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**INTRODUCTION:**

This web page provides insight as to how energy and forces arise from electric, magnetic and gravitational fields.

**FIELDS:**

An important concept in physics is that the total energy of a particle is finite but the energy distribution of that particle stretches out to infinity. Hence a high vacuum still contains energy due to net electric, magnetic and gravitational fields from distant particles.

The concept that electric, magnetic and gravitational vector fields contain energy leads to well known equations for electric, magnetic and gravitational forces as well as to better understanding of physical phenomena.

The concept of distributed energy in vector fields also gives students a qualitative understanding of certain general relativistic effects.

**ISOLATED PARTICLE:**

Assume that an isolated particle has surrounding vector fields that extend out to infinity and that contain a finite amount of distributed energy.

**FINITE CONSTANT CHARGE:**

An isolated particle has a finite constant electric charge Q. This charge is the same regardless of the distance of the observer. The electric field E is a result of the contained charge. The surface area at radius R is 4 Pi R^2. Hence the electric field E at radius R is given by:

E = Q / (Epsilon R^2)

where:

Epsilon = constant

**INTERACTING PARTICLES:**

When multiple particles co-exist their vector fields overlap everywhere in the universe. This field overlap causes a change in the rest energy content of every element of volume. The law of conservation of energy simultaneously causes an equal but opposite change in the kinetic energy content of the same element of volume. The apparent force between particle i and all the other particles in a cluster is really just the change in total cluster rest energy with respect to a change in the position of particle i with respect to the cluster position.

**ASSUMPTION:**

Assume that the gravitational, electric and magnetic field vectors of a particle are all mutually orthogonal. In effect this assumption means that field unit vectors have six dimensions. In addition to the normal **X, Y, Z** cartesian co-ordinate unit vectors there are co-ordinate **jX, jY, jZ** unit vectors, where **j = (-1)^0.5**. The additional imaginary unit vectors allow gravitational energy to be negative.

**ASSUMPTION:**

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

Then the net electric field vector **Fe** at position **X** is given by:

**Fe** = Sum of all **Fei**

where **Fei** denotes the electric field vector at **X** due to the ith particle.

Then the net magnetic field vector **Fm** at position **X** is given by:

**Fm** = Sum of all **Fmi**

where **Fmi** denotes the magnetic field vector at **X** due to the ith particle.

Then the net gravitational field vector **Fg** at position **X** is given by:

**Fg** = Sum of all **Fgi**

where **Fgi** denotes the gravitational field vector at **X** due to the ith particle.

**ASSUMPTION:**

Assume that for each field type **the rest energy density of the field is proportional to the net field vector squared**.

**FIELD ENERGY DENSITY:**

Then the element of rest energy **dEo** contained in an element of volume **dV = (dX dY dZ)** at **X** can be expressed as:

**dEo = ((Ke/2) |Fe|^2 + (Km / 2) |Fm|^2 - (Kg) |Fg|^2)dV**

where:

**Fe** = net electric field vector at **X**

**Fm** = net magnetic field vector at **X**

**Fg** = net gravitational field vector at **X**

**Ke** = positive natural constant

(Ke = Epsilon = electric permittivity of free space)

**Km** = positive natural constant

(Km = Mu = magnetic permeability of free space)

**Kg** = positive natural constant

(Kg is related to the Newton gravitational constant G)

Note that when the gravitational fields of normal matter particles overlap the gravitational field energy becomes more negative. It is hypothesized that when the gravitational fields of normal matter and antimatter particles overlap the gravitational field energy becomes more positive. It is further hypothesized that when the gravitational fields of two antimatter particles overlap the gravitational field energy becomes more negative. These two hypotheses, although consistent with observational data, have not been directly experimentally confirmed.

**PARTICLE CHARACTERIZATION:**

Each 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 particles electric, magnetic and gravitational fields.

**FIELD ENERGIES:**

The field energies are integrals over spacial volume of the field energy densities.

**MULTIPLE PARTICLES:**

Most real systems involve multiple particle. 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 rest energy the law of conservation of energy requires a corresponding opposite sign change in kinetic 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 occurrence of a three body interaction is usually extremely small.

**CLUSTER OF PARTICLES:**

A nearly isolated cluster of particles absorbs or emits 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 energy density is proportional to the net local vector field strength squared. Hence the local potential energy density changes nonlinearly as the field overlap changes.

**FORCES:**

A change in the total cluster potential energy **dEo** with respect to a change in a position d**Xi** of a particular particle is 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 causes a corresponding change in particle 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 vector fields from charges of opposite sign causes far field vector cancellation and hence a reduction in total positive potential energy, leading to an attractive force.

Overlap of vector fields from charges of the same sign causes far field vector addition and hence an increase in total positive potential energy, leading to 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 negative potential energy, which is a reduction in potential energy. Hence as normal masses approach each other they gain positive kinetic energy to keep the total energy constant, leading to an attractive force.

Hence gravitational force is simply a result of the law of conservation of energy.

Note that if anti-matter produces a gravitational field with a unit vector of:

(- i) = (- (-1)^0.5)

as expected from its negative energy, then the resulting gravitational field energy around antimatter will be negative like ordinary matter. Hence anti matter may be able to form gravitationally bound atoms with other antimatter particles. Recall that 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 with negative energy. 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 structure of the universe that have been observed by astronomers. 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.

**ELECTRIC FIELD RELATED FORCE:**

For an electric field:

**Ke = (Epsilon / 2)**

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**

**GRAVITATIONAL FIELD ENERGY DENSITY:**

The gravitaional field energy density is given by:

(gravitational energy per unit volume) = **Kg (gravitational field)^2**

Every element of volume has associated with it a radial gravitational vector field component that contains gravitational potential energy. The gravitational field vector is mathematically orthogonal to both electric and magnetic field vectors. The gravitational field has a vector flux that is proportional to contained energy.

However, because gravitational fields contain energy and gravitational fields are the result of energy the resulting forces are not exactly proportional to **(1 / R^2)**. Precise measurement of orbital motion near the surface of a large energy concentration such as the sun reveals this discrepancy. eg Perihelion of the planet Mercury.

**GRAVITATIONAL FIELD:**

The presence of rest energy **Eo** causes an imaginary radial gravitational vector field magnitude given by:

**Eo / 4 Pi R^2**

For gravity the total external radial vector flux from a particle is proportional to the contained energy and hence in the far field this vector flux is nearly constant because most of the particle energy is concentrated close to the nominal position of the particle. 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)**.

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

The total gravitational vector flux at **X** due to contained energy Ec is given by:

Flux = + i Ec

where:

i = (-1)^0.5

Ec = potential energy contained in sphere with radius **R**.

The gravitational field flux per unit area at radius **R** due to energy **Ec** contained inside radius **R** is given by:

**Gravitational Field Flux / area = i Ec / 4 Pi R^2**

Then the gravitational potential energy per unit volume at **R** is given by:

**Gravitational Potential Energy / unit volume = Kg (Gravitational Field Flux / unit area)^2**

Then:

**Gravitational Field Energy / unit volume
= Kg [i Ec / (4 Pi R^2)]^2
= - Kg [Ec / (4 Pi R^2)]^2**

Note that the potential energy contained in the gravitational field is negative.

Assume that the gravitational potential energy density is proportional to the square of the vector field flux per unit area. Thus the gravitational field energy density is given by:

**-Kg (Ec / 4 Pi R^2)^2**

Note that the gravitational field energy density is negative.

**dEc = - Kg (Ec / 4 Pi R^2)^2 (4 Pi R^2) dR**

or

**dEc / Ec^2 = - Kg dR / (4 Pi R^2)**

Integrating from **R = Ri, Ec = Eco to R= infinity, Ec = Eci** gives:

**(- 1 / Eci) - (-1 / Eco) = (- Kg / 4 Pi) ((-1 / infinity) - (-1 / Ro))**

or
**(1 / Eco) - (1 / Eci) = - Kg / (4 Pi Ro)**

or
**(1 / Eci) = (1 / Eco) + Kg / (4 Pi Ro)**

or
**Eci = 1 / [(1 / Eco) + Kg / (4 Pi Ro)]
= 4 Pi Ri Eco / (4 Pi Ro + Kg Eco)
= Eco / (1 + (Kg Eco / 4 Pi Ro))**

For most practical situations, excluding neutron stars, it can be shown that:

**|Kg Eco / 4 Pi Ro| << 1**

Hence the equation for Eci is of the form:

**Eci = Eco / (1 + X)**

where:

**|X| << 1**
**1 / (1+ X) ~ 1 + Ax + BX^2**

**d[1 / (1 + X)] / dX = - 1 / (1 + X)^2
= A + 2 B X **

At **X = 0, A = -1
d^2 [1 / (1 + X)] / dX^2
= d [ -1 / (1 + X)^2] / dX
= 2 (1 + X) / (1 + X)^4
= 2 B **

At **X = 0, B = 1**

Thus for **|X| << 1
1 / (1 + X) ~ 1 - X + X^2**

Hence for the case of **|Kg Eco / 4 Pi Ro| << 1
Ect = Eco / (1 + (Kg Eco / 4 Pi Ro))
~ Eco[ 1 - (Kg Eco / 4 Pi Ro) + (Kg Eco / 4 Pi Ro)^2]**

Consider two widely separated energy clusters **Eci** and **Ecj** that have identical Ro values:

The individual cluster energies are:

**Eci = Ecoi[ 1 - (Kg Ecoi / 4 Pi Ro) + (Kg Ecoi / 4 Pi Ro)^2]**

and

**Ecj = Ecoj[ 1 - (Kg Ecoj / 4 Pi Ro) + (Kg Ecoj / 4 Pi Ro)^2]**

If the two clusters are superimposed the combined within radius **Ro** the new cluster energy is:

**Ects = (Ecoi + Ecoj)[ 1 - (Kg (Ecoi + Ecoj) / 4 Pi Ro) + (Kg (Ecoi + Ecoj) / 4 Pi Ro)^2]**

The energy difference between the separated energy clusters and the combined cluster is:

**Ecti + Ectj - Ects
= Ecoi[ 1 - (Kg Ecoi / 4 Pi Ro) + (Kg Ecoi / 4 Pi Ro)^2]
+ Ecoj[ 1 - (Kg Ecoj / 4 Pi Ro) + (Kg Ecoj / 4 Pi Ro)^2]
- (Ecoi + Ecoj)[ 1 - (Kg (Ecoi + Ecoj) / 4 Pi Ro) + (Kg (Ecoi + Ecoj) / 4 Pi Ro)^2]
= Ecoi[ - (Kg Ecoi / 4 Pi Ro) + (Kg Ecoi / 4 Pi Ro)^2]
+ Ecoj[ - (Kg Ecoj / 4 Pi Ro) + (Kg Ecoj / 4 Pi Ro)^2]
- (Ecoi + Ecoj)[ - (Kg (Ecoi + Ecoj) / 4 Pi Ro) + (Kg (Ecoi + Ecoj) / 4 Pi Ro)^2]
= (Kg 2 Ecoi Ecoj / 4 Pi Ro) + (Ecoi^3 + Ecoj^3 - (Ecoi + Ecoj)^3) (Kg / 4 Pi Ro)^2
= (Kg 2 Ecoi Ecoj / 4 Pi Ro)+ (Ecoi^3 + Ecoj^3 - (Ecoi^3 +3 Ecoi Ecoj^2 + 3 Ecoi^2 Ecoj + Ecoj^3)) (Kg / 4 Pi Ro)^2
= (Kg 2 Ecoi Ecoj / 4 Pi Ro) - ( +3 Ecoi Ecoj^2 + 3 Ecoi^2 Ecoj) (Kg / 4 Pi Ro)^2
= (Kg Ecoi Ecoj / 2 Pi Ro) [1 - 3(Ecoj + Ecoi)(Kg / 2 (4 Pi Ro)]**

In Newton's approximation of gravity:

**F = G Mi Mj / R^2**

According to Newton the energy released in bringing Mi and Mj together from infinity is:

**Integral from R = infinity to R = Ro of: (-G Mi Mj / R^2) dR
= G Mi Mj / Ro**

Neglecting 2nd order terms, equating the two formulas for energy release gives:

**Kg Ecoi Ecoj / 2 Pi Ro = G Mi Mj / Ro**

However from Special Relativity:

**Mi = Ecti / C^2**

and

**Mj = Ectj / C^2**

Hence:

**Kg = G Mi Mj / (2 Pi Ecoi Ecoj)
= G (Eci / C^2) (Ectj / C^2) / (2 Pi Ecoi Ecoj)
~ G / (2 Pi C^4)**

Note that the gravitational force is not exactly proportional to 1 / R^2 due to the potential energy contained within the gravitational field itself. However, this effect is quite 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.

The exact gravitational force equation simplifies to Newton's gravitational force equation when the potential energy contained in the field volume where R > Ro is much less than the total potential energy contained in the field volume where R < Ro.

Hence the Gravitational force Fg between two masses Ma and Mb is approximately in accordance with Newtons formula:

**Fg = G Ma Mb / R^2**

where:

**G** = gravitational constant

**R** = center to center distance between mass a and mass b.

There is an additional general relativistic correction similar to that derived by Einstein.

The condition for validity of Newtons gravitational force equation is:

**|Kg Eco / 4 Pi Ro| << 1**

Substituting for **Kg** this condition becomes:

**[G Eco / (2 Pi C^4)] / 4 Pi Ro << 1**

or

**[G Mo / (2 Pi C^2)] / 4 Pi Ro << 1**

or

**[G Mo / (8 Pi^2 Ro C^2)] << 1**

Consider the condition 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

Substituting in the above formula gives:

[G Mo / (8 Pi^2 Ro C^2)]

= [6.67384 X 10^-11 m^3 kg^-1 s^-2 X 1.98892 X 10^30 kg] / [8 X 3.14 X 6.955 X 10^8 m X 9 X 10^16 m^2 / s^2]

= .00844 X 10^-5

= **8.44 X 10^-7**

This error fraction in **Eci** can be resolved by precise astronomical observation of the advance of the perihelion of the orbit of the planet Mercury.

This web page last updated April 1, 2015.

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