Re: point particles

[John Denker <jsd@AV8N.COM>]

Quoting Ludwik Kowalski <kowalskil@MAIL.MONTCLAIR.EDU>:

> In other words, the dipole moment of an object
> containing a single point charge is not an
> intrinsic property of that object,

Yes and no.

A lot of people have been talking past each other, i.e.
failing to distinguish
  -- a "mathematics" definition of dipole moment, versus
  -- a "physics" definition of dipole moment.

Let us choose the point r0 as our datum;  you could think of
it as a shift in the origin of coordinates, or (as I prefer)
you could leave the coordinate system alone and choose,
temporarily or otherwise, to measure certain things relative
to r0 rather than relative to the origin.

Now let me define my version of electric dipole moment to be
   M1q := integral (r - r0) dq            [1]
Those who aren't familiar with writing the measure as "dq" are
free to expand it using dq == rho dx dy dz ... but dq is the
more general notion.  In particular, for a collection of
discrete charges, [1] reduces to
   M1q = sum (r_i - r0) q_i               [2]

In contrast, the "mathematics" definition of dipole moment omits
the (- r0) from these expressions.

The total charge on the object is
   q := integral dq                        [3]
For a neutral object (q=0) the electric dipole moment is
independent of r0 (binomial theorem, remember?), in which
case the physics definition and the mathematics definition
coincide.

For a non-neutral object, we can _make_ the electric dipole
moment be _intrinsic_ to the object if we choose a datum
(r0) that is "attached" to the object.

You might think the 'natural' choice for r0 would be the
center of charge, i.e.
    cq := (integral r dq) / (integral dq)   [4]
But if you plug in r0 = cq in [1] you find that the dipole
moment defined thereby is identically zero.  That's
coordinate-independent, but not very useful.  Sigh.

So a better choice, and the conventional physics choice, is
to find the center of mass, i.e.
    cm := (integral r dm) / (integral dm)    [5]
and use that as our datum (r0).

Putting it all together, we get the the "physics" electric
dipole moment is
    integral r dq - (q/m) integral r dm
which is manifestly coordinate-independent and intrinsic to
the particle.