San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

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 E1(z) and E2(z) be the intensities of two fields at a point z. The energy density at that point is proportional to

( E1 + E2)·( E1 + E2) = E1·E1 + 2E1·E2 + E2·E2

Therefore 2E1·E2 is the energy density of the interaction of the two fields. When the two particles are antiparticles which are near each other then E2≅−E2 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 E0. The energy density of the combined fields is proportional to

(E+ + E + E0)·(E+ + E + E0)

There will be interactions of E+ with E0, E with E0, and E+ with E. To the extent that E≅−E+ the interaction of E with E0 will cancel out the interaction of E+ with E0. So the energy of the combined fields of the two antiparticles will decline primarily because of their interaction with each other.

(To be continued.)

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