Mathematics of the Pure Charge Theory
The charge will be surrounded by a magnetic field, which we call its field of motion, given by
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mass, inertia, electric charge, charge, kinetic energy, field of motion (1), (2)
Accelerating a charge
A charge q has a velocity and an electric field gives it an acceleration . If we assume that is not parallel to , then we can define Cartesian coordinates such that the X-axis is in the direction of . The point has polar coordinates such that its Cartesian coordinates are:
The acceleration produces a change in the magnetic field given by
The vector cross product is distributive across vector addition and subtraction while the process by which we prove that relies on vector addition and subtraction. We can therefore differentiate the cross product of times the constant vector giving
This rate of change of the magnetic intensity changes the energy density of the magnetic field and this results in a flow of magnetic energy density flux both into and/or out of of the surface of the charge depending on the relative direction of compared with and on the position on the surface of the spherical charge. The energy density at in its scalar and vector forms is
and we can expand these to get three more equations.
The effect of acceleration on a body is twofold. The speed of the body may change and its direction of motion may change. There are two special cases: acceleration parallel to the velocity changes only the speed; acceleration perpendicular to the velocity changes only the direction. When we consider a single spherical charge, the effects on the field of motion are a change in its energy content caused by a change in speed and a rotation caused by a change in direction. The movement of magnetic energy density flux is constrained so that it can only move parallel to the electric field of the charge. Furthermore, the electric field is fixed in direction and cannot rotate. So rotation of the magnetic field is accomplished by the movement of energy density flux both into and out of the charge.
A simplistic approach might be to differentiate the first equation for with respect to time to give
This does not however take into account the fact that magnetic energy density flux has vector like properties. If in a small time the magnetic intensity changes from to , the energy density vector changes from to . However, the vector addition of must have the same geometry as that of . If we draw vector triangles to represent the addition, they must be similar.
We look to the vector triangle from which the limiting result for is derived and scale it by a factor of .
This means that
and this corresponds to the following result.
At this point, it is best to enter into Cartesian coordinates and simplify the vector product because algebraic expansion would introduce extra sines and cosines of unknown angles. We have
The magnitude is easy enough to work out as giving
At this point, we are in danger of running into complications. We really need this vector in the form of a magnitude times a unit vector, but this expression is rather messy. So we will delay dealing with this problem by making a substitution.
Then we can write
This is a rate of change of the magnetic energy density vector. We need to divide the space surrounding the charge in such a way as to facilitate integration. We do this by dividing it into conic elements extending outwards from the sphere with rectangular cross section by . In a conic element, the rate of change of the magnetic energy density is inversely proportional to fourth power of the radius r. We need to integrate this over r to find the rate of flow of energy density flux into the conic element.
To find the velocity with which magnetic energy density flux is flowing from the surface of the charge, we must divide by the energy density times the area of the surface of the charge within the conic element. Note that at this point, we need the magnitude and so we omit the direction which is given by the unit vector .
We now know that magnetic energy density flux of density and in the direction of is flowing out of the surface at velocity . Since we have correctly calculated the velocity by considering changes in the magnetic energy density flux within the conic element, we may now use that velocity in a form of Faraday's law to calculate the electric field experienced by the surface of the charge.
What we need to understand here is that we are generating (or adsorbing) magnetic energy density flux at the surface of the charge. Its energy density is the same as the energy density that is already there, but its directional property is different. The value of to use is given by
Where the value of B is that surrounding the moving charge. We are more familiar with (where is the angle between and ) than with , but the angle between and is not known, so we use the modulus of the cross product. This gives us the resulting electric field experienced by the surface of the charge.
Now we find that and are reunited and we can substitute for .
The force felt by the charge acting on the surface within the conic element is
where the factor ½ is the penetration coefficient. This is needed because the surface of the charge has a finite thickness. Energy density flux is generated throughout the thickness of the surface. We have used the value of at the outer face of the surface, but this is its maximum value and we need to average it over the thickness of the surface.
This integration is not as hard as it looks because a lot of the integrals of individual terms are zero. It simplifies to
We find that effecting the change in the field of motion generates a force proportional to and opposing the acceleration. Thus a simple spherical charge with no other properties has an inherent electromagnetic reaction to acceleration. The charge has the property which we call inertial mass. We may write a familiar equation.
where is now the force required to accelerate the charge and
The inertial mass of a pure charge is thus dependent on its charge and its radius.
This result is fully consistent with the idea that the kinetic energy of the charge is stored in its field of motion.
So we find that we can account for the generation of a force resisting the acceleration, and of the correct magnitude to account for the changes in the energy stored in the magnetic field which surrounds a charge by virtue of its velocity. To do this, we have had to refine our understanding of the way magnetic flux behaves. Instead of measuring the bulk of a magnetic field as the integral over a surface of magnetic induction through that surface, we take the volume integral of the energy density of the field. When a field changes in size, it is the flow of energy into and out of the field which is calculated. We still need to take into account the directional properties of magnetic flux when considering it as an energy density field. We need also to specify that movement of energy density flux can only take place parallel to the electric field of the charge.
When we consider real electrons and take the charge on an electron together with the Bohr radius of the electron and Einstein's E = mc2, we can calculate that the mass equivalent of the energy of the electric field of an electron is
The similarity in the two suggests that we are close to explaining the inertial of electrons as a purely electromagnetic phenomena.
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© Copyright Bruce Harvey 1997.