San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

The Energies of the Electrical Fields of
Charged Particles and their Charge Radii


A motionless, spherically-shaped charge of radius R and magnitude Q establishes an electric field throughout space with an intensity E given by

E = KQ/s²
for s≥R

where K is a constant and s is the distance to the center of the charge. In the MKS system K is equal to 8.988×109 N·m²/C². Often the constant K is expressed in the form of

K = 1/(4πε0)

where ε0 is the permittivity of free space and its value is 8.854x10−12 farads per meter.

The intensity for s<R depends upon the distribution of the charge within the radius R.

The energy density H per unit volume of an electric field is given by

H = ½εE²

where ε is the permittivity of the medium in which the field exists. The permittivity can be taken to be that of free space. Thus the energy contained in a spherical shell of radius s and thickness ds is


The total enegy contained in the field for s≥R is then

H = ∫R[½εE = 2εK²Q²π ∫Rds/s²
= 2εK²Q²π[1/R]

Taking into account that K=1/(4πε0) this reduces to

H = Q²/[8πε0R]

Point Charges?

For a charged point particle R is 0 and thus the energy of a charged point particle is infinite. When an electron and a positron annihilate each other the fields of both disappear, but amount of energy released is not infinite. Therefore electrons and positrons are not point particles. There are similar annihilation of other particles and their anti-particles.

Thus point charges do not and cannot exist.

The energy released upon the annihilation of a particle is mc² where m is the particle mass and c is the speed of light. This means that

R = Q²/[8πε0mc²]

It is notable that the size of a particle is inversely proportional to its mass.

The Electron

The charge of an electron is 1.60217657×10-19 coulombs and its mass is 9.10938291×10-31 kilograms. This implies that the charge radius of an electron is

R = (1.602×10-19)²/[8*(3.1416)*(8.854x10−12)(9.10938×10-31)(3×108)²]
which reduces to
1.40674×10-15 m = 1.40674 fermi .

This is a reasonable value given the scale of atoms. The radius of the lowest orbit of an electron in a hydrogen atom is 5.2918×10−11 m. This is over 37 thousand times the above value for the radius of an electron.

The Proton

The mass of a proton is 1836 times that of an electron. If its mass-energy is entirely due to that of its electric field then its charge radius would have to be 1/1836 of the radius of an electron. The conventional estimate of the charge radius of a proton based upon scattering experiments usually electron beams is 0.8775 fermi. This means that a proton's electric field accounts for only 1.6 times the mass-energy of an electron. The rest of the 1836 ratio involves proton mass-energy from a different source.

(To be continued.)

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