# Energy of an electric quadrupole in an Electric Field

## Main Question or Discussion Point

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!

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

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

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

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