San José State University |
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energy of particles' fields when they annihilate? |
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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
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
The energy dU contained in a spherical shell of radius r and thickness dr is
If the charge is contained in a spherical shell of radius R then the total energy of the field between R and ∞ is
Thus if R→0 then U→∞.
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.
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
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.
(To be continued.)
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