# Energy of an electric quadrupole in an Electric Field

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!

hmm....thanks for the link friend, but it doesnt give me enough detail unfortunately :(

Meir Achuz
Homework Helper
Gold Member
$$U_Q = -\frac{1}{3}\bf[Q]:[{\nabla} E]$$.

thanks a lot clem, that really helps alot...do you have reference by any chance?

Meir Achuz
Homework Helper
Gold Member
Section 2.4 of Franklin's "Classical Electromagnetism" discusses electric quadrupoles.

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 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

Section 2.4 of Franklin's "Classical Electromagnetism" discusses electric quadrupoles.

Meir Achuz
$$U_Q = -\frac{1}{3}\bf[Q]:[{\nabla} E]$$.
$$U_Q = -\frac{1}{3}\bf[Q]:[{\nabla}\nabla E]$$
$$=-\frac{1}{3}[(\bf[Q]\cdot\nabla)\cdot\nabla] E$$.