# Energy of an electric quadrupole in an Electric Field

karanmohan
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!

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

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

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

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

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

SiggyYo
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.

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