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What happens to the
energy of particles' fields
when they annihilate?

Particles which annihilate each other are antiparticles of each other and thus of opposite charge so when their centers
coinncide at the instant of annihilation their fields exactly cancel. The energy of the particles' field cancels
partially in their journey toward each other. Like their potential energy the energies of their fields are converted into
kinetic energies

Energies of Fields

The energy density of an electrostatic field of field intensity E is ½εE²., where ε
is the permittivity of the material. More precisely it is ½εE·E, where E·E is the dot
product of the vector E with itself.

The field intensity created by charge Q at a distance r from its center is

E = kQ/r²

The energy dU contained in a spherical shell of radius r and thickness dr is

dU = ½ε(kQ/r²)²(4πr²)dr = 2πε(k²Q²/r²)dr

If the charge is contained in a spherical shell of radius R then the total energy of the field between R and ∞
is

U = 2πε(k²Q²/R

Thus if R→0 then U→∞.

The Interaction of Two Fields

Let E_{1}(z) and E_{2}(z) be the intensities of two fields at a point z. The energy density at that point is proportional to

Therefore 2E_{1}·E_{2} is the energy density of the interaction of the two fields.
When the two particles are antiparticles which are near each other then E_{2}≅−E_{2}
and the interaction energy is negative.

The Interaction of Three Fields

Suppose the fields of two antiparticles, E_{+} and E_{−}, are imposed upon a background field of E_{0}.
The energy density of the combined fields is proportional to

(E_{+} + E_{−} + E_{0})·(E_{+} + E_{−} + E_{0})

There will be interactions of
E_{+} with E_{0}, E_{−} with E_{0}, and E_{+} with E_{−}.
To the extent that E_{−}≅−E_{+} the interaction of E_{−} with E_{0} will
cancel out the interaction of E_{+} with E_{0}. So the energy of the combined fields of the two antiparticles will decline primarily
because of their interaction with each other.