Re: How can an electron have a large charge density?

Greetings, Jonathan:

Possibly your question should be reversed:  "Why does the proton have a 
lower charge density than the electron?"  The reason we might consider 
rephrasing it like that is because the proton isn't the only subatomic 
particle with the same magnitude of electric charge as the electron.  One 
of the others, the positron, is just like the electron in every way, 
EXCEPT that its electric charge is positive instead of negative.  So why 
should we consider the electron to be anomolous, and not the proton?

Yet there is a better way to answer your question.  Note that the exact 
nature of the thing which we call "electric charge" is something which 
nobody yet fully understands.  We are pretty sure, however, that just 
because electric charge happens to be associated with certain 
mass-possessing particles, that does not mean that mass and charge MUST be 
associated.  A particular particle that I have in mind, in support of this 
statement, is the 'Z' boson, one of the particles associated with the Weak 
Nuclear Force.  (Unstable atoms, which eventually spontaneously break down 
and release dangerous radiation, are unstable because of the Weak Nuclear 
Force.)  To the best of my knowledge, the 'Z' boson posseses a significant 
amount of mass (for a subatomic particle, that is), but no slightest iota 
of electric charge.  Other electrically neutral particles, such as the 
ordinary neutron, are made of smaller particles that DO possess electric 
charge; the total charge of the smaller particles just happens to cancel 
out.  A better particle even than the 'Z' boson would be the neutrino, 
which recent experiments seem to indicate has a very tiny mass (perhaps 
1/100,000 that of the electron), and also has no slightest iota of 
electric charge.  (The mass of a neutrino is so small -- and its other 
properties so minimal -- that it took experimental physicists about 20 
years to prove the particle really existed, and then 40 more years to 
measure it!)

The point of the preceding is simply this:  If mass is mass and charge is 
charge, and they are not required to co-exist, then it doesn't matter in 
the slightest what the charge-to-mass ratio of a given particle is.  Nor 
does it matter if the charge-to-mass ratio of some other particle is a 
different value.

On the other hand, there is still another way to answer your question.  
All the most fundamental electrically charged particles -- electrons and 
quarks -- appear indistiguishable from mathematical points.  Our best 
instruments cannot resolve width for either electron or quark.  This is 
not to say that they do not have a width; it is simply smaller than we can 
measure.  PERHAPS they are indeed mathematical points; most certainly the 
mathematics of Quantum Electrodynamics, which describes the behavior of 
simple electrically charged particles to at least 14 significant figures 
of measured accuracy, is perfectly comfortable with the notion.  (If I say 
that some particle of dust has a measured mass of 0.001038395434 grams, 
then that is only 10 significant figures of accuracy -- the leading zeros 
don't count.)

Now consider the idea that a proton is made of three quarks.  While each 
quark MAY be as small as a mathematical point, the proton has a measurable 
width because of the space between its constituent quarks.  Since each 
quark possesses only some of the proton's total electric charge, their 
separation naturally leads to a "spreading out" of that total charge.  
Meanwhile, the electron's charge remains concentrated, apparently like a 
mathematical point.  Perhaps this, at last, is the answer you were hoping 
to find....