# What is a dipole moment in classical electromagnetisim?

• hanson

#### hanson

I am rather confused about "dipole moment" in classical electromagnetisim. Since I have no previous background in this field of physics, I find it hard to understand the Maxwell's equations and other equations that involve the concept of dipole moment. Could anyone explain this to me in plain terms please?

Consider a body that has no charge. Now we bring a second body of negative charge close to any point on the first body. Although the first body has no charge, it has an equally distributed amount of both positve and negative charges that cancel out.

Because opposites attract, some of the distributed positive charge will move to the surface at the location of the second body (due to the attracted forces that draw them there). But because the overall charge of the first body is zero, there has to be a loss of positive charges on the other end of the body - it now has a negative charge.

The result is that we have induced a dipole on the body of no charge by bringing the negatively charged body close to it. One side now has a (small) net positive charge, and the other a net negative charge.

So an electric dipole moment of this scenario would be p=qd

where p is the moment created by the charges separated by a distance d. (on each end of the first body). It's an induced dipole moment.

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I am rather confused about "dipole moment" in classical electromagnetisim. Since I have no previous background in this field of physics, I find it hard to understand the Maxwell's equations and other equations that involve the concept of dipole moment. Could anyone explain this to me in plain terms please?

The long answer involves Legendre polynomials (q.v.) which are an orthogonal set involving monopole, dipole, quadripole, etc moments.

The short answer would consider a bar magnet. The dipole moment would be pole strength times pole separation.

Think about the electric field created by two point charges +q and and -q, separated by a (small) distance d.

If you reduce the distance and increase the charge so that qd remains constant, the field stays approximately the same, except for the region close to the charges.

A dipole is the limiting case of this, as d goes to zero and q goes to infinity but qd remains finite.

But the term "moment" gives me a feeling of "rotation", as in mechanics...
Is it essetially relevant to any rotational motion?

But the term "moment" gives me a feeling of "rotation", as in mechanics...
Is it essetially relevant to any rotational motion?

I agree, that's what it suggests to me as well. I'd be interested in understanding a connection between the two as well!

But the term "moment" gives me a feeling of "rotation", as in mechanics...
Is it essetially relevant to any rotational motion?

Put (say) an electric dipole (with dipole moment $\vec p$) in an external electric field $\vec E$. If the dipole moment is not aligned with the field, the dipole experiences a torque $\vec \tau = \vec p \times \vec E$ which tends to rotate it towards alignment.

A more practical example: a magnetic compass needle is a magnetic dipole, which rotates towards alignment in an external magnetic field (such as the Earth's).

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But the term "moment" gives me a feeling of "rotation", as in mechanics...
Is it essetially relevant to any rotational motion?

The term 'moment' can imply rotation: moment arm, for example. 'Moment' is taken to mean something like 'distribution'.

Take a macroscopic object. Newton's approach is to replace the extended body with a mass-point located at the center of mass. Similarly with a charged sphere- the electric field outside the body behaves as if the total charge is located at the center of the body.

However, we can distribute the mass or charge any way we would like, in real-life. The spatial distribution of mass and charge can then be represented mathematically as a series:

Total = monopole + dipole + quadrupole + octupole +...

Where the dipole moment, quadrupole moment, octupole moment, etc are all idealized geometries- in electrostatics, a dipole is two charged points separated by a certain distance. The quadrupole is four points, octupole 8 points, etc. In terms of mass, it's a little more complex but the mathematics is the same.

In mechanics, Euler introduced the term "moment of inertia" back in 1730, so the origin of the term 'moment' goes back at least that far. But again, the term is used as a way to describe the spatial distribution of mass of an extended body.