Re: Charged Particles: near_energy versus far_enegy

["John S. Denker" <jsd@MONMOUTH.COM>]

At 09:25 PM 4/7/01 -0400, Robert B Zannelli wrote:
> However one Physicist held the opinion that any
> particle with an infinite range interaction could not be massless.

I wouldn't have said that.

1) As a preliminary, to prevent possible misunderstanding, we should
clarify the distinction between mediating and sourcing:
 a) The photon has zero mass.  This is the hallmark of a particle that
*mediates* an infinite-range interaction.
 b) The electron has mass.  So does an alpha particle.  In each case,
presumably some of this mass can be attributed to the energy of the static
electric field surrounding the particle.  The *source* of this field is the
charge of the particle.  In contrast the photon is not the source of an
electric field.  To understand this precisely, apply Gauss's law to a box
containing only a bunch of electrons, and contrast this with a box
containing only a bunch of photons.

2) To be charitable, I will assume that the aforementioned Physicist was
talking about a charged particle that is the *source* of an infinite-range
interaction.  But even in that case, talking about the range of the
interaction doesn't seem relevant, for reasons we will now discuss.

A) The electrostatic energy density goes like field squared.  The field
goes like (1/r squared).  We must integrate this over all space.  Let's use
spherical polar coordinates.  The upper limit of the integral is
R=infinity.  The lower limit of the integral is R=0, if we are dealing with
a point particle.  The measure of volume is (r^2 dr).  That is,

        energy = integral (from 0 to infinity) (1/r^4) (r^2 dr)

It is instructive to break this into two pieces:

        near_energy = integral (from 0 to R1)       (1/r^4) (r^2 dr)
        far_energy = integral (from R1 to infinity) (1/r^4) (r^2 dr)

And for *any* choice of R1 we  have:

        energy = near_energy + far_energy

B) Remember,
 -- An infinite-range interaction potential goes like 1/r.
 -- A typical Yukawa-type finite-range interaction potential
    goes like exp(-r)*(1/r)

At short distances, the Yukawa potential is indistinguishable from the
(1/r) potential;  the exponential drops out.

C) Combining (A) and (B), we get to the key distinction:
 -- The far_energy is finite, no matter whether the interaction has finite
or infinite range.  The integral converges.
 -- The near_energy is infinite, for any point particle, no matter whether
the interaction has finite or infinite range.  The integral is strongly
divergent.

If we have a particle that is *not* a point particle, it can have a finite
electrostatic energy.  An example of this is an ideal conducting sphere,
that has a field on the outside but zero field inside.

Bottom line: When calculating the field energy of a particle, the important
thing is the near_energy.  The near_energy could be infinite.  The
far_energy is just a finite addendum to the near_energy.  The important
question is how the interaction behaves at short distances, not long distances.

We tend to know how things behave at long distances.  We tend not to know
with confidence what happens at really short distances.