Charged particles like the electron and the proton consist of two parts: a core and a field. The core consists of the distribution of charge over a limited range. Here is the empirically determined charge distributions for protons and neutrons.

As can be seen above, a neutron does have a charges distribution even though it has zero net charge and no field.

The Energy of the Field
of a Charged Particle

Consider a stationary particle with charge Q. The electrical field intensity E at a distance r from the center of that particle is given by

E(r) = (1/(4πε₀))Q/r²

where ε₀ is a constant called the permitivity of space.

For a particle in empty space the energy density U is

U = ½ε₀E² = (1/(32π²ε₀))Q²/r⁴

The energy in a spherical shell of radius r and thickness dr is equal to 4πr²(Q²/(32π²r⁴) which reduces to Q²/(8πε₀r²). The integration of these terms from R to ∞ gives a total energy T_R

T_R = ∫_R^∞U(4πr²)dr = Q²/(8πε₀R)

If R→0, then T_R → ∞ no matter how small is the magnitude of the charge so long as it is nonzero. Thus a charged point particle, if such existed, would have infinite energy.

The Absurdities of a Particle
with Infinite Energy

The absurdity of the notion of a single charged subatomic having more energy than all of kinetic energy, potential and mass energy of all particles in all of the galaxies of the Universe really needs no further elaboration but it is entertaining to consider the attempt to create a chared point particle. First suppose the energy of a speeding train is tried. It is not enough. Then the energy of an ocean liner is tried. No, still not enough. Then the energy of an exploding nuclear weapon is tried and found insufficient. Then the energy of the entire mass of the Earth is tried and it also is not enough. Then suppose the energy of the Sun and all of its planets is tried. It would bed found insufficient. Then surely the energy of all 800 billion stars of the Milkyway Galaxy could be ‏tried and it too would be found wanting. Even if the nearby Andomeda Galaxy is thrown in it is not enough. Surely if the energies of all 800 billion galaxies are included it should be enough. But no it is not enough.

But even so there is no energy left over for creating a second charged point particle. And there are zillions upon zillions of charged subatomic particles.

It is worth noting that there can be no conservation of energy if there is an infinite source of energy that can be drawn from.

There is also the Larmor Proposition that accelerated or decelerated charges radiate electromagnetic waves. It does not apply to spatially distributed charges but it does apply to charged point particles. Thus a charged point particle subjected to centripetal acceleration in a rotating structure or thermal aggitation would radiate electromagnetic waves. But because this radiation if drawing from an infinite source it would effectively be creating additional energy in the Universe. This is an additional absurdity of a charged point particle.

Since a particle and its antiparticle can annihilate each other and their fields, it might be thought that such annihilation of point particles should produce photons of infinite energy but such is not the case. As two particles of opposite charge approach each other they cancel out each others charge. The mutual annihilation of an electron and a positron produces gamma ray(s) of only finite energy. Therefore electrons and positrons cannot be point particles.

How Physical Evidence
of the Pointedness of
Particles Can Arise

There is a wonderful theorem in mathematical physics that says that the effect of a spherical charge uniformly distributed is the same as if the charge were concentrated at the center of the sphere. That is for points outside of sphere of the charge; inside that sphere the effect is zero. Therefore the deflection of probe particles impinging upon a spherical shell of charge is the same as if the charge of the shell were concentrated at its center provided that the probe particles do not have enough energy to penetrate within the shell. The effect on probe particles within the shell would be called asymptotic freedom. Thus evidence of a particle having the same effect over a limited range of energies as a point particle is not evidence of the existence of point particles.

The Consequence of the
Assummption in Quantum
Field Theories of Fields which
are those of Point Particles

Quantum Field Theories are plagued with infinities. Various devices such as renormalization group theory were developed to successfully eliminate those infinities. The interesting intellectual question is whether those infinitinities arise because the fields assumed in QFT are the same as those of point particles. But that is a topic for a different webpage.

(To be continued.)