Energy of an electric quadrupole in an Electric Field

• karanmohan
In summary, the conversation discusses the energy of an electric quadrupole moment in an electro-magnetic field and the use of first order perturbation theory to derive the matrix element of an atom interacting with a field. The expression for the energy of an electric quadrupole is U_Q = -\frac{1}{3}\bf[Q]:[{\nabla}\nabla E] and references are provided for further reading. The conversation also touches on the operation : and its definition in this context.
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!

hmm...thanks for the link friend, but it doesn't give me enough detail unfortunately :(

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

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

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

JJfortherear said:
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

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

clem said:
$$U_Q = -\frac{1}{3}\bf[Q]:[{\nabla} E]$$.
That is a misprint.
The correct equation is
$$U_Q = -\frac{1}{3}\bf[Q]:[{\nabla}\nabla E]$$
$$=-\frac{1}{3}[(\bf[Q]\cdot\nabla)\cdot\nabla] E$$.

Last edited:

1. What is an electric quadrupole?

An electric quadrupole is a system of two equal and opposite electric dipoles placed close together, resulting in a net charge of zero but a non-zero electric field.

2. How is the energy of an electric quadrupole in an electric field calculated?

The energy of an electric quadrupole in an electric field is calculated as the product of the electric field strength and the dipole moment, multiplied by a factor of 1/3.

3. What is the significance of the electric quadrupole moment?

The electric quadrupole moment measures the deviation of the charge distribution from a spherical shape, and is an important parameter in determining the behavior of an electric quadrupole in an electric field.

4. How does the energy of an electric quadrupole in an electric field change with the orientation of the quadrupole?

The energy of an electric quadrupole in an electric field is dependent on the orientation of the quadrupole. When the quadrupole is aligned with the electric field, the energy is minimized, whereas when the quadrupole is perpendicular to the electric field, the energy is maximized.

5. Can the energy of an electric quadrupole in an electric field be negative?

Yes, the energy of an electric quadrupole in an electric field can be negative. This occurs when the quadrupole is aligned in a direction opposite to the electric field, resulting in a negative value for the energy.

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