11. Inductance
In this section we will finally come up with an electro-magnetic model of mass. Two derivations are done, a couple of simple inductance problems, which are arranged to demonstrate one point. Mass is merely the self-inductance of an electric field.
Imagine a statically charged dielectric ring that is physically torqued in the plane of the ring.
Fig.4: Ring with angular acceleration.
Since a current is introduced, there is a change in flux and thus an induced electric field which creates a force that opposes the torque. We use Stokes' theorem on Eq.(1-1) and obtain
(1)
The integral on the right is the flux and substituting in the expression of flux in relation to the inductance1 yields
(2)
The current in this case is
(3)
where v is the velocity of the rim and l is the linear charge density in the direction of motion. Substituting this into Eq.(2) gives
(4)
where a is the acceleration of the rim. Since the induced electric field is uniform around the rim and always in the direction of dl, we can bring E out of the integral, which then resolves to the circumference. Solving for E then yields
(5)
where s is the circumference of the ring. The reactive force is thus
(6)
Since the reactive force is proportional to the acceleration, it can be brought to the other side of Newton's formula and emulate a mass increase such that
(7)
This indicates that the inductance is not just the electromagnetic analogy of mechanical mass. It is actually mass in every sense of the word (or more precisely, Ll2 is the actual mass).
Now consider Eq.(9-6) again. It is possible to use the energy developed by a current (or motion of charge) to calculate the inductance2, that is by:
(8)
Substituting into Eq.(9-6) yields
(9)
Remember that the right hand side of this equation is the energy of the fields developed during an increment of velocity in relation to the rest energy of the particle. Using the relationship of Eq.(3) we obtain
(10)
Substituting in for the rest energy (from Eq.(9-8)) gives3
(11)
This indicates that, indeed, mass is an electromagnetic property, ultimately arising from the self- inductance of the electric fields that make up the particle. When mass is present, this inductance must also exist. If there is inductance, it always represents an actual mass (Eq.(7)).
The implications here are pervasive. Eq.(11) is literally the definition of mass, thus any particle that has mass must also have an associated electric field. If the external electric field does not appear to represent the total mass of a particle then there must exist internal fields that supply the rest of the mass. This is not impossible as we have already seen that as long as there is spherical symmetry, electric fields can vary widely. There apparently are further constraints on exactly what charge fields can actually exist (see Section 2), but these specific conditions have yet to be determined. As a boundary condition the fields always ultimately (as distance increases) fall off as at least 1/r2, but within the region of non-zero divergence (the charge field), the electric fields can vary. This consequently disallows Earnshaw's theorem (in these specific regions) and thereby has significant impact on both nuclear and atomic theory.
1. Griffiths1 295.
2. Griffiths1 300 or Jackson2 261.
3. Note that the process in this section could be used to derive Eq.(9-8). We could use Eq.(7) to establish the inductance- mass relation and with substitution of Eq.(11) into Eq.(10), the mass-energy relation of Eq.(9-8) would again be derived yielding the same result as before.
Contents
Next