XYLENE POWER LTD.

FIELD THEORY

By Charles Rhodes, P. Eng., Ph.D.

CHARGE, CHARGE MOTION AND ENERGY QUANTA:
The universe may be thought of as being composed of charge quanta, circulating current quanta and radiant energy quanta. Each quantum has an influence field in space surrounding the quantum's nominal relative position. The nominal relative position of each quantum can have relative motion. Changes in overlap of influence field vectors causes changes in the total potential energy.

On a microscopic scale each charge quantum consists of a string of charge that circulates around a closed path at the speed of light. That charge circulation produces a magnetic field that enables particle structure and stability. Energy exists as field potential energy, particle kinetic energy or as propagating radiation.
 

FORCE AT A DISTANCE:
Electric charge, magnetism and gravitation each seem to independently exert force at a distance. The force on particle i is actually a change in potential energy with respect to a change in particle relative position (Xi - Xo). The existence of separate independent forces at a distance requires that electric charge quanta, magnetic quanta and gravitation quanta each have separate potential energy distributions.
 

INFLUENCE VECTOR FIELD:
Influence fields are vector constructs that enable calculation of potential energy distributions. Associated with each electric (charge), magnetic (charge motion) and radiant energy quantum is a vector field that defines the influence of the charge, charge motion or radiant energy quantum at a distance. An electric field is a result of local charge presence. A magnetic field is a result of local charge circulation. A gravitional field is a result of local energy presence. Radiant energy is propagating fluctuations in these vectors. At every point in relative space and time each influence field has both a magnitude and a direction.

Extending outward from each isolated element of electric charge is a spherical influence vector flux that is proportional to the amount of charge. Extending outward from each element of electric current is a cylindrical magnetic influence field vector that is proportional to the current. Extending outward from each element of potential energy (mass) is a spherical gravitational influence vector flux that is proportional to the amount of mass (energy). The energy density at a point caused by a field is proportional to the vector field squared at that point.

The radial unit vectors are:
(+1)
for positive charge,
(-1)
for negative charge,
(i)
for normal matter and:
(-i)
for hypothesized anti-matter.

The circumference magnetic cross product unit vectors are (+1) and (-1)

Let x, y, z be unit vectors in a cartesian co-ordinate system. Let r be a unit vector along the straight line joining point (X,Y, Z) to point (Xoi, Yoi, Zoi). Let (Xo, Yo, Zo) be the position of a charge in that co-ordinate system. Let (X, Y, Z) be the position of a point in that co-ordiate system.

The electric field dEi at point (X, Y, Z) caused by charge dQi at point (Xoi, Yoi, Zoi) is given by:
dEi = dQi / {4 Pi Epsilono [(X - Xoi)^2 + (Y - Yoi)^2 + (Z - Zoi)^2]}
where:
dEi = [dExi^2 + dEyi^2 + dEzi^2]^0.5
and
dExi = dEi (X - Xoi)x / [(X - Xoi)^2 + (Y - Yoi)^2 + (Z - Zoi)^2]^0.5
and
dEyi = dEi (Y - Yoi)y / [(X - Xoi)^2 + (Y - Yoi)^2 + (Z - Zoi)^2]^0.5
and
dEzi = dEi (Z - Zoi)z / [(X - Xoi)^2 + (Y - Yoi)^2 + (Z - Zoi)^2]^0.5

For multiple charges i located at different points Xoi, Yoi, Zoi the net electric field at (X, Y, Z) is given by:
Ex x = sum over all i of dExi
Ey y = sum over all i of dEyi
Ez z = sum over all i of dEzi

The magnetic field vector dBi at point (X, Y, Z) caused by current Np I flowing along element of length dLi at (Xoi, Yoi, Zoi) is given by:
dBi = Muo Np I dLi X ri / {4 Pi Ri^2
or
= Muo Np I dLi X ri / {4 Pi[(X - Xoi)^2 + (Y - Yoi)^2 + (Z - Zoi)^2]}
where:
dBi = [dBxi x + dByi y + dBzi z
where:
dLi = dLxix + dLyiy + dLziz
and
ri = [rxix + ryiy + rziz] / [rxi^2 + ryi^2 + rzi^2] and
Ri = (X - Xoi)x + (Y - Yoi)y + (Z - Zoi)z / [(X - Xo)^2 + (Y - Yoi)^2 + (Z - Zoi)^2]^0.5

For multiple current elements i located at different points Xoi, Yoi, Zoi the net magnetic field at (X, Y, Z) is given by:
Bx x = sum over all i of dBxi
By y = sum over all i of dByi
Bz z = sum over all i of dBzi

Recall that from definitions of dot and cross products:
x.x = 1
y.y = 1
z.z = 0
x.y = y.x = 0
y.z = z.y = 0
x.z = z.x = 0
x X y = z
y X z = x
z X x = y
y X x = -z
z X y = -x
x X z = -y

The unit vectors of charge influence (electric field), magnetic influence (magnetic field) and energy influence (gravitational field) are all mathematically orthogonal. Charge can be either positive or negative. Charge motion can be either clockwise (CW) or counter clockwise (CCW).

These unit vectors result in electric and magnetic fields contributing positive potential energy density and gravitational fields from norml matter contributing negative potential energy density.
The unit vector representation corresponds to experimentally observed potential energy distributions and gradients.

Note that for a particular isolated element of charge the electric influence vector flux resulting from integration over a surrounding spherical surface at radius R is constant independent of R. Thus the influence vector flux per unit area at that spherical surface at radius R from the element is:
E = Q r / (4 Pi Epsilono R^2).

If there are multiple distributed charge quanta then at each point in space the various charge quanta influence vectors add. If there are multiple magnetic quanta then at each point in space the magnetic influence vectors add. Similarly, at each point in space the energy influence (gravity) vectors add.

The radial electric field vector distribution produced by a single point charge is of the form:
dEi = dQi r / 4 Pi Epsilono R^2

The potential energy density of an electric vector flux field is:
(Epsilono / 2) [Q^2 r.r / (4 Pi Epsilono R^2)]^2
where:
Epsilono is the permittivity of free space,
ri is a unit vector pointing away from the charge along radius R
ri.ri = 1
 

ELECTRIC FIELD:
Consider the special case of an isolated electric charge. Assume that charge only exists in the region where:
R < Ro

Then for an isolated electric charge the total electric field potential energy in the region R > Ro is:
Integral from R = Ro to R = infinity of:
(Epsilono / 2) [Q / (4 Pi Epsilono R^2)]^2 4 Pi R^2 dR

= Integral from R = Ro to R = infinity of:
[(Q^2 / 8 Pi Epsilono)] [dR / R^2]

= Q^2 / (8 Pi Epsilono Ro)

This result is the well known expression for the electric field energy around an isolated electric charge. Note that for a theoretical point charge:
Ro = 0
which causes the theoretical electric field potential energy for a point charge to go to infinity. This problem can be avoided by use of spheromaks which have combined electric and magnetic field structure.
 

ENERGY DENSITY:
The potential energy density at any point (X - Xo) in relative space at any time (t - to) is the sum of the squares of the orthogonal net influence vectors at that point and time. The total energy density at any point (X - Xo) in space and at any time (t - to) is the sum of the potential energy density plus the kinetic energy density plus the field free vacuum energy density. At each point in relative space and time the energy density has a magnitude but no direction.

Energy density is a relative quantitiy. Until recently the field free vacuum energy density was presumed to be zero. However, astromonical observations of distant galaxies seem to indicate that in deep space the vacuum energy density may be greater than zero. It is possible that the vacuum energy density is partially a result of low level propagating gravitation waves that present instrumentation cannot resolve.
 

ENERGY CONSERVATION:
Energy is a conserved constituant of the universe. In any isolated system interaction the total isolated system energy before and after that interaction remains unchanged.
 

RADIATION:
Radiation photons are propagating fluctuations in the influence vector fields that convey energy and momentum.
 

MOMENTUM:
Momentum indicates a movement of energy at a particular rate in a particular direction at a particular position and at particular time. Momentum is a vector quantity.
 

PARTICLE:
A particle is a concentration of energy at a nominal position in relative space and relative time. A particle has a non-zero rest mass. Each particle has corresponding vector fields. Although a particle has a nominal position, due to its associated vector fields the particle's field energy density distribution actually extends to infinity. However, the field energy density (energy per unit volume) decreases sufficiently rapidly with increasing distance R from the particle's nominal position that the total particle energy is finite.

Note that energy density calculations are normally made relative to an assumed field free vacuum.

Momentum is used to characterize the net motion of all of the energy conveyed by a particle.
 

FORCES:
When multiple particles co-exist their vector fields overlap everywhere in the universe. Outside of nuclei, at every point in the universe at any instant in time t there is a net electric field vector, a net magnetic field vector and a net gravitational field vector. These vectors are mathematically mutually orthogonal. There is a resulting potential energy density for every element of volume. If this potential energy changes the law of conservation of energy simultaneously causes an equal but opposite change in the kinetic energy content of this same element of volume and/or the emission/absorption of radiation photons. The apparent force between particle i and all the other particles in a cluster is really just the change in total potential energy dEpi with respect to a change in the position of particle i in the cluster.

The fact that electric, magnetic and gravitational fields contain potential energy leads to well known equations for apparent electric, magnetic and gravitational forces.
 

INFLUENCE VECTOR FIELD ADDITION:
For electric field strength in a central system is:
(electric field) = E = Q / (4 Pi Epsilono R^2) = (1 / 4 Pi Epsilono)[Q / R^2] = K [Q / R^2]
where:
K = 1 / 4 Pi Epsilono

The corresponding electric field energy density is:
(Epsilono / 2)(electric field)^2 = (Epsilono / 2) E^2 = (1 / 8 Pi K) E^2

The gravity field strength in a central system is:
G i M / R^2 = G (i M / R^2)

By analogy, the gravitational field energy density will be:
Rhog = - (1 / 8 Pi G)[M / R^2]^2

For a spherical mass:
dM = Rhog 4 Pi R^2 dR
= - (1 / 8 Pi G)[M / R^2]^2 4 Pi R^2 dR
= - (1 / 2 G) M^2 dR / R^2

Note that due to the gravity field M will decrease as R increases. Thus:
dM / M^2 = - (1 / 2 G) dR / R^2

For magnetic fields the field energy density is:
(1 / 2 Muo)(magnetic field)^2 = B^2 / 2 Muo

Assume that at every point in space the electric, magnetic and gravitational field vectors from different particles linearly add to give position dependent net electric, net magnetic and net gravitational field vectors.

The net magnetic field vector Fm = B at position (X - Xo) is given by:
Fm = Sum of all Fmi
where Fmi denotes the magnetic field vector at (X - Xo) due to motion of the ith particle.

The net electric field vector Fe at position (X - Xo) is given by:
Fe = E = Sum of all Fei
where Fei denotes the electric field vector at (X - Xo) due to the ith particle.

The net gravitational field vector Fg at position (X - Xo) is given by:
Fg = Sum of all Fgi
where Fgi denotes the gravitational field vector at (X - Xo) due to the energy of the ith particle.
Note that all the Fgi are imaginary so that the product (Fgi)(Fgj) is always negative.
 

ORTHOGONALITY ASSUMPTION:
Assume that the magnetic, electric and gravitational field vectors of a particle are all mutually mathematically orthogonal. In addition to the normal x, y, z cartesian co-ordinate unit vectors for electric and magnetic fields there are imaginary co-ordinate jx, jy, jz unit vectors, where j = (-1)^0.5 for gravitation. The additional imaginary unit vectors cause the gravitational field energy density for normal matter and for anti-matter to be negative. In addition, normal matter and anti-matter tend to gravitationally repel each other.
 

TOTAL FIELD ENERGY DENSITY:
Recall that for each field type the potential energy density of the field is proportional to the net influence vector flux density squared.

Then the total field energy dEo contained in an element of volume dV = (dX dY dZ) at (X - Xo)X can be expressed as:
dEo|(X - Xo) = {+ (Km / 2) [Fm]^2 + (Ke / 2) [Fe]^2 + (Kg / 2) [Fg]^2 + (vacuum energy density)} dV
where:
Fm = net magnetic field vector at (X - Xo)
Fe = net electric field vector at (X - Xo)
Fg = net gravitational field vector at (X - Xo)
Note that Fg is an imaginary quantity so that the gravitational energy density term is negative.
For a point mass:
Fg = i M / 4 Pi R^2

dM = Rho(R) 4 Pi R^2 dR

Note that the gravitational field energy density of normal matter is negative. When the field energy density at position (X - Xo) becomes more negative the kinetic energy density at position (X - Xo)
instantaneously becomes more positive by the same amount so that the total energy density at position (X - Xo) is unchanged.
 

FIELD ENERGIES:
The field potential energy of a particle is the sum of integrals over spacial volume of the various field potential energy densities.
 

RELATIVE SIZE OF PARTICLE ENERGY COMPONENTS:
In most practical situations:
Particle Core Energy Density >> Electromagnetic Field Energy Density >> Gravity Field Energy Density

Thus changes in gravity field energy density are only important when both the core energy density and the electromagnetic field energy density components are stable. Similarly changes in electromagnetic field energy density are only important when the core energy density component is stable.

The core energy density decays very quickly (exponentially) with increasing radius from the nominal particle position. The magnetic field energy density component decreases less quickly with increasing radius. The electric and negative gravitational field energy density components both decrease slower and in proportion to:
(1 / R)^4
 

PARTICLE CHARACTERIZATION:
Each stationary particle i can be characterized as having a charge Qi, a magnetic moment Mi, a radius Ri and a rest energy Eoi. The rest energy Eoi includes the energy content of the particle's electric, magnetic and gravitational fields.
 

MULTIPLE PARTICLES:
Most real systems involve multiple particles. A real system has a Centre of Momentum (CM), which serves as a system position reference point.
 

CONSERVATION OF ENERGY FOR AN ISOLATED CLUSTER OF INTERACTING PARTICLES:
The law of conservation of energy requires that for any fully isolated system the total system energy with respect to an inertial observer is constant. Hence, if progressive overlap of vector fields causes a change in field energy the law of conservation of energy requires a corresponding opposite sign change in kinetic energy and photon energy to keep the total energy constant.
 

CONSERVATION OF ENERGY FOR TWO ISOLATED INTERACTING PARTICLES:
If two particles forming an isolated system interact the individual particle energies can remain unchanged or an element of energy dE can be transferred from one particle to the other or a third particle can be formed but the total system energy remains unchanged. Note that a process involving creation of a third particle from an interaction between two particles is usually not reversible because such reversal requires a three body interaction. The probability of a random three body interaction is usually extremely small compared to the probability of a random two body interaction.

A common example is two bodies interacting and liberating a photon in circumstances where the probability of photon capture is very small. Thus conservation of energy in combination with photon emission leads to formation of assemblies of particles mutually bound in potential wells and determines the direction of evolution of many processes in the local universe. eg Condensation of water vapor to form liquid water.
 

CLUSTER OF PARTICLES:
A nearly isolated cluster of particles can absorb or emit photons, which increase or decrease the total cluster energy. Hence the total cluster energy will gradually change until the rate of energy absorption equals the rate of energy emission. The Earth, in its orbit in space around the sun, is an example of a nearly isolated cluster of particles.
 

CHANGE IN POTENTIAL ENERGY DUE TO CHANGE IN FIELD OVERLAP:
Vector fields from different particles add linearly. Hence, overlap of vector fields from multiple particles changes the net local vector field strength linearly. However, the local field energy density is proportional to the net local vector field strength squared. Hence the local field energy density changes nonlinearly as the vector field overlap changes.
 

UNDERSTANDING FIELD OVERLAP:
Conventional Electric Field Force Theory:
For an isolated particle with charge Q:
[(Electric Field)|R] = Q /(4 Pi Epsilon R^2)

Assume that charge Q is distributed over the surface of a sphere of radius Ro. the radial force on charge dQ at radius R is:
Q dQ / (4 Pi Epsilon R^2).
The kinetic energy released in taking dQ from Ro to infinity is:
Integral from R = Ro to R = infinity of:
Q dQ dR / (4 Pi Epsilon R^2)
= Q dQ / (4 Pi Epsilon Ro)

The total energy required to assemble charge Q is:
Integral from Q = 0 to Q = Q of:
Q dQ / (4 Pi Epsilon Ro)

= Q^2 / (8 Pi Epsilon Ro)
 

Electric Vector Field Theory:
The energy contained in the electric field is:
= Integral from R = Ro to R = infinity of:
(Ke / 2)(electric field)^2 4 Pi R^2 dR

= Integral from R = Ro to R = infinity of:
(Ke / 2) [Q /(4 Pi Epsilon R^2)]^2 (4 Pi R^2 dR)

= Integral from R = Ro to R = infinity of:
(Ke / 2) (Q / Epsilon)^2 [dR / (4 Pi R^2)]

= (Ke / 2) (Q / Epsilon)^2 [1 / 4 Pi Ro]

Equating the above two expressions for total electric field energy gives:
(Ke / 2) (Q / Epsilon)^2 [1 / 4 Pi Ro] = Q^2 / (8 Pi Epsilon Ro)
or
(Ke / 2)(1 / Epsilon) = (1 / 2)
or
Ke = Epsilon

Thus:
Electric field energy density is:
(Ke / 2) (Electric Field)^2 = (Epsilon / 2)(Electric Field)^2
= conventional expression for electric field energy density
 

CONVENTIAL GRAVITY FORCE THEORY:
The release of kinetic energy in taking dM from R = infinity to R = Ro is:
Integral from R = infinity to R = Ro of:
- G M dM dR / (R^2)

= [- G M dM / R]|R = Ro - [- G M dM / R]|R = infinity
= - G M dM / Ro

The total total release of kinetic energy in assembly of mass M is:
Integral from M = 0 to M = M of:
- G M dM / Ro

= - G M^2 / (2 Ro)
 

GRAVITY FIELD THEORY:
For an isolated particle with core mass Mo the gravity field at Ro is given by:
[(Gravity Field)|Ro] = i G Mo / (Ro^2)

For R > Ro:
The gravity field at radius R is:
i G M / R^2

The energy contained in the gravity field is:
= Integral from R = Ro to R = infinity of:
(Kg / 2)(gravity field)^2 4 Pi R^2 dR

= Integral from R = Ro to R = infinity of:
(Kg / 2) [i G M(R) / (R^2)]^2 (4 Pi R^2 dR)

= Integral from R = Ro to R = infinity of:
- (Kg / 2) (G M(R))^2 [4 Pi dR / (R^2)]

~ - (Kg / 2) (G Mo)^2 [4 Pi / Ro]

FIX THIS INTEGRATION!

Equating the two expressions for total gravity field energy gives:
- (Kg / 2) (G M)^2 [4 Pi / Ro] = - G M^2 / (2 Ro)
or
(Kg / 2)(G 4 Pi) = (1 / 2)
or
Kg = 1 / (4 G Pi)

Thus:
Gravity field energy density at radius R is:
- (Kg / 2) (Gravity Field)^2 = - (1 / 8 G Pi)(Gravity Field)^2
= - (1 / 8 G Pi)(G M / R^2)^2
= - G M^2 / (8 Pi R^4)
= gravity field energy density

Summary:
Total gravity field energy = - G M^2 / (2 Ro)
Total energy = M C^2
Ratio of (Total gravity field energy / Total energy
= - G M^2 / (2 Ro M C^2)
= - G M / (2 Ro C^2)

Consider our sun:
G = 6.6743 X 10^-11 m^3 / kg s^2
Msun = 1.9891 X 10^30 kg
Rosun = 6.96340 X 10^8 m

For our sun the ratio is:
- G M / (2 Ro C^2)
= -6.6743 X 10^-11 m^3 / kg s^2 X 1.9891 X 10^30 kg s^2/ (2 X 6.96340 X 10^8 m X 9 X 10^16 m^2)

= - 6.6743 X 1.9891 X 10^-5 / (2 X 6.96340 X 9)

= - 0.1059 X 10^5

= 1.059 X 10^-6

For a neutron star mass M remains comparable while Ro becomes much smaller, which causes fraction of total energy contained in the gravitational field to be much larger.
 

Note that for a neutral particle, at the nominal particle location the core energy density is positive with respect to a vacuum, in the surrounding space the gravitational field energy density is negative with respect to a field free vacuum and distant from the nominal particle location the energy density is that of a field free vacuum. Thus adjacent to each mass having positive core energy is a shallow negative gravitational potential energy well

In the gravitational potential energy well the speed of light, as seen by an outside observer, is less than in field free space. Recall that:
(Lamda)(Frequency) = C = speed of light
A decrease in C causes a decrease in (Lamda) which bends a photon beam around the central core region of high energy density.

At an extremely high central energy density the curvature of the photon path will be so high that light cannot escape from the potential energy well. That condition describes a black hole.

In the region of negative gravitational field energy density clocks run slower than outside this region so that the apparent speed of light to an observer inside the region remains unchanged. This is a region of high acceleration.
 

SOLAR SYSTEM:
Inside the sun's surface the sun's potential energy density is highly positive with respect to a vacuum. Outside the sun's surface the potential energy density is relatively small but is negative with respect to a vacuum. At larger radii from the sun the potential energy density approaches the potential energy density of a vacuum. When the system includes one or more planets the total energy is less than for a widely separated sun and planets. Hence the planets are bound to the sun by a mutual potential energy well.
 

ATOM:
The total potential energy of widely separated nuclear protons and free electrons decreases as the particles approach other. As the particles approach each other part of the vector field potential energy converts into electron kinetic energy. Part of this electron kinetic energy is radiated away leaving the electrons and protons bound in a mutual potential energy well.
 

FORCES:
A change in the total cluster field energy dEo with respect to a change in a position dXi of a particular particle is expressed as a force on that particle. If the velocity of a particle affects its potential energy (eg an electrically charged particle moving in a magnetic field which creates a secondary magnetic field) then there is an additional dynamic force component.

A force on a particle due to a potential energy gradient causes a corresponding change in energy motion. Energy motion is also known as momentum.

Let Xo be an observers position vector. The force Fi on particle i causes a change in kinetic energy dEki during a change in particle position vector:
d(Xi - Xo).
Hence:
dEki = Fi * d((Xi - Xo))
 

FORCE SIGNS:
Electric and magnetic vector fields have real unit vectors.

Overlap of electric vector fields from charges of opposite sign causes net far field vector cancellation and hence a reduction in total positive field energy, leading to an increase in kinetic energy which appears to be the result of an attractive force. Similarly, overlap of electric vector fields from charges of the same sign causes net far field vector addition and hence an increase in total positive field energy, leading to a reduction in kinetic energy which appears to be the result of a repulsive force.

Magnetic fields, when viewed as originating from small electric current loops, also diminish in proportion to (1 / |X - Xi|^2) but are orientation dependent with respect to sign. As with electric charges, opposite signs lead to an attractive force whereas same signs lead to a repulsive force.

Gravitational fields from normal matter have unit vectors that are proportional to (-1)^0.5, which when squared causes negative potential energy / unit volume. Overlap of gravitational vector fields from mutual proximity of normal matter masses causes far field vector addition and hence an increase in magnitude of the negative field energy. Hence as normal masses approach each other they gain positive kinetic energy to keep the total energy constant, leading to the appearance of an attractive force.

Hence forces are simply a result of the law of conservation of energy.

ANTI-MATTER ISSUES:
Pair production lifts energy from below the field free vacuum state to above the field free vacuum state. Hence an anti-matter particle is in effect an energy hole. The gravitational field related to this hole should reverse direction. When an electron-positron pair annihilate each other the change in rest mass energy is twice the electron rest mass energy. Thus the energy of the positron is negative with respect to the field free vacuum reference.

The form of the vector field equations suggests that overlap of the gravitational vector field from ordinary matter with the gravitational vector field from antimatter results in the far field vector cancellation which makes the total potential energy less negative and hence causes a force that is repulsive. Hence normal matter and antimatter will gravitationally repel each other. Hence we do not expect to find any free antimatter in our solar system.

The deduced repulsive gravitational force between normal matter and antimatter may explain certain aspects of the intergalactic expansion of the universe that have been observed by astronomers. Two galaxies, one composed of normal matter and the other composed of anti-matter will likely repel each other. Some parties attribute this repulsion to "dark energy". The gravitational interaction between matter and antimatter is presently impossible to resolve in the laboratory because the electric and magnetic forces affecting single particles are many orders of magnitude larger than the gravitational force.
 

FINITE CONSTANT CHARGE:
An isolated particle has a finite constant electric charge Q. This charge is the same regardless of its distance from the observer. The electric field E is a result of the contained charge. The surface area at radius R from the nominal position of a particle is (4 Pi R^2). Hence the electric field E at radius R is given by:
E = Q / (4 Pi Epsilon R^2)
where:
Epsilon = natural constant
 

ELECTRIC FIELD RELATED FORCE:
For an electric field:
Ke = (Epsilon)
where:
Epsilon = permittivity of free space
= 8.85 X 10^-12 coulomb^2 newton^-1 m^-2

Epsilon is one of a handfull of independent natural constants that can only be determined by experimental measurement.

Consider a particle with charge Qi with radius Ri. For R > Ri the electric field Fe around isolated charged particle i is given by:
Fe = Qi / (4 Pi R^2 Epsilon)
where:
Pi = 3.14159
R = radius from the particle center

The corresponding electric field energy density is given by:
(Ke / 2) Fe^2 = (Epsilon / 2)[Qi / (4 Pi R^2 Epsilon)]^2

Then the electric field energy Ee surrounding the charge Qi is given by:
Ee = Integral from Ri to infinity of:
[Epsilon / 2][Qi / 4 Pi R^2 Epsilon]^2] 4 Pi R^2 dR
= (Qi^2 / 8 Pi Epsilon Ri)
This Ee is the electric field rest energy associated with an isolated charge of radius Ri.

Thus an isolated charge Qi has an electric field rest energy Eei of:
Eei = (Qi^2 / 8 Pi Epsilon Ri)

Similarly an isolated charge Qj has an electric field rest energy Eej of:
Eej = (Qj^2 / 8 Pi Epsilon Rj)

If the two charges are both within radius Ro the total electric field rest energy Eet is given by:
Eet = ((Qi + Qj)^2 / 8 Pi Epsilon Ro)
= ((Qi^2 + Qj^2 + 2 Qi Qj) / (8 Pi Epsilon Ro)

Hence the change in total electric field potential energy required to bring two isolated charges together is:
Eet - Eei - Eej
= (2 Qi Qj) / (8 Pi Epsilon Ro)

Differentiating this expression with respect to Ro gives:
F = d(Et - Ei - Ej) / dRo
= - (1 / 4 Pi Epsilon)( Qi Qj / Ro^2)

Recall that the electrostatic force Fe attracting two charges Qi and Qj separated by distance Ro is given by:
Fe = - (1 / 4 Pi Epsilon)(Qi Qj / Ro^2)

Hence we have shown that the electric force is simply the change in electric field energy with respect to position that results from overlap of electric fields.
 

MAGNETIC FIELD ENERGY:
For a magnetic field the constant Km is given by: Km = (1 / Mu)
where:
Mu = permiability of free space
= 4 Pi X 10^-7 webers / amp-m
= 4 Pi X 10^-7 T-m / amp

1 Tesla (T) = 1 weber / m^-2

Consider a toroidal solenoid. The magnetic field inside the solenoid is given by:
B = (Mu N I) / L
where:
B = magnetic field strength N = number of turns
I = current through each turn
L = average magnetic path length

The magnetic field volume within the toroidal solenoid is:
L A
where A is the cross sectional area of the magnetic field.

The solenoid self inductance is:
N B A / I
= (N A / I) (Mu N I / L)
= (Mu N^2 A) / L

The magnetic field energy Em is:
Em = (inductance) I^2 / 2
= (Mu N^2 I^2 A) / 2 L

The magnetic field energy density is:
(Mu N^2 I^2 A) / (2 L (L A))
= (Mu N^2 I^2) / 2 L^2
= (Mu / 2)(N I / L)^2
=(1 / 2 Mu) B^2
= (Mu / 2) H^2
= (Km / 2) H^2

Hence the magnetic field energy density is given by:
(Km / 2) Fm^2 = (Mu / 2) H^2
where:
Fm = H = (B / Mu)

Thus we have shown the relationship between toroidal solenoid parameters and magnetic field energy density.
 

MAGNETIC FORCE:
In an electrical contactor the closing force results from reducing the length of a magnetic circuit of approximately uniform cross-section. The stored magnetic energy is:
Em = (Mu N^2 I^2 A) / 2 L

The contactor closing force is given by:
dEm / dL = - (Mu N^2 I^2 A) / 2 L^2
 

GRAVITY FIELD:
Every element of volume has associated with it a gravitational vector field component that contains gravitational field energy. The gravitational field vector is mathematically orthogonal to both electric and magnetic field vectors. Like the electric and magnetic fields the gravitational field has a net vector flux per unit area that when squared is proportional to gravitational field energy density.

However, because gravitational fields contain energy and gravitational fields are themselves the result of energy the resulting forces are not exactly proportional to (1 / R^2).
 

Assume that each particle is surrounded by a radial gravitational vector field.

For gravity the total external radial vector flux from a particle is proportional to the contained energy.

The total gravitational vector flux due to contained energy Ec is given by:
Flux = + j Ec / Kg C^2
where:
j = (-1)^0.5
Kg = natural constant
C = speed of light
Ec = total energy contained in sphere with radius R.

The surface area of a sphere of radius R is:
(4 Pi R^2).
Hence, for a single isolated particle the external local vector field strength diminishes approximately in proportion to:
1 / (R^2).

The gravitational field flux per unit area at radius R due to energy E contained inside radius R is given by:
Gravitational Field Flux / area = Fg
= j Ec / (C^2 Kg 4 Pi R^2)

Assume:
Ec = M C^2
so that M = mass contained within radius R.

Assume that the gravitational field energy density is proportional to the square of the vector field flux per unit area. Then the gravitational field energy density at R is given by:
(Gravitational Field Energy / unit volume) = (Kg / 2) (Gravitational Field Flux / unit area)^2

Note that the gravitational field energy density is negative.

Let R = Ro at the surface of a solid sphere of mass Mo. The gravitational field energy between radius R = Ro and R = infinity is:
Integral from R = Ro to R = infinity of:
- [1 / (32 Kg)] [(M^2) / (Pi^2 R^4)] 4 Pi R^2 dR
= Integral from R = Ro to R = infinity of:
- [1 / (8 Kg Pi)] [(M^2) / (R^2)] dR

Define:
Mi = mass inside a sphere of radius Ri where Ri > Ro
and
Mo = same mass inside radius Ro

However: Mi = Mo + Integral from R = Ro to R = Ri of (dM / dR) dR.

Mass density = (energy density / C^2). Then:
dM / dR = (Field energy density) 4 Pi R^2 / C^2
= - [1 / (32 Kg)] [(M^2) / (Pi^2 R^4)] [4 Pi R^2 / C^2]
= - [1 / (8 Kg)] [(M^2) / (Pi R^2 C^2)]

Thus:
[dM / M^2] = - [1 / (8 Kg Pi C^2)] [dR / R^2]

Integrate from R = Ro, M = Mo to R = Ri, M = Mi giving:
{[- 1 / Mi] - [- 1 / Mo]} = - [1 / (8 Kg Pi C^2)]{[- 1 / Ri] - [- 1 / Ro]}
or
{[1 / Mi] - [1 / Mo]} = [1 / (8 Kg Pi C^2)][(1 / Ro) - (1 / Ri)]
or
{(Mo - Mi) / Mo Mi} = [1 / (8 Kg Pi C^2)] [(1 / Ro) - (1 / Ri)]
or
(Mo - Mi) = [(Mo Mi) / (8 Kg Pi C^2)][(1 / Ro) - (1 / Ri)]
or
(Eo - Ei) / C^2 = [(Mo Mi) / (8 Kg Pi C^2)][(1 / Ro) - (1 / Ri)]
or
(Eo - Ei) = [(Mo Mi) / (8 Kg Pi)] [(1 / Ro) - (1 / Ri)]

Assume:
Kg = 1 / (G 4 Pi)

Then:
(Eo - Ei) = [(G Mo Mi) / (2)][(1 / Ro) - (1 / Ri)]
or
(Ei - Eo) = - [(G Mo Mi) / 2][(1 / Ro) - (1 / Ri)]
which is the gravitational field energy between R = Ri and R = Ro.

In order to evaluate the full gravitational field energy we let Ri go to infinity. Then:
(Ei - Eo) = - [(G Mo Mi) / 2 Ro]

Remember that in this formula Mo and Mi refer to the same object except that Mi includes its gravity field energy which is negative. Hence (Ei - Eo) is the gravity field energy formed by bringing bits of matter from R = infinity to R = Ro. Note that there is simultaneous creation of an equal amount of positive kinetic energy.

Mi = Mo + (Ei - Eo) / C^2

Hence:
(Ei - Eo) = - [(G Mo Mo) / 2 Ro] - [(G Mo (Mi - Mo) / 2 Ro]
= - [(G Mo Mo) / 2 Ro] - [(G Mo (Ei - Eo) / 2 C^2 Ro]

Thus:
(Ei - Eo)[1 + (G Mo / 2 C^2 Ro)] = - [(G Mo Mo) / 2 Ro]
or
(Ei - Eo) = - [(G Mo^2) / 2 Ro] / [1 + (G Mo / 2 C^2 Ro)]
 

GRAVITATIONAL FORCE:
Gravitational force
= (change in gravitational energy with respect to a change in Ro)
= d(Ei - Eo) / dRo = + {[(G Mo^2) / 2 Ro^2] / [1 + (G Mo / 2 C^2 Ro)]}
- {[(G Mo^2) / 2 Ro](G Mo / 2 C^2 Ro^2) / [1 + (G Mo / 2 C^2 Ro)]^2}
 
= + {[(G Mo^2) / 2 Ro^2] / [1 + (G Mo / 2 C^2 Ro)]}
- {[(G^2 Mo^3) / 4 C^2 Ro^3] / [1 + (G Mo / 2 C^2 Ro)]^2}
 
=+ {[(G Mo^2) / 2 Ro^2] / [1 + (G Mo / 2 C^2 Ro)]}
- {[G Mo / 2 C^2 Ro][(G Mo^2) / 2 Ro^2] / [1 + (G Mo / 2 C^2 Ro)]^2}
 
= {[(G Mo^2) / 2 Ro^2] / [1 + (G Mo / 2 C^2 Ro)]} {1 - ([G Mo / 2 C^2 Ro] / [1 + (G Mo / 2 C^2 Ro)])}

For:
[(G Mo) / (2 C^2 Ro)] << 1
this gravitational force expression simplifies to:
d(Ei - Eo) / dRo ~ [(G Mo^2) / 2 Ro^2] [1 - (G Mo / C^2 Ro)]

Note that the gravitational force is not exactly proportional to 1 / R^2 due to the energy contained within the gravitational field itself. However, this effect is very small and is difficult to detect with nearly circular planetary orbits. Precise measurement of the advance of the perihelion of the elliptical orbit of Mercury was required to experimentally observe this effect.
 

NEWTONIAN METHOD:
Newtons gravitational force equation is:
Force = G Ma Mb / R^2

In Newtons method there are forces at a distance but there is no concept of a vector field containing potential energy.

According to Newton the potential energy released in bringing mass dM from infinity to Ro is:
Integral from R = infinity to R = Ro of: dE = (G M dM) dR / R^2 = - G M dM / Ro

Then the potential energy released in building up M at Ro from M = 0 to M = Mo is:
DeltaE = - G M^2 / 2 Ro

Thus according to Newton moving mass Mo from distributed particles at infinity to a spherical shell at R = Ro causes a change in potential energy :
DeltaE = - G Mo^2 / 2 Ro

Thus the field method and Newton's method give similar results if:
(G Mo / C^2 Ro) << 1
or if
(G Mo / Ro) << C^2

Note that (2 G Mo / Ro)^0.5 is the theoretical escape velocity from a Newtonian system. This parameter becomes significant when the escape velocity is a significant fraction of the speed of light C as is the case in the proximity of a large black hole.
 

ESCAPE VELOCITY AND BLACK HOLES:
The escape velocity is the minimum radial velocity that an object must have to escape from a gravitational potential well. In Newtonian gravitation:
Gravitational acceleration is:
dV / dT = - G Mo / R^2
or
dV = - (G Mo) dT / R^2
= - (G Mo) (dT / dR) dR / R^2
= - (G Mo) (1 / V) dR / R^2

Hence:
V dV = - (G Mo) dR / R^2

Integrating from V = Vo, R = Ro to V = Vf, R = Rf gives:
(Vf^2 / 2) - (Vo^2 / 2) = (G Mo) [(1 / Rf) - (1 / Ro)]

Escape is prevented if Vf = 0 at Rf = infinity
or
- (Vo^2 / 2) = (G Mo) [ - (1 / Ro)]
or
Vo^2 = (2 G Mo / Ro)

Thus a particle cannot directly escape from a gravitational potential well if its initial velocity Vo is:
Vo < (2 G Mo / Ro)^0.5

Hence if:
(2 G Mo / Ro) > C^2
then nothing can escape and there is a black hole.

Consider the value of (G Mo / C^2 Ro) near the surface of the sun:
G = 6.67384 X 10^-11 m^3 kg^-1 s^-2
Ms = 1.98892 X 10^30 kg
Rs = 6.955 X 10^8 m
C = 3 X 10^8 m / s

Substituting in the above formula gives:
[G Mo / (Ro C^2)]
= [6.67384 X 10^-11 m^3 kg^-1 s^-2 X 1.98892 X 10^30 kg] / [6.955 X 10^8 m X 9 X 10^16 m^2 / s^2]
= .212 X 10^-5
= 2.12 X 10^-6

Under favorable circumstances this error fraction of two parts per million in apparent gravitational force can be observed by precise astronomical observations.
 

ELECTROMAGNETIC RADIATION:
Consider 2 equal masses with opposite charges Q and - Q. Assume that these two charges orbit in the X-Y plane around a central point. Assume that the distance between these two charges is D. Assume that there is an observer in the x-Y plane at distance Ro from the central point.

When D is parallel to Ro the electric field seen by the observer for an orbital period that is long compared to [Ro / C] is:
Q / (4 Pi Epsilono)[(Ro + (D / 2))]^2 + - Q / (4 Pi Epsilono)[(Ro - (D / 2))]^2

= {[Q / (4 Pi Epsilono)]} {[1 / [(Ro + (D / 2))]^2] - [1 /[(Ro - (D / 2))]^2}

= {[Q / (4 Pi Epsilono)]} {[(Ro - (D / 2))]^2 - [(Ro + (D / 2))]^2} /[(Ro - (D / 2))]^2[(Ro + (D / 2))]^2]}

= {[Q / (4 Pi Epsilono)]} {- Ro D - Ro D} /{[(Ro^2 - (D / 2)^2)]^2}

= {[Q / (2 Pi Epsilono)]} {- Ro D /[(Ro^2 - (D / 2)^2)]^2}
and the tangential electric field is zero.

When D is perpendicular to Ro the radial electric field seen by the observer for an orbital period that is long compared to [Ro / C] is:
= {[Q / (4 Pi Epsilono)]} {[Ro / {[(Ro^2 + (D / 2)^2)]^1.5] - [Ro /[(Ro^2 - (D / 2)^2)]^1.5]}}
= 0 and the tangential electric field is:
= {[Q / (4 Pi Epsilono)]} {[(D / 2) / {[(Ro^2 + (D / 2)^2)]^1.5] - [- (D / 2) /[(Ro^2 - (D / 2)^2)]^1.5]}}

= {Q / (4 Pi Epsilono)} {D / [(Ro^2 + (D / 2)^2)]^1.5}

Thus at distance Ro there is a rotating electric field vector related to propagating electromagnetic radiation. This thermal radiation is expected from normal matter. This radiation emission will cause the emitter to cool off unless it is in an environment from which it absorbs radiation as fast as it emits it.
 

GRAVITATIONAL RADIATION:
If one of the orbiting charged particles is normal matter and the other is anti-matter, as in positronium, there is potentially significant emission of gravitational radiation as well as electromagnetic radiation. The gravitational radiation emission will be relatively small if both of the orbiting particles are normal matter.
 

QUANTUM MECHANICS:
Radiation emission can only occur when there is an energy exchange between kinetic energy and radiant energy. This energy exchange can only occur in circumstances that:
dE = h F
where:
dE = change in particle energy
and
F = frequency of radiation photon
and
h = Planck constant.
 

This web page last updated June 11, 2024.