagnetism" are wise words in this context :P

 Quote by clem Section 2.4 of Franklin's "Classical Electromagnetism" discusses electric quadrupoles.
HW Helper
 Quote by clem $$U_Q = -\frac{1}{3}\bf[Q]:[{\nabla} E]$$.
$$U_Q = -\frac{1}{3}\bf[Q]:[{\nabla}\nabla E]$$
$$=-\frac{1}{3}[(\bf[Q]\cdot\nabla)\cdot\nabla] E$$.