10. Mass
In the last section we calculated the energy mass relation for a point charge and did not find the expected E=mc2. In fact this problem is well known in classical electromagnetism and it turns out that other charge configurations also yield different results. In this section we will derive the mass energy for two of these different charge configurations. The same equipartition of energy method will be used as in the last section. This is interesting in the easy way the method works with these kind of problems. It will yield the same results as other methods. After reminding the reader of this famous problem, we will simply claim that these solutions are in fact all correct and that mass and energy are not connected by a single relation, but the value does, in fact, depend on the geometry of the electric field of the object that is accelerated. But if this is the case, what is mass?
Let us now determine the energy in the magnetic field of the charge pair described in the Section 7 (assuming that they are rigidly attached). First imagine that it has a velocity v in the z-direction. The energy in the magnetic field can be obtained from Eq.(9-4) which results in
(1)
since the radial electric field is the sum of each radial field, we have
(2)
and then
(3)
Note that the two first terms represent the magnetic field energy in each charge independently (from which we derived the kinetic energy in Section 9). The last cross term represents the magnetic energy in relation to the potential energy between the charges. This energy is
(4)
Substituting Eq.(7-12) into this expression yields
(5)
Since the magnetic field represents a kinetic energy, we substitute such that
(6)
and thus
(7)
Therefore the potential energy represents a resistance to acceleration that is equivalent to the mass given in Eq.(7).
Now imagine that it has a velocity v in the y-direction. It will be necessary in this case to divide out the potential energy into rectilinear components which yields
(8)
Note that each component contributes equally to the energy thus we can write
(9)
where E1 and E2 represent either the y-component or the x-component of the electric fields of each charge. The energy in the magnetic field can again be obtained from Eq.(9-4) which yields
(10)
Utilizing the contributions from both charges and selecting only the cross terms (as before) yields
(11)
since
(12)
we have
(13)
By substituting in from Eq.(9) we obtain
(14)
Since the energy in the magnetic field is the kinetic energy we can substitute such that
(15)
and obtain
(16)
Therefore the potential energy represents a resistance to acceleration that is equivalent to the mass1 given in Eq.(16). We have now derived three different mass energy relations (Eq.(7), Eq.(16) and Eq.(9-8)). However, these derivations are all valid. The fact is that energy is not equivalent to mass. We will find in the next section that mass is related to inductance. It is not surprising thus that a different arrangement of charge has a different resistance to acceleration. It turns out that we can use the energy in the electric fields to find what this resistance is, but there is not a real equivalence, because all we are doing is using the energy in the electric field to find the energy in the magnetic field and thus the inductance. Since the mass related to the potential energy is quite small, the mass of the spherical distribution of the elementary charges is dominant. Note that the value of this mass (from Eq.(9-8)) is between the values of the charge distributions extended along the direction of motion and extended perpendicular to the motion as one would expect. Note also that in Eq.(9-8) we assumed that each dimension of electric field contributed equally to the energy, an assumption that does not apply to the electric field of the charge pair.
1. The results of Eq.(7) and Eq.(16) can also be obtained by considering the forces directly in charged pairs that are accelerated for short periods of time (see for example Griffiths1 436 and references). The method requires the general electric field from the charge pair's retarded position. A Taylor expansion is then applied where the first term yields the above results. A second term dependent on the derivative of acceleration and independent of separation of the charges constitutes the radiation reaction. For the pair oriented perpendicular to the acceleration:
(a)
and for the pair oriented parallel to the acceleration:
(b)
where s is the distance between the charges and a is the acceleration. Griffiths' use of the charge pair is particularly ingenious as it avoids the problem of dealing with charge that is infinitely close (as opposed to the Abraham-Lorentz model). We have avoided the problem here by utilizing the energy in the magnetic field and solving the problem indirectly using conservation of energy. The different results of these various calculations has been considered a notorious paradox, but I claim now that the results are simply correct except in the Abraham-Lorentz model where the infinite closeness of the charge introduced some error. There does, in fact, exist a proof here that Coulomb's law is actually inconsistent with Maxwell's Equations. Using energies should have produced an equivalent mass-energy for the Abraham-Lorentz model (as it did successfully for Griffith's double charge models), but since they obtained a different result, it implies that their assumption that infinitesimal charge elements produced force by Coulomb's law was incorrect.
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